{"id":694,"date":"2017-10-27T16:30:34","date_gmt":"2017-10-27T16:30:34","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/chapter\/the-ideal-gas-law\/"},"modified":"2017-11-08T03:25:13","modified_gmt":"2017-11-08T03:25:13","slug":"the-ideal-gas-law","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/chapter\/the-ideal-gas-law\/","title":{"raw":"The Ideal Gas Law","rendered":"The Ideal Gas Law"},"content":{"raw":"\n<div class=\"textbox learning-objectives\">\n<h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>\n<ul>\n<li>State the ideal gas law in terms of molecules and in terms of moles.<\/li>\n<li>Use the ideal gas law to calculate pressure change, temperature change, volume change, or the number of molecules or moles in a given volume.<\/li>\n<li>Use Avogadro\u2019s number to convert between number of molecules and number of moles.<\/li>\n<\/ul>\n<\/div>\n<div class=\"bc-figure figure\" id=\"import-auto-id1563742\">\n<div class=\"bc-figcaption figcaption\">The air inside this hot air balloon flying over Putrajaya, Malaysia, is hotter than the ambient air. As a result, the balloon experiences a buoyant force pushing it upward. (credit: Kevin Poh, Flickr)<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id2600697\" data-alt=\"\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_14_03_00.jpg\" data-media-type=\"image\/jpg\" alt=\"\" width=\"200\"><\/span><\/p><\/div>\n<p id=\"import-auto-id2409403\">In this section, we continue to explore the thermal behavior of gases. In particular, we examine the characteristics of atoms and molecules that compose gases. (Most gases, for example nitrogen, [latex]{\\text{N}}_{2}[\/latex], and oxygen, [latex]{\\text{O}}_{2}[\/latex], are composed of two or more atoms. We will primarily use the term \u201cmolecule\u201d in discussing a gas because the term can also be applied to monatomic gases, such as helium.)<\/p>\n<p id=\"import-auto-id2596139\">Gases are easily compressed. We can see evidence of this in <a href=\"\/contents\/0b6d0d23-e96f-4f34-8051-643361f8c7c7@4#import-auto-id1814176\" class=\"autogenerated-content\">(Figure)<\/a>, where you will note that gases have the <em data-effect=\"italics\">largest<\/em> coefficients of volume expansion. The large coefficients mean that gases expand and contract very rapidly with temperature changes. In addition, you will note that most gases expand at the <em data-effect=\"italics\">same<\/em> rate, or have the same [latex]\\beta [\/latex]. This raises the question as to why gases should all act in nearly the same way, when liquids and solids have widely varying expansion rates.<\/p>\n<p id=\"import-auto-id1590402\">The answer lies in the large separation of atoms and molecules in gases, compared to their sizes, as illustrated in <a href=\"#import-auto-id2233613\" class=\"autogenerated-content\">(Figure)<\/a>. Because atoms and molecules have large separations, forces between them can be ignored, except when they collide with each other during collisions. The motion of atoms and molecules (at temperatures well above the boiling temperature) is fast, such that the gas occupies all of the accessible volume and the expansion of gases is rapid. In contrast, in liquids and solids, atoms and molecules are closer together and are quite sensitive to the forces between them.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id2233613\">\n<div class=\"bc-figcaption figcaption\">Atoms and molecules in a gas are typically widely separated, as shown. Because the forces between them are quite weak at these distances, the properties of a gas depend more on the number of atoms per unit volume and on temperature than on the type of atom.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id2600397\" data-alt=\"Spheres representing atoms and molecules; the spheres are relatively far apart and are distributed randomly.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_14_03_01.jpg\" data-media-type=\"image\/jpg\" alt=\"Spheres representing atoms and molecules; the spheres are relatively far apart and are distributed randomly.\" width=\"250\"><\/span><\/p><\/div>\n<p id=\"import-auto-id1856142\">To get some idea of how pressure, temperature, and volume of a gas are related to one another, consider what happens when you pump air into an initially deflated tire. The tire\u2019s volume first increases in direct proportion to the amount of air injected, without much increase in the tire pressure. Once the tire has expanded to nearly its full size, the walls limit volume expansion. If we continue to pump air into it, the pressure increases. The pressure will further increase when the car is driven and the tires move. Most manufacturers specify optimal tire pressure for cold tires. (See <a href=\"#import-auto-id2102337\" class=\"autogenerated-content\">(Figure)<\/a>.)<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id2102337\">\n<div class=\"bc-figcaption figcaption\">(a) When air is pumped into a deflated tire, its volume first increases without much increase in pressure. (b) When the tire is filled to a certain point, the tire walls resist further expansion and the pressure increases with more air. (c) Once the tire is inflated, its pressure increases with temperature.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1498134\" data-alt=\"The figure has three parts, each part showing a pair of tires, and each tire connected to a pressure gauge. Each pair of tires represents the before and after images of a single tire, along with a change in pressure in that tire. In part a, the tire pressure is initially zero. After some air is added, represented by an arrow labeled Add air, the pressure rises to slightly above zero. In part b, the tire pressure is initially at the half-way mark. After some air is added, represented by an arrow labeled Add air, the pressure rises to the three-fourths mark. In part c, the tire pressure is initially at the three-fourths mark. After the temperature is raised, represented by an arrow labeled Increase temperature, the pressure rises to nearly the full mark.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_14_03_02.jpg\" data-media-type=\"image\/jpg\" alt=\"The figure has three parts, each part showing a pair of tires, and each tire connected to a pressure gauge. Each pair of tires represents the before and after images of a single tire, along with a change in pressure in that tire. In part a, the tire pressure is initially zero. After some air is added, represented by an arrow labeled Add air, the pressure rises to slightly above zero. In part b, the tire pressure is initially at the half-way mark. After some air is added, represented by an arrow labeled Add air, the pressure rises to the three-fourths mark. In part c, the tire pressure is initially at the three-fourths mark. After the temperature is raised, represented by an arrow labeled Increase temperature, the pressure rises to nearly the full mark.\" width=\"500\"><\/span><\/p><\/div>\n<p id=\"import-auto-id1149559\">At room temperatures, collisions between atoms and molecules can be ignored. In this case, the gas is called an ideal gas, in which case the relationship between the pressure, volume, and temperature is given by the equation of state called the ideal gas law.<\/p>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id2683246\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">Ideal Gas Law<\/div>\n<p id=\"import-auto-id1652885\">The <span data-type=\"term\" id=\"import-auto-id1530949\">ideal gas law<\/span> states that<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id2374115\">[latex]\\text{PV}=\\text{NkT},[\/latex]<\/div>\n<p id=\"import-auto-id921222\">where [latex]P[\/latex] is the absolute pressure of a gas, [latex]V[\/latex] is the volume it occupies, [latex]N[\/latex] is the number of atoms and molecules in the gas, and [latex]T[\/latex] is its absolute temperature. The constant [latex]k[\/latex] is called the <span data-type=\"term\" id=\"import-auto-id1634900\">Boltzmann constant<\/span> in honor of Austrian physicist Ludwig Boltzmann (1844\u20131906) and has the value<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id1586104\">[latex]k=1\\text{.}\\text{38}\u00d7{\\text{10}}^{-\\text{23}}\\phantom{\\rule{0.25em}{0ex}}\\text{J}\/\\text{K}.[\/latex]<\/div>\n<\/div>\n<p id=\"import-auto-id2238750\">The ideal gas law can be derived from basic principles, but was originally deduced from experimental measurements of Charles\u2019 law (that volume occupied by a gas is proportional to temperature at a fixed pressure) and from Boyle\u2019s law (that for a fixed temperature, the product [latex]\\text{PV}[\/latex] is a constant). In the ideal gas model, the volume occupied by its atoms and molecules is a negligible fraction of [latex]V[\/latex]. The ideal gas law describes the behavior of real gases under most conditions. (Note, for example, that [latex]N[\/latex] is the total number of atoms and molecules, independent of the type of gas.)<\/p>\n<p id=\"import-auto-id2248763\">Let us see how the ideal gas law is consistent with the behavior of filling the tire when it is pumped slowly and the temperature is constant. At first, the pressure [latex]P[\/latex] is essentially equal to atmospheric pressure, and the volume [latex]V[\/latex] increases in direct proportion to the number of atoms and molecules [latex]N[\/latex] put into the tire. Once the volume of the tire is constant, the equation [latex]\\text{PV}=\\text{NkT}[\/latex] predicts that the pressure should increase in proportion to <em data-effect=\"italics\">the number N of atoms and molecules<\/em>. <\/p>\n<div data-type=\"example\" class=\"textbox examples\" id=\"fs-id1667893\">\n<div data-type=\"title\" class=\"title\">Calculating Pressure Changes Due to Temperature Changes: Tire Pressure<\/div>\n<p id=\"import-auto-id3224621\">Suppose your bicycle tire is fully inflated, with an absolute pressure of [latex]7\\text{.}\\text{00}\u00d7{\\text{10}}^{5}\\phantom{\\rule{0.25em}{0ex}}\\text{Pa}[\/latex] (a gauge pressure of just under [latex]\\text{90}\\text{.}0\\phantom{\\rule{0.25em}{0ex}}{\\text{lb\/in}}^{2}[\/latex]) at a temperature of [latex]\\text{18}\\text{.}0\\text{\u00ba}\\text{C}[\/latex]. What is the pressure after its temperature has risen to [latex]\\text{35}\\text{.}0\\text{\u00ba}\\text{C}[\/latex]? Assume that there are no appreciable leaks or changes in volume.<\/p>\n<p id=\"import-auto-id2194007\"><strong>Strategy<\/strong><\/p>\n<p id=\"import-auto-id2631292\">The pressure in the tire is changing only because of changes in temperature. First we need to identify what we know and what we want to know, and then identify an equation to solve for the unknown.<\/p>\n<p id=\"import-auto-id1084477\">We know the initial pressure [latex]{P}_{0}=7\\text{.00}\u00d7{\\text{10}}^{5}\\phantom{\\rule{0.25em}{0ex}}\\text{Pa}[\/latex], the initial temperature<br>\n[latex]{T}_{0}=\\text{18}\\text{.}0\u00baC[\/latex], and the final temperature<br>\n[latex]{T}_{\\text{f}}=35\\text{.}0\u00baC[\/latex]. We must find the final pressure<br>\n[latex]{P}_{\\text{f}}[\/latex]. How can we use the equation<br>\n[latex]\\text{PV}=\\text{NkT}[\/latex]? At first, it may seem that not enough information is given, because the volume<br>\n[latex]V[\/latex] and number of atoms<br>\n[latex]N[\/latex] are not specified. What we can do is use the equation twice:<br>\n[latex]{P}_{0}{V}_{0}={\\text{NkT}}_{0}[\/latex] and<br>\n[latex]{P}_{\\text{f}}{V}_{\\text{f}}={\\text{NkT}}_{\\text{f}}[\/latex]. If we divide<br>\n[latex]{P}_{\\text{f}}{V}_{\\text{f}}[\/latex] by<br>\n[latex]{P}_{0}{V}_{0}[\/latex] we can come up with an equation that allows us to solve for<br>\n[latex]{P}_{\\text{f}}[\/latex]. <\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id2084395\">[latex]\\frac{{P}_{\\text{f}}{V}_{\\text{f}}}{{P}_{0}{V}_{0}}=\\frac{{N}_{\\text{f}}{\\text{kT}}_{\\text{f}}}{{N}_{0}{\\text{kT}}_{0}}[\/latex]<\/div>\n<p id=\"import-auto-id1585921\">Since the volume is constant, [latex]{V}_{\\text{f}}[\/latex] and [latex]{V}_{0}[\/latex] are the same and they cancel out. The same is true for [latex]{N}_{\\text{f}}[\/latex] and [latex]{N}_{0}[\/latex], and [latex]k[\/latex], which is a constant. Therefore,<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id1807145\">[latex]\\frac{{P}_{\\text{f}}}{{P}_{0}}=\\frac{{T}_{\\text{f}}}{{T}_{0}}\\text{.}[\/latex]<\/div>\n<p id=\"import-auto-id2084638\">We can then rearrange this to solve for [latex]{P}_{\\text{f}}[\/latex]:<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id2084642\">[latex]{P}_{\\text{f}}={P}_{0}\\frac{{T}_{\\text{f}}}{{T}_{0}},[\/latex]<\/div>\n<p id=\"import-auto-id2798629\">where the temperature must be in units of kelvins, because [latex]{T}_{0}[\/latex] and [latex]{T}_{\\text{f}}[\/latex] are absolute temperatures.<\/p>\n<p id=\"import-auto-id2673782\"><strong>Solution<\/strong><\/p>\n<p id=\"import-auto-id2387499\">1. Convert temperatures from Celsius to Kelvin.<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id2387502\">[latex]\\begin{array}{}{T}_{0}=\\left(\\text{18}\\text{.}0+\\text{273}\\right)\\text{K}=\\text{291 K}\\\\ {T}_{\\text{f}}=\\left(\\text{35}\\text{.}0+\\text{273}\\right)\\text{K}=\\text{308 K}\\end{array}[\/latex]<\/div>\n<p id=\"import-auto-id2677636\">2. Substitute the known values into the equation.<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id2677638\">[latex]{P}_{\\text{f}}={P}_{0}\\frac{{T}_{\\text{f}}}{{T}_{0}}=7\\text{.}\\text{00}\u00d7{\\text{10}}^{5}\\phantom{\\rule{0.25em}{0ex}}\\text{Pa}\\left(\\frac{\\text{308 K}}{\\text{291 K}}\\right)=7\\text{.}\\text{41}\u00d7{\\text{10}}^{5}\\phantom{\\rule{0.25em}{0ex}}\\text{Pa}[\/latex]<\/div>\n<p id=\"import-auto-id2919467\"><strong>Discussion<\/strong><\/p>\n<p id=\"import-auto-id2919469\">The final temperature is about 6% greater than the original temperature, so the final pressure is about 6% greater as well. Note that <em data-effect=\"italics\"><em data-effect=\"italics\">absolute<\/em><\/em> pressure and <em data-effect=\"italics\"><em data-effect=\"italics\">absolute<\/em><\/em> temperature must be used in the ideal gas law.<\/p>\n<\/div>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id1387947\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">Making Connections: Take-Home Experiment\u2014Refrigerating a Balloon<\/div>\n<p id=\"import-auto-id3224623\">Inflate a balloon at room temperature. Leave the inflated balloon in the refrigerator overnight. What happens to the balloon, and why? <\/p>\n<\/div>\n<div data-type=\"example\" class=\"textbox examples\" id=\"fs-id1843505\">\n<div data-type=\"title\" class=\"title\">Calculating the Number of Molecules in a Cubic Meter of Gas<\/div>\n<p id=\"import-auto-id2919471\">How many molecules are in a typical object, such as gas in a tire or water in a drink? We can use the ideal gas law to give us an idea of how large [latex]N[\/latex] typically is.<\/p>\n<p id=\"import-auto-id2677390\">Calculate the number of molecules in a cubic meter of gas at standard temperature and pressure (STP), which is defined to be [latex]0\\text{\u00ba}\\text{C}[\/latex] and atmospheric pressure.<\/p>\n<p id=\"import-auto-id2919757\"><strong>Strategy<\/strong><\/p>\n<p id=\"import-auto-id2717709\">Because pressure, volume, and temperature are all specified, we can use the ideal gas law [latex]\\text{PV}=\\text{NkT}[\/latex], to find [latex]N[\/latex].<\/p>\n<p id=\"import-auto-id2409089\"><strong>Solution<\/strong><\/p>\n<p id=\"import-auto-id2409092\">1. Identify the knowns.<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id2409094\">[latex]\\begin{array}{lll}T&amp; =&amp; 0\\text{\u00ba}\\text{C}=\\text{273 K}\\\\ P&amp; =&amp; 1\\text{.}\\text{01}\u00d7{\\text{10}}^{5}\\phantom{\\rule{0.25em}{0ex}}\\text{Pa}\\\\ V&amp; =&amp; 1\\text{.}\\text{00}\\phantom{\\rule{0.25em}{0ex}}{\\text{m}}^{3}\\\\ k&amp; =&amp; 1\\text{.}\\text{38}\u00d7{\\text{10}}^{-\\text{23}}\\phantom{\\rule{0.25em}{0ex}}\\text{J\/K}\\end{array}[\/latex]<\/div>\n<p id=\"import-auto-id3224551\">2. Identify the unknown: number of molecules, [latex]N[\/latex].<\/p>\n<p id=\"import-auto-id3218613\">3. Rearrange the ideal gas law to solve for [latex]N[\/latex].<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id3224643\">[latex]\\begin{array}{}\\text{PV}=\\text{NkT}\\\\ N=\\frac{\\text{PV}}{\\text{kT}}\\end{array}[\/latex]<\/div>\n<p id=\"import-auto-id2743296\">4. Substitute the known values into the equation and solve for [latex]N[\/latex]. <\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id2887530\">[latex]N=\\frac{\\text{PV}}{\\text{kT}}=\\frac{\\left(1\\text{.}\\text{01}\u00d7{\\text{10}}^{5}\\phantom{\\rule{0.25em}{0ex}}\\text{Pa}\\right)\\left(1\\text{.}{\\text{00 m}}^{3}\\right)}{\\left(1\\text{.}\\text{38}\u00d7{\\text{10}}^{-\\text{23}}\\phantom{\\rule{0.25em}{0ex}}\\text{J\/K}\\right)\\left(\\text{273 K}\\right)}=2\\text{.}\\text{68}\u00d7{\\text{10}}^{\\text{25}}\\phantom{\\rule{0.25em}{0ex}}\\text{molecules}[\/latex]<\/div>\n<p id=\"import-auto-id2583233\"><strong>Discussion<\/strong><\/p>\n<p id=\"import-auto-id3224684\">This number is undeniably large, considering that a gas is mostly empty space. [latex]N[\/latex] is huge, even in small volumes. For example, [latex]1\\phantom{\\rule{0.25em}{0ex}}{\\text{cm}}^{3}[\/latex] of a gas at STP has [latex]2\\text{.}\\text{68}\u00d7{\\text{10}}^{\\text{19}}[\/latex] molecules in it. Once again, note that [latex]N[\/latex] is the same for all types or mixtures of gases.<\/p>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id2669976\">\n<h1 data-type=\"title\">Moles and Avogadro\u2019s Number<\/h1>\n<p id=\"import-auto-id1725013\">It is sometimes convenient to work with a unit other than molecules when measuring the amount of substance. A <span data-type=\"term\" id=\"import-auto-id2677643\">mole<\/span> (abbreviated mol) is defined to be the amount of a substance that contains as many atoms or molecules as there are atoms in exactly 12 grams (0.012 kg) of carbon-12. The actual number of atoms or molecules in one mole is called <span data-type=\"term\" id=\"import-auto-id2677648\">Avogadro\u2019s number<\/span>[latex]\\left({N}_{\\text{A}}\\right)[\/latex], in recognition of Italian scientist Amedeo Avogadro (1776\u20131856). He developed the concept of the mole, based on the hypothesis that equal volumes of gas, at the same pressure and temperature, contain equal numbers of molecules. That is, the number is independent of the type of gas. This hypothesis has been confirmed, and the value of Avogadro\u2019s number is <\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id2677659\">[latex]{N}_{\\text{A}}=6\\text{.}\\text{02}\u00d7{\\text{10}}^{\\text{23}}\\phantom{\\rule{0.25em}{0ex}}{\\text{mol}}^{-1}\\text{.}[\/latex]<\/div>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id2403668\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">Avogadro\u2019s Number<\/div>\n<p id=\"import-auto-id2936405\">One mole always contains [latex]6\\text{.}\\text{02}\u00d7{\\text{10}}^{\\text{23}}[\/latex] particles (atoms or molecules), independent of the element or substance. A mole of any substance has a mass in grams equal to its molecular mass, which can be calculated from the atomic masses given in the periodic table of elements.<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id2919405\">[latex]{N}_{\\text{A}}=6\\text{.}\\text{02}\u00d7{\\text{10}}^{\\text{23}}\\phantom{\\rule{0.25em}{0ex}}{\\text{mol}}^{-1}[\/latex]<\/div>\n<\/div>\n<div class=\"bc-figure figure\" id=\"import-auto-id2936416\">\n<div class=\"bc-figcaption figcaption\">How big is a mole? On a macroscopic level, one mole of table tennis balls would cover the Earth to a depth of about 40 km.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id2936418\" data-alt=\"The illustration shows relatively flat land with a solitary mountain, labeled Mt. Everest, and blue sky above. A double-headed vertical arrow stretches between the land and a point in the sky that is well above the peak of the mountain. The arrow, labeled table tennis balls, serves to indicate that a column of one mole of table tennis balls would reach a point in the sky that is much higher than the peak of Mt. Everest.\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_14_03_03.jpg\" data-media-type=\"image\/jpg\" alt=\"The illustration shows relatively flat land with a solitary mountain, labeled Mt. Everest, and blue sky above. A double-headed vertical arrow stretches between the land and a point in the sky that is well above the peak of the mountain. The arrow, labeled table tennis balls, serves to indicate that a column of one mole of table tennis balls would reach a point in the sky that is much higher than the peak of Mt. Everest.\" width=\"400\"><\/span><\/p><\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2696745\" data-element-type=\"check-understanding\" data-label=\"\">\n<div data-type=\"title\">Check Your Understanding<\/div>\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1379188\">\n<p id=\"import-auto-id2763671\">The active ingredient in a Tylenol pill is 325 mg of acetaminophen [latex]\\left({\\text{C}}_{8}{\\text{H}}_{9}{\\text{NO}}_{2}\\right)[\/latex]. Find the number of active molecules of acetaminophen in a single pill.<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1490941\">\n<p id=\"import-auto-id2763690\">We first need to calculate the molar mass (the mass of one mole) of acetaminophen. To do this, we need to multiply the number of atoms of each element by the element\u2019s atomic mass.<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id2919489\">[latex]\\begin{array}{}\\left(\\text{8 moles of carbon}\\right)\\left(\\text{12 grams\/mole}\\right)+\\left(\\text{9 moles hydrogen}\\right)\\left(\\text{1 gram\/mole}\\right)\\\\ +\\left(\\text{1 mole nitrogen}\\right)\\left(\\text{14 grams\/mole}\\right)+\\left(\\text{2 moles oxygen}\\right)\\left(\\text{16 grams\/mole}\\right)=\\text{151 g}\\end{array}[\/latex]<\/div>\n<p id=\"import-auto-id2763682\">Then we need to calculate the number of moles in 325 mg.<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id2386722\">[latex]\\left(\\frac{\\text{325 mg}}{\\text{151 grams\/mole}}\\right)\\left(\\frac{1 gram}{\\text{1000 mg}}\\right)=2.15\u00d7{\\text{10}}^{-3}\\phantom{\\rule{0.25em}{0ex}}\\text{moles}[\/latex]<\/div>\n<p id=\"import-auto-id2386714\">Then use Avogadro\u2019s number to calculate the number of molecules.<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id2627386\">[latex]N=\\left(\\text{2.15}\u00d7{\\text{10}}^{-3}\\phantom{\\rule{0.25em}{0ex}}\\text{moles}\\right)\\left(\\text{6.02}\u00d7{\\text{10}}^{\\text{23}}\\phantom{\\rule{0.25em}{0ex}}\\text{molecules\/mole}\\right)=\\text{1.30}\u00d7{\\text{10}}^{\\text{21}}\\phantom{\\rule{0.25em}{0ex}}\\text{molecules}[\/latex]<\/div>\n<\/div>\n<\/div>\n<div data-type=\"example\" class=\"textbox examples\" id=\"fs-id2322275\">\n<div data-type=\"title\" class=\"title\">Calculating Moles per Cubic Meter and Liters per Mole<\/div>\n<p id=\"import-auto-id2870639\">Calculate: (a) the number of moles in [latex]1\\text{.}\\text{00}\\phantom{\\rule{0.25em}{0ex}}{\\text{m}}^{3}[\/latex] of gas at STP, and (b) the number of liters of gas per mole.<\/p>\n<p id=\"import-auto-id2870635\"><strong>Strategy and Solution<\/strong><\/p>\n<p id=\"import-auto-id2717734\">(a) We are asked to find the number of moles per cubic meter, and we know from <a href=\"#fs-id1843505\" class=\"autogenerated-content\">(Figure)<\/a> that the number of molecules per cubic meter at STP is [latex]2\\text{.}\\text{68}\u00d7{\\text{10}}^{\\text{25}}[\/latex]. The number of moles can be found by dividing the number of molecules by Avogadro\u2019s number. We let [latex]n[\/latex] stand for the number of moles,<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id2564703\">[latex]n\\phantom{\\rule{0.25em}{0ex}}{\\text{mol\/m}}^{3}=\\frac{N\\phantom{\\rule{0.25em}{0ex}}{\\text{molecules\/m}}^{3}}{6\\text{.}\\text{02}\u00d7{\\text{10}}^{\\text{23}}\\phantom{\\rule{0.25em}{0ex}}\\text{molecules\/mol}}=\\frac{2\\text{.}\\text{68}\u00d7{\\text{10}}^{\\text{25}}\\phantom{\\rule{0.25em}{0ex}}{\\text{molecules\/m}}^{3}}{6\\text{.}\\text{02}\u00d7{\\text{10}}^{\\text{23}}\\phantom{\\rule{0.25em}{0ex}}\\text{molecules\/mol}}=\\text{44}\\text{.}5\\phantom{\\rule{0.25em}{0ex}}{\\text{mol\/m}}^{3}\\text{.}[\/latex]<\/div>\n<p id=\"import-auto-id2595510\">(b) Using the value obtained for the number of moles in a cubic meter, and converting cubic meters to liters, we obtain<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id2919356\">[latex]\\frac{\\left({\\text{10}}^{3}\\phantom{\\rule{0.25em}{0ex}}{\\text{L\/m}}^{3}\\right)}{44\\text{.}5\\phantom{\\rule{0.25em}{0ex}}{\\text{mol\/m}}^{3}}=\\text{22}\\text{.}5\\phantom{\\rule{0.25em}{0ex}}\\text{L\/mol}\\text{.}[\/latex]<\/div>\n<p id=\"import-auto-id2167136\"><strong>Discussion<\/strong><\/p>\n<p id=\"import-auto-id2883491\">This value is very close to the accepted value of 22.4 L\/mol. The slight difference is due to rounding errors caused by using three-digit input. Again this number is the same for all gases. In other words, it is independent of the gas.<\/p>\n<p id=\"import-auto-id2919351\">The (average) molar weight of air (approximately 80% [latex]{\\text{N}}_{2}[\/latex] and 20% [latex]{\\text{O}}_{2}[\/latex] is [latex]M=\\text{28}\\text{.}8\\phantom{\\rule{0.25em}{0ex}}\\text{g}\\text{.}[\/latex] Thus the mass of one cubic meter of air is 1.28 kg. If a living room has dimensions [latex]5\\phantom{\\rule{0.25em}{0ex}}\\text{m}\u00d7\\text{5 m}\u00d7\\text{3 m,}[\/latex] the mass of air inside the room is 96 kg, which is the typical mass of a human. <\/p>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2710286\" data-element-type=\"check-understanding\" data-label=\"\">\n<div data-type=\"title\">Check Your Understanding<\/div>\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2322677\">\n<p id=\"import-auto-id2017983\">The density of air at standard conditions [latex]\\left(P=1\\phantom{\\rule{0.25em}{0ex}}\\text{atm}[\/latex] and [latex]T=\\text{20}\\text{\u00ba}\\text{C}\\right)[\/latex] is [latex]1\\text{.}\\text{28}\\phantom{\\rule{0.25em}{0ex}}{\\text{kg\/m}}^{3}[\/latex]. At what pressure is the density [latex]0\\text{.}{\\text{64 kg\/m}}^{3}[\/latex] if the temperature and number of molecules are kept constant?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1509649\">\n<p id=\"import-auto-id2183997\">The best way to approach this question is to think about what is happening. If the density drops to half its original value and no molecules are lost, then the volume must double. If we look at the equation [latex]\\text{PV}=\\text{NkT}[\/latex], we see that when the temperature is constant, the pressure is inversely proportional to volume. Therefore, if the volume doubles, the pressure must drop to half its original value, and [latex]{P}_{\\text{f}}=0\\text{.}\\text{50}\\phantom{\\rule{0.25em}{0ex}}\\text{atm}\\text{.}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id2248571\">\n<h1 data-type=\"title\">The Ideal Gas Law Restated Using Moles<\/h1>\n<p id=\"import-auto-id2797949\">A very common expression of the ideal gas law uses the number of moles, [latex]n[\/latex], rather than the number of atoms and molecules, [latex]N[\/latex]. We start from the ideal gas law,<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id2689659\">[latex]\\text{PV}=\\text{NkT,}[\/latex]<\/div>\n<p id=\"import-auto-id2889310\">and multiply and divide the equation by Avogadro\u2019s number [latex]{N}_{\\text{A}}[\/latex]. This gives<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id2889307\">[latex]\\text{PV}=\\frac{N}{{N}_{\\text{A}}}{N}_{\\text{A}}\\text{kT}\\text{.}[\/latex]<\/div>\n<p id=\"import-auto-id2084701\">Note that [latex]n=N\/{N}_{\\text{A}}[\/latex] is the number of moles. We define the universal gas constant [latex]R={N}_{\\text{A}}k[\/latex], and obtain the ideal gas law in terms of moles.<\/p>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id2798942\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">Ideal Gas Law (in terms of moles)<\/div>\n<p id=\"import-auto-id2386604\">The ideal gas law (in terms of moles) is<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]\\text{PV}=\\text{nRT}.[\/latex]<\/div>\n<p id=\"import-auto-id2736663\">The numerical value of [latex]R[\/latex] in SI units is<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id3226853\">[latex]R={N}_{\\text{A}}k=\\left(6\\text{.}\\text{02}\u00d7{\\text{10}}^{\\text{23}}\\phantom{\\rule{0.25em}{0ex}}{\\text{mol}}^{-1}\\right)\\left(1\\text{.}\\text{38}\u00d7{\\text{10}}^{-\\text{23}}\\phantom{\\rule{0.25em}{0ex}}\\text{J\/K}\\right)=8\\text{.}\\text{31}\\phantom{\\rule{0.25em}{0ex}}\\text{J}\/\\text{mol}\\cdot \\text{K}.[\/latex]<\/div>\n<p id=\"import-auto-id2919773\">In other units, <\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id2919790\">[latex]\\begin{array}{lll}R&amp; =&amp; 1\\text{.}\\text{99}\\phantom{\\rule{0.25em}{0ex}}\\text{cal\/mol}\\cdot \\text{K}\\\\ R&amp; =&amp; \\text{0}\\text{.}\\text{0821 L}\\cdot \\text{atm\/mol}\\cdot \\text{K}\\text{.}\\end{array}[\/latex]<\/div>\n<p id=\"import-auto-id3227661\">You can use whichever value of [latex]R[\/latex] is most convenient for a particular problem. <\/p>\n<\/div>\n<div data-type=\"example\" class=\"textbox examples\" id=\"fs-id1444855\">\n<div data-type=\"title\" class=\"title\">Calculating Number of Moles: Gas in a Bike Tire<\/div>\n<p id=\"import-auto-id3224716\">How many moles of gas are in a bike tire with a volume of [latex]2\\text{.}\\text{00}\u00d7{\\text{10}}^{\u20133}\\phantom{\\rule{0.25em}{0ex}}{\\text{m}}^{3}\\left(2\\text{.}\\text{00 L}\\right),[\/latex] a pressure of [latex]7\\text{.}\\text{00}\u00d7{\\text{10}}^{5}\\phantom{\\rule{0.25em}{0ex}}\\text{Pa}[\/latex] (a gauge pressure of just under [latex]\\text{90}\\text{.}0\\phantom{\\rule{0.25em}{0ex}}{\\text{lb\/in}}^{2}[\/latex]), and at a temperature of [latex]\\text{18}\\text{.}0\\text{\u00ba}\\text{C}[\/latex]?<\/p>\n<p id=\"import-auto-id2600714\"><strong>Strategy<\/strong><\/p>\n<p id=\"import-auto-id2396040\">Identify the knowns and unknowns, and choose an equation to solve for the unknown. In this case, we solve the ideal gas law, [latex]\\text{PV}=\\text{nRT}[\/latex], for the number of moles [latex]n[\/latex]. <\/p>\n<p id=\"import-auto-id2124325\"><strong>Solution<\/strong><\/p>\n<p id=\"import-auto-id2124328\">1. Identify the knowns.<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id2124330\">[latex]\\begin{array}{lll}P&amp; =&amp; 7\\text{.}\\text{00}\u00d7{\\text{10}}^{5}\\phantom{\\rule{0.25em}{0ex}}\\text{Pa}\\\\ V&amp; =&amp; 2\\text{.}\\text{00}\u00d7{\\text{10}}^{-3}\\phantom{\\rule{0.25em}{0ex}}{\\text{m}}^{3}\\\\ T&amp; =&amp; \\text{18}\\text{.}0\\text{\u00ba}\\text{C}=\\text{291 K}\\\\ R&amp; =&amp; 8\\text{.}\\text{31}\\phantom{\\rule{0.25em}{0ex}}\\text{J\/mol}\\cdot \\text{K}\\end{array}[\/latex]<\/div>\n<p id=\"import-auto-id2402615\">2. Rearrange the equation to solve for [latex]n[\/latex] and substitute known values.<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id2402631\">[latex]\\begin{array}{lll}n&amp; =&amp; \\frac{\\text{PV}}{\\text{RT}}=\\frac{\\left(7\\text{.}\\text{00}\u00d7{\\text{10}}^{5}\\phantom{\\rule{0.25em}{0ex}}\\text{Pa}\\right)\\left(2\\text{.}00\u00d7{\\text{10}}^{-3}\\phantom{\\rule{0.25em}{0ex}}{\\text{m}}^{3}\\right)}{\\left(8\\text{.}\\text{31}\\phantom{\\rule{0.25em}{0ex}}\\text{J\/mol}\\cdot \\text{K}\\right)\\left(\\text{291}\\phantom{\\rule{0.25em}{0ex}}\\text{K}\\right)}\\\\ \\text{}&amp; =&amp; \\text{0}\\text{.}\\text{579}\\phantom{\\rule{0.25em}{0ex}}\\text{mol}\\end{array}[\/latex]<\/div>\n<p id=\"import-auto-id2862630\"><strong>Discussion<\/strong><\/p>\n<p id=\"import-auto-id2862614\">The most convenient choice for [latex]R[\/latex] in this case is [latex]8\\text{.}\\text{31}\\phantom{\\rule{0.25em}{0ex}}\\text{J\/mol}\\cdot \\text{K,}[\/latex] because our known quantities are in SI units. The pressure and temperature are obtained from the initial conditions in <a href=\"#fs-id1667893\" class=\"autogenerated-content\">(Figure)<\/a>, but we would get the same answer if we used the final values.<\/p>\n<\/div>\n<p id=\"import-auto-id2862600\">The ideal gas law can be considered to be another manifestation of the law of conservation of energy (see <a href=\"\/contents\/c5adb012-fbab-41a7-b5b2-4644984097fc@5\">Conservation of Energy<\/a>). Work done on a gas results in an increase in its energy, increasing pressure and\/or temperature, or decreasing volume. This increased energy can also be viewed as increased internal kinetic energy, given the gas\u2019s atoms and molecules.<\/p>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1542832\">\n<h1 data-type=\"title\">The Ideal Gas Law and Energy<\/h1>\n<p id=\"import-auto-id2717771\">Let us now examine the role of energy in the behavior of gases. When you inflate a bike tire by hand, you do work by repeatedly exerting a force through a distance. This energy goes into increasing the pressure of air inside the tire and increasing the temperature of the pump and the air.<\/p>\n<p id=\"import-auto-id2388350\">The ideal gas law is closely related to energy: the units on both sides are joules. The right-hand side of the ideal gas law in [latex]\\text{PV}=\\text{NkT}[\/latex] is [latex]\\text{NkT}[\/latex]. This term is roughly the amount of translational kinetic energy of [latex]N[\/latex] atoms or molecules at an absolute temperature [latex]T[\/latex], as we shall see formally in <a href=\"\/contents\/3fffbab5-5b0d-4622-be83-eb65111f9c3e@5\">Kinetic Theory: Atomic and Molecular Explanation of Pressure and Temperature<\/a>. The left-hand side of the ideal gas law is [latex]\\text{PV}[\/latex], which also has the units of joules. We know from our study of fluids that pressure is one type of potential energy per unit volume, so pressure multiplied by volume is energy. The important point is that there is energy in a gas related to both its pressure and its volume. The energy can be changed when the gas is doing work as it expands\u2014something we explore in <a href=\"\/contents\/7892cddd-833e-42b8-bcaa-a1e94fb9eb9c@3\">Heat and Heat Transfer Methods<\/a>\u2014similar to what occurs in gasoline or steam engines and turbines.<\/p>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id2254366\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">Problem-Solving Strategy: The Ideal Gas Law<\/div>\n<p id=\"import-auto-id2564676\"><strong>Step 1<\/strong> Examine the situation to determine that an ideal gas is involved. Most gases are nearly ideal.<\/p>\n<p id=\"import-auto-id1747449\"><strong>Step 2<\/strong> Make a list of what quantities are given, or can be inferred from the problem as stated (identify the known quantities). Convert known values into proper SI units (K for temperature, Pa for pressure, [latex]{\\text{m}}^{3}[\/latex] for volume, molecules for [latex]N[\/latex], and moles for [latex]n[\/latex]).<\/p>\n<p id=\"import-auto-id2731891\"><strong>Step 3<\/strong> Identify exactly what needs to be determined in the problem (identify the unknown quantities). A written list is useful.<\/p>\n<p id=\"import-auto-id3227648\"><strong>Step 4<\/strong> Determine whether the number of molecules or the number of moles is known, in order to decide which form of the ideal gas law to use. The first form is [latex]\\text{PV}=\\text{NkT}[\/latex] and involves [latex]N[\/latex], the number of atoms or molecules. The second form is [latex]\\text{PV}=\\text{nRT}[\/latex] and involves [latex]n[\/latex], the number of moles.<\/p>\n<p><strong>Step 5<\/strong> Solve the ideal gas law for the quantity to be determined (the unknown quantity). You may need to take a ratio of final states to initial states to eliminate the unknown quantities that are kept fixed.<\/p>\n<p id=\"import-auto-id2688944\"><strong>Step 6<\/strong> Substitute the known quantities, along with their units, into the appropriate equation, and obtain numerical solutions complete with units. Be certain to use absolute temperature and absolute pressure.<\/p>\n<p id=\"import-auto-id2688950\"><strong>Step 7<\/strong> Check the answer to see if it is reasonable: Does it make sense?<\/p>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1445894\" data-element-type=\"check-understanding\" data-label=\"\">\n<div data-type=\"title\">Check Your Understanding<\/div>\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1445900\">\n<p id=\"import-auto-id2688959\">Liquids and solids have densities about 1000 times greater than gases. Explain how this implies that the distances between atoms and molecules in gases are about 10 times greater than the size of their atoms and molecules. <\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1677733\">\n<p id=\"import-auto-id2688965\">Atoms and molecules are close together in solids and liquids. In gases they are separated by empty space. Thus gases have lower densities than liquids and solids. Density is mass per unit volume, and volume is related to the size of a body (such as a sphere) cubed. So if the distance between atoms and molecules increases by a factor of 10, then the volume occupied increases by a factor of 1000, and the density decreases by a factor of 1000.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"section-summary\" data-depth=\"1\" id=\"fs-id1698840\">\n<h1 data-type=\"title\">Section Summary<\/h1>\n<ul id=\"fs-id2398625\">\n<li id=\"import-auto-id739591\">The ideal gas law relates the pressure and volume of a gas to the number of gas molecules and the temperature of the gas.<\/li>\n<li id=\"import-auto-id739594\">The ideal gas law can be written in terms of the number of molecules of gas:\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id739597\">[latex]\\text{PV}=\\text{NkT},[\/latex]<\/div>\n<p>where [latex]P[\/latex] is pressure, [latex]V[\/latex] is volume, [latex]T[\/latex] is temperature, [latex]N[\/latex] is number of molecules, and [latex]k[\/latex] is the Boltzmann constant<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id2564573\">[latex]k=1\\text{.}\\text{38}\u00d7{\\text{10}}^{\u2013\\text{23}}\\phantom{\\rule{0.25em}{0ex}}\\text{J\/K}.[\/latex]<\/div>\n<\/li>\n<li id=\"import-auto-id2402831\">A mole is the number of atoms in a 12-g sample of carbon-12. <\/li>\n<li id=\"import-auto-id2402833\">The number of molecules in a mole is called Avogadro\u2019s number [latex]{N}_{\\text{A}}[\/latex],\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id2402820\">[latex]{N}_{\\text{A}}=6\\text{.}\\text{02}\u00d7{\\text{10}}^{\\text{23}}\\phantom{\\rule{0.25em}{0ex}}{\\text{mol}}^{-1}.[\/latex]<\/div>\n<\/li>\n<li id=\"import-auto-id2599009\">A mole of any substance has a mass in grams equal to its molecular weight, which can be determined from the periodic table of elements.<\/li>\n<li id=\"import-auto-id2599036\">The ideal gas law can also be written and solved in terms of the number of moles of gas:\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id2599038\">[latex]\\text{PV}=\\text{nRT},[\/latex]<\/div>\n<p>where [latex]n[\/latex] is number of moles and [latex]R[\/latex] is the universal gas constant, <\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id2862692\">[latex]R=8\\text{.}\\text{31}\\phantom{\\rule{0.25em}{0ex}}\\text{J\/mol}\\cdot \\text{K}.[\/latex]<\/div>\n<\/li>\n<li id=\"import-auto-id2883325\">The ideal gas law is generally valid at temperatures well above the boiling temperature.<\/li>\n<\/ul>\n<\/div>\n<div class=\"conceptual-questions\" data-depth=\"1\" id=\"fs-id2719827\" data-element-type=\"conceptual-questions\">\n<h1 data-type=\"title\">Conceptual Questions<\/h1>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2591367\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2591368\">\n<p id=\"import-auto-id2883312\">Find out the human population of Earth. Is there a mole of people inhabiting Earth? If the average mass of a person is 60 kg, calculate the mass of a mole of people. How does the mass of a mole of people compare with the mass of Earth?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2799079\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2799080\">\n<p id=\"import-auto-id2236451\">Under what circumstances would you expect a gas to behave significantly differently than predicted by the ideal gas law?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2378151\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2378152\">\n<p id=\"import-auto-id2236458\">A constant-volume gas thermometer contains a fixed amount of gas. What property of the gas is measured to indicate its temperature?<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"problems-exercises\" data-depth=\"1\" id=\"fs-id2682273\" data-element-type=\"problems-exercises\">\n<h1 data-type=\"title\">Problems &amp; Exercises<\/h1>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2682279\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2682280\">\n<p id=\"import-auto-id2236470\">The gauge pressure in your car tires is [latex]2\\text{.}\\text{50}\u00d7{\\text{10}}^{5}\\phantom{\\rule{0.25em}{0ex}}{\\text{N\/m}}^{2}[\/latex] at a temperature of [latex]\\text{35}\\text{.}0\\text{\u00ba}\\text{C}[\/latex] when you drive it onto a ferry boat to Alaska. What is their gauge pressure later, when their temperature has dropped to [latex]\u2013\\text{40}\\text{.}0\\text{\u00ba}\\text{C}[\/latex]?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id2684234\">\n<p id=\"import-auto-id2677607\">1.62 atm<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2798567\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2798568\">\n<p id=\"import-auto-id2677616\">Convert an absolute pressure of [latex]7\\text{.}\\text{00}\u00d7{\\text{10}}^{5}\\phantom{\\rule{0.25em}{0ex}}{\\text{N\/m}}^{2}[\/latex] to gauge pressure in [latex]{\\text{lb\/in}}^{2}\\text{.}[\/latex] (This value was stated to be just less than [latex]\\text{90}\\text{.}{\\text{0 lb\/in}}^{2}[\/latex] in <a href=\"#fs-id1444855\" class=\"autogenerated-content\">(Figure)<\/a>. Is it?) <\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2804147\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2804148\">\n<p id=\"import-auto-id2681625\">Suppose a gas-filled incandescent light bulb is manufactured so that the gas inside the bulb is at atmospheric pressure when the bulb has a temperature of [latex]\\text{20}\\text{.}0\\text{\u00ba}\\text{C}[\/latex]. (a) Find the gauge pressure inside such a bulb when it is hot, assuming its average temperature is [latex]\\text{60}\\text{.}0\\text{\u00ba}\\text{C}[\/latex] (an approximation) and neglecting any change in volume due to thermal expansion or gas leaks. (b) The actual final pressure for the light bulb will be less than calculated in part (a) because the glass bulb will expand. What will the actual final pressure be, taking this into account? Is this a negligible difference?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id2736663\">\n<p id=\"import-auto-id2386001\">(a) 0.136 atm <\/p>\n<p id=\"import-auto-id2386017\">(b) 0.135 atm. The difference between this value and the value from part (a) is negligible.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2687223\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2687224\">\n<p id=\"import-auto-id2386026\">Large helium-filled balloons are used to lift scientific equipment to high altitudes. (a) What is the pressure inside such a balloon if it starts out at sea level with a temperature of [latex]\\text{10}\\text{.}0\\text{\u00ba}\\text{C}[\/latex] and rises to an altitude where its volume is twenty times the original volume and its temperature is [latex]\u2013\\text{50}\\text{.}0\\text{\u00ba}\\text{C}[\/latex]? (b) What is the gauge pressure? (Assume atmospheric pressure is constant.) <\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1746230\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1746231\">\n<p id=\"import-auto-id3224535\">Confirm that the units of [latex]\\text{nRT}[\/latex] are those of energy for each value of [latex]R[\/latex]: (a) [latex]8\\text{.}\\text{31}\\phantom{\\rule{0.25em}{0ex}}\\text{J\/mol}\\cdot \\text{K}[\/latex], (b) [latex]1\\text{.}\\text{99 cal\/mol}\\cdot \\text{K}[\/latex], and (c) [latex]0\\text{.}\\text{0821 L}\\cdot \\text{atm\/mol}\\cdot \\text{K}[\/latex].<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id2404519\">\n<p id=\"import-auto-id2926437\">(a) [latex]\\text{nRT}=\\left(\\text{mol}\\right)\\left(\\text{J\/mol}\\cdot \\text{K}\\right)\\left(\\text{K}\\right)=\\text{J}[\/latex]<\/p>\n<p id=\"import-auto-id1563389\">(b) [latex]\\text{nRT}=\\left(\\text{mol}\\right)\\left(\\text{cal\/mol}\\cdot \\text{K}\\right)\\left(\\text{K}\\right)=\\text{cal}[\/latex]<\/p>\n<p id=\"import-auto-id2409504\">(c) [latex]\\begin{array}{lll}\\text{nRT}&amp; =&amp; \\left(\\text{mol}\\right)\\left(\\text{L}\\cdot \\text{atm\/mol}\\cdot \\text{K}\\right)\\left(\\text{K}\\right)\\\\ &amp; =&amp; \\text{L}\\cdot \\text{atm}=\\left({\\text{m}}^{3}\\right)\\left({\\text{N\/m}}^{2}\\right)\\\\ &amp; =&amp; \\text{N}\\cdot \\text{m}=\\text{J}\\end{array}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2094656\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2094657\">\n<p id=\"import-auto-id2637302\">In the text, it was shown that [latex]N\/V=2\\text{.}\\text{68}\u00d7{\\text{10}}^{\\text{25}}\\phantom{\\rule{0.25em}{0ex}}{\\text{m}}^{-3}[\/latex]  for gas at STP. (a) Show that this quantity is equivalent to [latex]N\/V=2\\text{.}\\text{68}\u00d7{\\text{10}}^{\\text{19}}\\phantom{\\rule{0.25em}{0ex}}{\\text{cm}}^{-3},[\/latex]  as stated. (b) About how many atoms are there in one [latex]{\\text{\u03bcm}}^{3}[\/latex] (a cubic micrometer) at STP? (c) What does your answer to part (b) imply about the separation of atoms and molecules? <\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2631403\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2631404\">\n<p id=\"import-auto-id2883481\">Calculate the number of moles in the 2.00-L volume of air in the lungs of the average person. Note that the air is at [latex]\\text{37}\\text{.}0\\text{\u00ba}\\text{C}[\/latex] (body temperature).<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id2683493\">\n<p id=\"eip-id1318898\">[latex]7\\text{.}\\text{86}\u00d7{\\text{10}}^{-2}\\phantom{\\rule{0.25em}{0ex}}\\text{mol}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2932069\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2932070\">\n<p id=\"import-auto-id3184957\">An airplane passenger has [latex]\\text{100}\\phantom{\\rule{0.25em}{0ex}}{\\text{cm}}^{3}[\/latex] of air in his stomach just before the plane takes off from a sea-level airport. What volume will the air have at cruising altitude if cabin pressure drops to [latex]7\\text{.}\\text{50}\u00d7{\\text{10}}^{4}\\phantom{\\rule{0.25em}{0ex}}{\\text{N\/m}}^{2}?[\/latex]<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2739655\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2739656\">\n<p id=\"import-auto-id2631677\">(a) What is the volume (in [latex]{\\text{km}}^{3}[\/latex]) of Avogadro\u2019s number of sand grains if each grain is a cube and has sides that are 1.0 mm long? (b) How many kilometers of beaches in length would this cover if the beach averages 100 m in width and 10.0 m in depth? Neglect air spaces between grains.<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id2734849\">\n<p id=\"import-auto-id2393401\">(a) [latex]6\\text{.}\\text{02}\u00d7{\\text{10}}^{5}\\phantom{\\rule{0.25em}{0ex}}{\\text{km}}^{3}[\/latex]<\/p>\n<p id=\"import-auto-id2393424\">(b) [latex]6\\text{.}\\text{02}\u00d7{\\text{10}}^{8}\\phantom{\\rule{0.25em}{0ex}}\\text{km}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1908858\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1908859\">\n<p id=\"import-auto-id2597680\">An expensive vacuum system can achieve a pressure as low as [latex]1\\text{.}\\text{00}\u00d7{\\text{10}}^{\u20137}\\phantom{\\rule{0.25em}{0ex}}{\\text{N\/m}}^{2}[\/latex] at [latex]\\text{20}\\text{\u00ba}\\text{C}[\/latex]. How many atoms are there in a cubic centimeter at this pressure and temperature? <\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2328150\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2328151\">\n<p id=\"import-auto-id3227463\">The number density of gas atoms at a certain location in the space above our planet is about [latex]1\\text{.}\\text{00}\u00d7{\\text{10}}^{\\text{11}}\\phantom{\\rule{0.25em}{0ex}}{\\text{m}}^{-3},[\/latex] and the pressure is [latex]2\\text{.}\\text{75}\u00d7{\\text{10}}^{\u2013\\text{10}}\\phantom{\\rule{0.25em}{0ex}}{\\text{N\/m}}^{2}[\/latex] in this space. What is the temperature there?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id2631515\">\n<p id=\"eip-id3084073\">[latex]-\\text{73}\\text{.}9\\text{\u00ba}\\text{C}[\/latex]<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1804845\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1804846\">\n<p id=\"import-auto-id2688601\">A bicycle tire has a pressure of [latex]7\\text{.}\\text{00}\u00d7{\\text{10}}^{5}\\phantom{\\rule{0.25em}{0ex}}{\\text{N\/m}}^{2}[\/latex] at a temperature of [latex]\\text{18}\\text{.}0\\text{\u00ba}\\text{C}[\/latex] and contains 2.00 L of gas. What will its pressure be if you let out an amount of air that has a volume of [latex]\\text{100}\\phantom{\\rule{0.25em}{0ex}}{\\text{cm}}^{3}[\/latex] at atmospheric pressure? Assume tire temperature and volume remain constant. <\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2889733\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2409195\">\n<p id=\"import-auto-id2698879\">A high-pressure gas cylinder contains 50.0 L of toxic gas at a pressure of [latex]1\\text{.}\\text{40}\u00d7{\\text{10}}^{7}\\phantom{\\rule{0.25em}{0ex}}{\\text{N\/m}}^{2}[\/latex] and a temperature of [latex]\\text{25}\\text{.}0\\text{\u00ba}\\text{C}[\/latex]. Its valve leaks after the cylinder is dropped. The cylinder is cooled to dry ice temperature [latex]\\left(\u2013\\text{78}\\text{.}5\\text{\u00ba}\\text{C}\\right)[\/latex] to reduce the leak rate and pressure so that it can be safely repaired. (a) What is the final pressure in the tank, assuming a negligible amount of gas leaks while being cooled and that there is no phase change? (b) What is the final pressure if one-tenth of the gas escapes? (c) To what temperature must the tank be cooled to reduce the pressure to 1.00 atm (assuming the gas does not change phase and that there is no leakage during cooling)? (d) Does cooling the tank appear to be a practical solution? <\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1695248\">\n<p id=\"import-auto-id2887240\">(a) [latex]9\\text{.}\\text{14}\u00d7{\\text{10}}^{6}\\phantom{\\rule{0.25em}{0ex}}{\\text{N\/m}}^{2}[\/latex]<\/p>\n<p id=\"import-auto-id2385013\">(b) [latex]8\\text{.}\\text{23}\u00d7{\\text{10}}^{6}\\phantom{\\rule{0.25em}{0ex}}{\\text{N\/m}}^{2}[\/latex]<\/p>\n<p id=\"import-auto-id2692716\">(c) 2.16 K<\/p>\n<p id=\"import-auto-id2385055\">(d) No. The final temperature needed is much too low to be easily achieved for a large object.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2084983\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2084984\">\n<p id=\"import-auto-id2692718\">Find the number of moles in 2.00 L of gas at [latex]\\text{35}\\text{.}0\\text{\u00ba}\\text{C}[\/latex] and under [latex]7\\text{.}\\text{41}\u00d7{\\text{10}}^{7}\\phantom{\\rule{0.25em}{0ex}}{\\text{N\/m}}^{2}[\/latex] of pressure. <\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2735156\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2735157\">\n<p id=\"import-auto-id2681492\">Calculate the depth to which Avogadro\u2019s number of table tennis balls would cover Earth. Each ball has a diameter of 3.75 cm. Assume the space between balls adds an extra 25.0% to their volume and assume they are not crushed by their own weight.<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id2735166\">\n<p id=\"import-auto-id2681473\">41 km<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1544154\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1544155\">\n<p id=\"import-auto-id2681482\">(a) What is the gauge pressure in a [latex]\\text{25}\\text{.}0\\text{\u00ba}\\text{C}[\/latex] car tire containing 3.60 mol of gas in a 30.0 L volume? (b) What will its gauge pressure be if you add 1.00 L of gas originally at atmospheric pressure and [latex]\\text{25}\\text{.}0\\text{\u00ba}\\text{C}[\/latex]? Assume the temperature returns to [latex]\\text{25}\\text{.}0\\text{\u00ba}\\text{C}[\/latex] and the volume remains constant. <\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2869966\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2869967\">\n<p id=\"import-auto-id2883532\">(a) In the deep space between galaxies, the density of atoms is as low as [latex]{\\text{10}}^{6}\\phantom{\\rule{0.25em}{0ex}}{\\text{atoms\/m}}^{3},[\/latex] and the temperature is a frigid 2.7 K. What is the pressure? (b) What volume (in [latex]{\\text{m}}^{3}[\/latex]) is occupied by 1 mol of gas? (c) If this volume is a cube, what is the length of its sides in kilometers?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id2403810\">\n<p id=\"import-auto-id2393393\">(a) [latex]3\\text{.}7\u00d7{\\text{10}}^{-\\text{17}}\\phantom{\\rule{0.25em}{0ex}}\\text{Pa}[\/latex]<\/p>\n<p id=\"import-auto-id3177718\">(b) [latex]6\\text{.}0\u00d7{\\text{10}}^{\\text{17}}\\phantom{\\rule{0.25em}{0ex}}{\\text{m}}^{3}[\/latex]<\/p>\n<p id=\"import-auto-id2677758\">(c) [latex]8\\text{.}4\u00d7{\\text{10}}^{2}\\phantom{\\rule{0.25em}{0ex}}\\text{km}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"glossary\" class=\"textbox shaded\">\n<h2 data-type=\"glossary-title\">Glossary<\/h2>\n<dl class=\"definition\" id=\"import-auto-id2744320\">\n<dt>ideal gas law<\/dt>\n<dd id=\"fs-id2084747\">the physical law that relates the pressure and volume of a gas to the number of gas molecules or number of moles of gas and the temperature of the gas<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id2744329\">\n<dt>Boltzmann constant<\/dt>\n<dd id=\"fs-id2699740\">[latex]k[\/latex] , a physical constant that relates energy to temperature; [latex]k=\\text{1.38}\u00d7{\\text{10}}^{\\text{\u201323}}\\phantom{\\rule{0.25em}{0ex}}\\text{J\/K}[\/latex]<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id2626622\">\n<dt>Avogadro\u2019s number<\/dt>\n<dd id=\"fs-id2818566\">[latex]{N}_{\\text{A}}[\/latex] , the number of molecules or atoms in one mole of a substance;<br>\n[latex]{N}_{\\text{A}}=6\\text{.}\\text{02}\u00d7{\\text{10}}^{\\text{23}}[\/latex]<br>\n     particles\/mole<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"fs-id1609952\">\n<dt>mole<\/dt>\n<dd id=\"fs-id1609957\">the quantity of a substance whose mass (in grams) is equal to its molecular mass<\/dd>\n<\/dl>\n<\/div>\n\n","rendered":"<div class=\"textbox learning-objectives\">\n<h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>\n<ul>\n<li>State the ideal gas law in terms of molecules and in terms of moles.<\/li>\n<li>Use the ideal gas law to calculate pressure change, temperature change, volume change, or the number of molecules or moles in a given volume.<\/li>\n<li>Use Avogadro\u2019s number to convert between number of molecules and number of moles.<\/li>\n<\/ul>\n<\/div>\n<div class=\"bc-figure figure\" id=\"import-auto-id1563742\">\n<div class=\"bc-figcaption figcaption\">The air inside this hot air balloon flying over Putrajaya, Malaysia, is hotter than the ambient air. As a result, the balloon experiences a buoyant force pushing it upward. (credit: Kevin Poh, Flickr)<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id2600697\" data-alt=\"\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_14_03_00.jpg\" data-media-type=\"image\/jpg\" alt=\"\" width=\"200\" \/><\/span><\/p>\n<\/div>\n<p id=\"import-auto-id2409403\">In this section, we continue to explore the thermal behavior of gases. In particular, we examine the characteristics of atoms and molecules that compose gases. (Most gases, for example nitrogen, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ce901c2fc7f90b06af1a774799a4fa18_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#78;&#125;&#125;&#95;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"20\" style=\"vertical-align: -3px;\" \/>, and oxygen, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-0deb39be3c7640742283a1b426c549fd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#79;&#125;&#125;&#95;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"21\" style=\"vertical-align: -3px;\" \/>, are composed of two or more atoms. We will primarily use the term \u201cmolecule\u201d in discussing a gas because the term can also be applied to monatomic gases, such as helium.)<\/p>\n<p id=\"import-auto-id2596139\">Gases are easily compressed. We can see evidence of this in <a href=\"\/contents\/0b6d0d23-e96f-4f34-8051-643361f8c7c7@4#import-auto-id1814176\" class=\"autogenerated-content\">(Figure)<\/a>, where you will note that gases have the <em data-effect=\"italics\">largest<\/em> coefficients of volume expansion. The large coefficients mean that gases expand and contract very rapidly with temperature changes. In addition, you will note that most gases expand at the <em data-effect=\"italics\">same<\/em> rate, or have the same <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-39ffee81b79fbfa10c128d48495e8b8b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#116;&#97;&#32;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"11\" style=\"vertical-align: -4px;\" \/>. This raises the question as to why gases should all act in nearly the same way, when liquids and solids have widely varying expansion rates.<\/p>\n<p id=\"import-auto-id1590402\">The answer lies in the large separation of atoms and molecules in gases, compared to their sizes, as illustrated in <a href=\"#import-auto-id2233613\" class=\"autogenerated-content\">(Figure)<\/a>. Because atoms and molecules have large separations, forces between them can be ignored, except when they collide with each other during collisions. The motion of atoms and molecules (at temperatures well above the boiling temperature) is fast, such that the gas occupies all of the accessible volume and the expansion of gases is rapid. In contrast, in liquids and solids, atoms and molecules are closer together and are quite sensitive to the forces between them.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id2233613\">\n<div class=\"bc-figcaption figcaption\">Atoms and molecules in a gas are typically widely separated, as shown. Because the forces between them are quite weak at these distances, the properties of a gas depend more on the number of atoms per unit volume and on temperature than on the type of atom.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id2600397\" data-alt=\"Spheres representing atoms and molecules; the spheres are relatively far apart and are distributed randomly.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_14_03_01.jpg\" data-media-type=\"image\/jpg\" alt=\"Spheres representing atoms and molecules; the spheres are relatively far apart and are distributed randomly.\" width=\"250\" \/><\/span><\/p>\n<\/div>\n<p id=\"import-auto-id1856142\">To get some idea of how pressure, temperature, and volume of a gas are related to one another, consider what happens when you pump air into an initially deflated tire. The tire\u2019s volume first increases in direct proportion to the amount of air injected, without much increase in the tire pressure. Once the tire has expanded to nearly its full size, the walls limit volume expansion. If we continue to pump air into it, the pressure increases. The pressure will further increase when the car is driven and the tires move. Most manufacturers specify optimal tire pressure for cold tires. (See <a href=\"#import-auto-id2102337\" class=\"autogenerated-content\">(Figure)<\/a>.)<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id2102337\">\n<div class=\"bc-figcaption figcaption\">(a) When air is pumped into a deflated tire, its volume first increases without much increase in pressure. (b) When the tire is filled to a certain point, the tire walls resist further expansion and the pressure increases with more air. (c) Once the tire is inflated, its pressure increases with temperature.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1498134\" data-alt=\"The figure has three parts, each part showing a pair of tires, and each tire connected to a pressure gauge. Each pair of tires represents the before and after images of a single tire, along with a change in pressure in that tire. In part a, the tire pressure is initially zero. After some air is added, represented by an arrow labeled Add air, the pressure rises to slightly above zero. In part b, the tire pressure is initially at the half-way mark. After some air is added, represented by an arrow labeled Add air, the pressure rises to the three-fourths mark. In part c, the tire pressure is initially at the three-fourths mark. After the temperature is raised, represented by an arrow labeled Increase temperature, the pressure rises to nearly the full mark.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_14_03_02.jpg\" data-media-type=\"image\/jpg\" alt=\"The figure has three parts, each part showing a pair of tires, and each tire connected to a pressure gauge. Each pair of tires represents the before and after images of a single tire, along with a change in pressure in that tire. In part a, the tire pressure is initially zero. After some air is added, represented by an arrow labeled Add air, the pressure rises to slightly above zero. In part b, the tire pressure is initially at the half-way mark. After some air is added, represented by an arrow labeled Add air, the pressure rises to the three-fourths mark. In part c, the tire pressure is initially at the three-fourths mark. After the temperature is raised, represented by an arrow labeled Increase temperature, the pressure rises to nearly the full mark.\" width=\"500\" \/><\/span><\/p>\n<\/div>\n<p id=\"import-auto-id1149559\">At room temperatures, collisions between atoms and molecules can be ignored. In this case, the gas is called an ideal gas, in which case the relationship between the pressure, volume, and temperature is given by the equation of state called the ideal gas law.<\/p>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id2683246\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">Ideal Gas Law<\/div>\n<p id=\"import-auto-id1652885\">The <span data-type=\"term\" id=\"import-auto-id1530949\">ideal gas law<\/span> states that<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id2374115\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-589efb3659e66211d221a525587ee725_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#80;&#86;&#125;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#78;&#107;&#84;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"88\" style=\"vertical-align: -4px;\" \/><\/div>\n<p id=\"import-auto-id921222\">where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-650eb7688af6737ac325425b5c9a5982_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#80;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/> is the absolute pressure of a gas, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-63ada879859a9e41fd935f035b7313bc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#86;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/> is the volume it occupies, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-5793832f979c2268e3694c246d53b1bb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#78;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"16\" style=\"vertical-align: 0px;\" \/> is the number of atoms and molecules in the gas, and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f9ed275b0bf1633b7ee83b78fcc28273_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#84;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"13\" style=\"vertical-align: 0px;\" \/> is its absolute temperature. The constant <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-3422b6bb5c160593658b7c39425d9880_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#107;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"9\" style=\"vertical-align: 0px;\" \/> is called the <span data-type=\"term\" id=\"import-auto-id1634900\">Boltzmann constant<\/span> in honor of Austrian physicist Ludwig Boltzmann (1844\u20131906) and has the value<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id1586104\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-68d877ad2c8f18a65673448be4c189ac_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#107;&#61;&#49;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#56;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#51;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#74;&#125;&#47;&#92;&#116;&#101;&#120;&#116;&#123;&#75;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"148\" style=\"vertical-align: -5px;\" \/><\/div>\n<\/div>\n<p id=\"import-auto-id2238750\">The ideal gas law can be derived from basic principles, but was originally deduced from experimental measurements of Charles\u2019 law (that volume occupied by a gas is proportional to temperature at a fixed pressure) and from Boyle\u2019s law (that for a fixed temperature, the product <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-bdc1ef9bc8ee074a1147649d94f3db01_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#80;&#86;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"25\" style=\"vertical-align: -1px;\" \/> is a constant). In the ideal gas model, the volume occupied by its atoms and molecules is a negligible fraction of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-63ada879859a9e41fd935f035b7313bc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#86;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/>. The ideal gas law describes the behavior of real gases under most conditions. (Note, for example, that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-5793832f979c2268e3694c246d53b1bb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#78;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"16\" style=\"vertical-align: 0px;\" \/> is the total number of atoms and molecules, independent of the type of gas.)<\/p>\n<p id=\"import-auto-id2248763\">Let us see how the ideal gas law is consistent with the behavior of filling the tire when it is pumped slowly and the temperature is constant. At first, the pressure <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-650eb7688af6737ac325425b5c9a5982_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#80;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/> is essentially equal to atmospheric pressure, and the volume <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-63ada879859a9e41fd935f035b7313bc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#86;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/> increases in direct proportion to the number of atoms and molecules <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-5793832f979c2268e3694c246d53b1bb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#78;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"16\" style=\"vertical-align: 0px;\" \/> put into the tire. Once the volume of the tire is constant, the equation <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-5d32b74a5df6abe7863a6f8bd9523f19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#80;&#86;&#125;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#78;&#107;&#84;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"84\" style=\"vertical-align: -1px;\" \/> predicts that the pressure should increase in proportion to <em data-effect=\"italics\">the number N of atoms and molecules<\/em>. <\/p>\n<div data-type=\"example\" class=\"textbox examples\" id=\"fs-id1667893\">\n<div data-type=\"title\" class=\"title\">Calculating Pressure Changes Due to Temperature Changes: Tire Pressure<\/div>\n<p id=\"import-auto-id3224621\">Suppose your bicycle tire is fully inflated, with an absolute pressure of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-3607f165288a4edbdc16988644623465_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#48;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#53;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#80;&#97;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"82\" style=\"vertical-align: -1px;\" \/> (a gauge pressure of just under <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-282961fcce3340d63ebd8fe6d131dc30_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#57;&#48;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#48;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#98;&#47;&#105;&#110;&#125;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"81\" style=\"vertical-align: -4px;\" \/>) at a temperature of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c272f96dee470023e23bc8f50037fede_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#56;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&ordm;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"43\" style=\"vertical-align: -1px;\" \/>. What is the pressure after its temperature has risen to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c143655cb1029e983b8af3ef07ba377f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#53;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&ordm;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"44\" style=\"vertical-align: 0px;\" \/>? Assume that there are no appreciable leaks or changes in volume.<\/p>\n<p id=\"import-auto-id2194007\"><strong>Strategy<\/strong><\/p>\n<p id=\"import-auto-id2631292\">The pressure in the tire is changing only because of changes in temperature. First we need to identify what we know and what we want to know, and then identify an equation to solve for the unknown.<\/p>\n<p id=\"import-auto-id1084477\">We know the initial pressure <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-cdc1c2419e41c8b1ee7d884946d6703b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#80;&#125;&#95;&#123;&#48;&#125;&#61;&#55;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#48;&#48;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#53;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#80;&#97;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"125\" style=\"vertical-align: -3px;\" \/>, the initial temperature<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2f6eee50dd545fb05941f71f15849ad2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#84;&#125;&#95;&#123;&#48;&#125;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#56;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#48;&ordm;&#67;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"87\" style=\"vertical-align: -3px;\" \/>, and the final temperature<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-83fc9295c31c7d31e3266fa1d7546ea6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#84;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#102;&#125;&#125;&#61;&#51;&#53;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#48;&ordm;&#67;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"85\" style=\"vertical-align: -4px;\" \/>. We must find the final pressure<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f1beaab217615c5de8eb83aba60d7538_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#80;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#102;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"16\" style=\"vertical-align: -4px;\" \/>. How can we use the equation<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-5d32b74a5df6abe7863a6f8bd9523f19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#80;&#86;&#125;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#78;&#107;&#84;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"84\" style=\"vertical-align: -1px;\" \/>? At first, it may seem that not enough information is given, because the volume<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-63ada879859a9e41fd935f035b7313bc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#86;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/> and number of atoms<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-5793832f979c2268e3694c246d53b1bb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#78;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"16\" style=\"vertical-align: 0px;\" \/> are not specified. What we can do is use the equation twice:<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-dfcecf9fa108841f5a7e7b65760933f4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#80;&#125;&#95;&#123;&#48;&#125;&#123;&#86;&#125;&#95;&#123;&#48;&#125;&#61;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#78;&#107;&#84;&#125;&#125;&#95;&#123;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"103\" style=\"vertical-align: -3px;\" \/> and<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-6c07ba2c504e1f3b5ab54eee515a33f1_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#80;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#102;&#125;&#125;&#123;&#86;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#102;&#125;&#125;&#61;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#78;&#107;&#84;&#125;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#102;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"96\" style=\"vertical-align: -4px;\" \/>. If we divide<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-9863aa2fb82aff139cd3aea93ad5cacf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#80;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#102;&#125;&#125;&#123;&#86;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#102;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"31\" style=\"vertical-align: -4px;\" \/> by<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-6334653d4e390c0ed48edaf9d962c4ca_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#80;&#125;&#95;&#123;&#48;&#125;&#123;&#86;&#125;&#95;&#123;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"36\" style=\"vertical-align: -3px;\" \/> we can come up with an equation that allows us to solve for<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f1beaab217615c5de8eb83aba60d7538_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#80;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#102;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"16\" style=\"vertical-align: -4px;\" \/>. <\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id2084395\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-e3431b40fd178bcc0553d0581f2e4fe2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#80;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#102;&#125;&#125;&#123;&#86;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#102;&#125;&#125;&#125;&#123;&#123;&#80;&#125;&#95;&#123;&#48;&#125;&#123;&#86;&#125;&#95;&#123;&#48;&#125;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#78;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#102;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#107;&#84;&#125;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#102;&#125;&#125;&#125;&#123;&#123;&#78;&#125;&#95;&#123;&#48;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#107;&#84;&#125;&#125;&#95;&#123;&#48;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"26\" width=\"100\" style=\"vertical-align: -8px;\" \/><\/div>\n<p id=\"import-auto-id1585921\">Since the volume is constant, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-523962ec104cb01b9eaa85a89dd474f0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#86;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#102;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"15\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-16f33352c457c46135b0862350da2bd4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#86;&#125;&#95;&#123;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"17\" style=\"vertical-align: -3px;\" \/> are the same and they cancel out. The same is true for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-fb90817851b317b4b01270074e3ce076_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#78;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#102;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"19\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-00ca9887822a4bb1cb0c78c59b728d35_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#78;&#125;&#95;&#123;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"21\" style=\"vertical-align: -3px;\" \/>, and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-3422b6bb5c160593658b7c39425d9880_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#107;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"9\" style=\"vertical-align: 0px;\" \/>, which is a constant. Therefore,<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id1807145\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d4eb620f1273b43fa1d2a96a73b2d641_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#80;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#102;&#125;&#125;&#125;&#123;&#123;&#80;&#125;&#95;&#123;&#48;&#125;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#84;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#102;&#125;&#125;&#125;&#123;&#123;&#84;&#125;&#95;&#123;&#48;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"64\" style=\"vertical-align: -8px;\" \/><\/div>\n<p id=\"import-auto-id2084638\">We can then rearrange this to solve for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f1beaab217615c5de8eb83aba60d7538_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#80;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#102;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"16\" style=\"vertical-align: -4px;\" \/>:<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id2084642\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ad64bd21e4d7b82d04a41e444a07a71f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#80;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#102;&#125;&#125;&#61;&#123;&#80;&#125;&#95;&#123;&#48;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#84;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#102;&#125;&#125;&#125;&#123;&#123;&#84;&#125;&#95;&#123;&#48;&#125;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"82\" style=\"vertical-align: -8px;\" \/><\/div>\n<p id=\"import-auto-id2798629\">where the temperature must be in units of kelvins, because <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-a34c57f8ebcd49f83055933c0468e06c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#84;&#125;&#95;&#123;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"17\" style=\"vertical-align: -3px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-65da1c364cb6d155b10ea2cf2504a7b8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#84;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#102;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"15\" style=\"vertical-align: -4px;\" \/> are absolute temperatures.<\/p>\n<p id=\"import-auto-id2673782\"><strong>Solution<\/strong><\/p>\n<p id=\"import-auto-id2387499\">1. Convert temperatures from Celsius to Kelvin.<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id2387502\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-7077822bec81b6618a867e8c19202bd5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#125;&#123;&#84;&#125;&#95;&#123;&#48;&#125;&#61;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#56;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#48;&#43;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#55;&#51;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#116;&#101;&#120;&#116;&#123;&#75;&#125;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#57;&#49;&#32;&#75;&#125;&#92;&#92;&#32;&#123;&#84;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#102;&#125;&#125;&#61;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#53;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#48;&#43;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#55;&#51;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#116;&#101;&#120;&#116;&#123;&#75;&#125;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#48;&#56;&#32;&#75;&#125;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"40\" width=\"225\" style=\"vertical-align: -15px;\" \/><\/div>\n<p id=\"import-auto-id2677636\">2. Substitute the known values into the equation.<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id2677638\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b8cd75bb5ab442f20522df58bc77856d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#80;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#102;&#125;&#125;&#61;&#123;&#80;&#125;&#95;&#123;&#48;&#125;&#92;&#102;&#114;&#97;&#99;&#123;&#123;&#84;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#102;&#125;&#125;&#125;&#123;&#123;&#84;&#125;&#95;&#123;&#48;&#125;&#125;&#61;&#55;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#48;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#53;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#80;&#97;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#48;&#56;&#32;&#75;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#57;&#49;&#32;&#75;&#125;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#55;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#52;&#49;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#53;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#80;&#97;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"348\" style=\"vertical-align: -8px;\" \/><\/div>\n<p id=\"import-auto-id2919467\"><strong>Discussion<\/strong><\/p>\n<p id=\"import-auto-id2919469\">The final temperature is about 6% greater than the original temperature, so the final pressure is about 6% greater as well. Note that <em data-effect=\"italics\"><em data-effect=\"italics\">absolute<\/em><\/em> pressure and <em data-effect=\"italics\"><em data-effect=\"italics\">absolute<\/em><\/em> temperature must be used in the ideal gas law.<\/p>\n<\/div>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id1387947\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">Making Connections: Take-Home Experiment\u2014Refrigerating a Balloon<\/div>\n<p id=\"import-auto-id3224623\">Inflate a balloon at room temperature. Leave the inflated balloon in the refrigerator overnight. What happens to the balloon, and why? <\/p>\n<\/div>\n<div data-type=\"example\" class=\"textbox examples\" id=\"fs-id1843505\">\n<div data-type=\"title\" class=\"title\">Calculating the Number of Molecules in a Cubic Meter of Gas<\/div>\n<p id=\"import-auto-id2919471\">How many molecules are in a typical object, such as gas in a tire or water in a drink? We can use the ideal gas law to give us an idea of how large <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-5793832f979c2268e3694c246d53b1bb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#78;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"16\" style=\"vertical-align: 0px;\" \/> typically is.<\/p>\n<p id=\"import-auto-id2677390\">Calculate the number of molecules in a cubic meter of gas at standard temperature and pressure (STP), which is defined to be <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-7eb23a0f0414aab9de988c35ceebe163_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#92;&#116;&#101;&#120;&#116;&#123;&ordm;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"21\" style=\"vertical-align: 0px;\" \/> and atmospheric pressure.<\/p>\n<p id=\"import-auto-id2919757\"><strong>Strategy<\/strong><\/p>\n<p id=\"import-auto-id2717709\">Because pressure, volume, and temperature are all specified, we can use the ideal gas law <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-5d32b74a5df6abe7863a6f8bd9523f19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#80;&#86;&#125;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#78;&#107;&#84;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"84\" style=\"vertical-align: -1px;\" \/>, to find <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-5793832f979c2268e3694c246d53b1bb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#78;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"16\" style=\"vertical-align: 0px;\" \/>.<\/p>\n<p id=\"import-auto-id2409089\"><strong>Solution<\/strong><\/p>\n<p id=\"import-auto-id2409092\">1. Identify the knowns.<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id2409094\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-4c71ef29a3f6cbc17c65e55861bb3c0e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#108;&#108;&#108;&#125;&#84;&#38;&#32;&#61;&#38;&#32;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&ordm;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#125;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#55;&#51;&#32;&#75;&#125;&#92;&#92;&#32;&#80;&#38;&#32;&#61;&#38;&#32;&#49;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#49;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#53;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#80;&#97;&#125;&#92;&#92;&#32;&#86;&#38;&#32;&#61;&#38;&#32;&#49;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#48;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#125;&#125;&#94;&#123;&#51;&#125;&#92;&#92;&#32;&#107;&#38;&#32;&#61;&#38;&#32;&#49;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#56;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#51;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#74;&#47;&#75;&#125;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"83\" width=\"171\" style=\"vertical-align: -37px;\" \/><\/div>\n<p id=\"import-auto-id3224551\">2. Identify the unknown: number of molecules, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-5793832f979c2268e3694c246d53b1bb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#78;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"16\" style=\"vertical-align: 0px;\" \/>.<\/p>\n<p id=\"import-auto-id3218613\">3. Rearrange the ideal gas law to solve for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-5793832f979c2268e3694c246d53b1bb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#78;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"16\" style=\"vertical-align: 0px;\" \/>.<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id3224643\">\n<pre class=\"ql-errors\">*** QuickLaTeX cannot compile formula:\n&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#80;&#86;&#125;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#78;&#107;&#84;&#125;&#92;&#92;&#32;&#78;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#80;&#86;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#107;&#84;&#125;&#125;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\n\n*** Error message:\n&#77;&#105;&#115;&#115;&#105;&#110;&#103;&#32;&#35;&#32;&#105;&#110;&#115;&#101;&#114;&#116;&#101;&#100;&#32;&#105;&#110;&#32;&#97;&#108;&#105;&#103;&#110;&#109;&#101;&#110;&#116;&#32;&#112;&#114;&#101;&#97;&#109;&#98;&#108;&#101;&#46;\r\n&#108;&#101;&#97;&#100;&#105;&#110;&#103;&#32;&#116;&#101;&#120;&#116;&#58;&#32;&#36;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#125;\r\n&#77;&#105;&#115;&#115;&#105;&#110;&#103;&#32;&#36;&#32;&#105;&#110;&#115;&#101;&#114;&#116;&#101;&#100;&#46;\r\n&#108;&#101;&#97;&#100;&#105;&#110;&#103;&#32;&#116;&#101;&#120;&#116;&#58;&#32;&#46;&#46;&#46;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#78;&#107;&#84;&#125;&#92;&#92;&#32;&#78;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#80;&#86;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#107;&#84;&#125;&#125;\r\n&#69;&#120;&#116;&#114;&#97;&#32;&#125;&#44;&#32;&#111;&#114;&#32;&#102;&#111;&#114;&#103;&#111;&#116;&#116;&#101;&#110;&#32;&#36;&#46;\r\n&#108;&#101;&#97;&#100;&#105;&#110;&#103;&#32;&#116;&#101;&#120;&#116;&#58;&#32;&#46;&#46;&#46;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#78;&#107;&#84;&#125;&#92;&#92;&#32;&#78;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#80;&#86;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#107;&#84;&#125;&#125;\r\n&#77;&#105;&#115;&#115;&#105;&#110;&#103;&#32;&#125;&#32;&#105;&#110;&#115;&#101;&#114;&#116;&#101;&#100;&#46;\r\n&#108;&#101;&#97;&#100;&#105;&#110;&#103;&#32;&#116;&#101;&#120;&#116;&#58;&#32;&#46;&#46;&#46;&#92;&#92;&#32;&#78;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#80;&#86;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#107;&#84;&#125;&#125;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\r\n&#69;&#120;&#116;&#114;&#97;&#32;&#125;&#44;&#32;&#111;&#114;&#32;&#102;&#111;&#114;&#103;&#111;&#116;&#116;&#101;&#110;&#32;&#36;&#46;\r\n&#108;&#101;&#97;&#100;&#105;&#110;&#103;&#32;&#116;&#101;&#120;&#116;&#58;&#32;&#46;&#46;&#46;&#92;&#92;&#32;&#78;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#80;&#86;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#107;&#84;&#125;&#125;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\r\n&#77;&#105;&#115;&#115;&#105;&#110;&#103;&#32;&#125;&#32;&#105;&#110;&#115;&#101;&#114;&#116;&#101;&#100;&#46;\r\n&#108;&#101;&#97;&#100;&#105;&#110;&#103;&#32;&#116;&#101;&#120;&#116;&#58;&#32;&#46;&#46;&#46;&#92;&#92;&#32;&#78;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#80;&#86;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#107;&#84;&#125;&#125;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\r\n&#69;&#120;&#116;&#114;&#97;&#32;&#125;&#44;&#32;&#111;&#114;&#32;&#102;&#111;&#114;&#103;&#111;&#116;&#116;&#101;&#110;&#32;&#36;&#46;\r\n&#108;&#101;&#97;&#100;&#105;&#110;&#103;&#32;&#116;&#101;&#120;&#116;&#58;&#32;&#46;&#46;&#46;&#92;&#92;&#32;&#78;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#80;&#86;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#107;&#84;&#125;&#125;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\r\n&#77;&#105;&#115;&#115;&#105;&#110;&#103;&#32;&#125;&#32;&#105;&#110;&#115;&#101;&#114;&#116;&#101;&#100;&#46;\r\n&#108;&#101;&#97;&#100;&#105;&#110;&#103;&#32;&#116;&#101;&#120;&#116;&#58;&#32;&#46;&#46;&#46;&#92;&#92;&#32;&#78;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#80;&#86;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#107;&#84;&#125;&#125;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\r\n&#69;&#120;&#116;&#114;&#97;&#32;&#125;&#44;&#32;&#111;&#114;&#32;&#102;&#111;&#114;&#103;&#111;&#116;&#116;&#101;&#110;&#32;&#36;&#46;\r\n&#108;&#101;&#97;&#100;&#105;&#110;&#103;&#32;&#116;&#101;&#120;&#116;&#58;&#32;&#46;&#46;&#46;&#92;&#92;&#32;&#78;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#80;&#86;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#107;&#84;&#125;&#125;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\r\n&#77;&#105;&#115;&#115;&#105;&#110;&#103;&#32;&#125;&#32;&#105;&#110;&#115;&#101;&#114;&#116;&#101;&#100;&#46;\r\n\n<\/pre>\n<\/div>\n<p id=\"import-auto-id2743296\">4. Substitute the known values into the equation and solve for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-5793832f979c2268e3694c246d53b1bb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#78;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"16\" style=\"vertical-align: 0px;\" \/>. <\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id2887530\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-939f4fd7baff27b8d4388b455fb1f517_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#78;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#80;&#86;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#107;&#84;&#125;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#49;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#53;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#80;&#97;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#48;&#32;&#109;&#125;&#125;&#94;&#123;&#51;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#56;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#51;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#74;&#47;&#75;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#55;&#51;&#32;&#75;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#61;&#50;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#54;&#56;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#53;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#111;&#108;&#101;&#99;&#117;&#108;&#101;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"38\" width=\"407\" style=\"vertical-align: -14px;\" \/><\/div>\n<p id=\"import-auto-id2583233\"><strong>Discussion<\/strong><\/p>\n<p id=\"import-auto-id3224684\">This number is undeniably large, considering that a gas is mostly empty space. <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-5793832f979c2268e3694c246d53b1bb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#78;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"16\" style=\"vertical-align: 0px;\" \/> is huge, even in small volumes. For example, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-639f82da3032bf313bd575d7b8760b08_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#109;&#125;&#125;&#94;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"42\" style=\"vertical-align: -1px;\" \/> of a gas at STP has <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-6c8336eca4a3070ecebddfb19831e7c0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#54;&#56;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#57;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"63\" style=\"vertical-align: -1px;\" \/> molecules in it. Once again, note that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-5793832f979c2268e3694c246d53b1bb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#78;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"16\" style=\"vertical-align: 0px;\" \/> is the same for all types or mixtures of gases.<\/p>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id2669976\">\n<h1 data-type=\"title\">Moles and Avogadro\u2019s Number<\/h1>\n<p id=\"import-auto-id1725013\">It is sometimes convenient to work with a unit other than molecules when measuring the amount of substance. A <span data-type=\"term\" id=\"import-auto-id2677643\">mole<\/span> (abbreviated mol) is defined to be the amount of a substance that contains as many atoms or molecules as there are atoms in exactly 12 grams (0.012 kg) of carbon-12. The actual number of atoms or molecules in one mole is called <span data-type=\"term\" id=\"import-auto-id2677648\">Avogadro\u2019s number<\/span><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-4d616388c9e19120093e7d10c9b92ec2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#78;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#65;&#125;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"37\" style=\"vertical-align: -4px;\" \/>, in recognition of Italian scientist Amedeo Avogadro (1776\u20131856). He developed the concept of the mole, based on the hypothesis that equal volumes of gas, at the same pressure and temperature, contain equal numbers of molecules. That is, the number is independent of the type of gas. This hypothesis has been confirmed, and the value of Avogadro\u2019s number is <\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id2677659\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-1b5e42d8df78fbbfe0db8c78ebf53ef6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#78;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#65;&#125;&#125;&#61;&#54;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#50;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#51;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#111;&#108;&#125;&#125;&#94;&#123;&#45;&#49;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"168\" style=\"vertical-align: -4px;\" \/><\/div>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id2403668\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">Avogadro\u2019s Number<\/div>\n<p id=\"import-auto-id2936405\">One mole always contains <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f8e07161d3153aae11d34ed3fc36a16b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#50;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#51;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"63\" style=\"vertical-align: -1px;\" \/> particles (atoms or molecules), independent of the element or substance. A mole of any substance has a mass in grams equal to its molecular mass, which can be calculated from the atomic masses given in the periodic table of elements.<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id2919405\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-e12b0831b31eea5543c7afb00de95231_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#78;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#65;&#125;&#125;&#61;&#54;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#50;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#51;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#111;&#108;&#125;&#125;&#94;&#123;&#45;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"163\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/div>\n<div class=\"bc-figure figure\" id=\"import-auto-id2936416\">\n<div class=\"bc-figcaption figcaption\">How big is a mole? On a macroscopic level, one mole of table tennis balls would cover the Earth to a depth of about 40 km.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id2936418\" data-alt=\"The illustration shows relatively flat land with a solitary mountain, labeled Mt. Everest, and blue sky above. A double-headed vertical arrow stretches between the land and a point in the sky that is well above the peak of the mountain. The arrow, labeled table tennis balls, serves to indicate that a column of one mole of table tennis balls would reach a point in the sky that is much higher than the peak of Mt. Everest.\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/Figure_14_03_03.jpg\" data-media-type=\"image\/jpg\" alt=\"The illustration shows relatively flat land with a solitary mountain, labeled Mt. Everest, and blue sky above. A double-headed vertical arrow stretches between the land and a point in the sky that is well above the peak of the mountain. The arrow, labeled table tennis balls, serves to indicate that a column of one mole of table tennis balls would reach a point in the sky that is much higher than the peak of Mt. Everest.\" width=\"400\" \/><\/span><\/p>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2696745\" data-element-type=\"check-understanding\" data-label=\"\">\n<div data-type=\"title\">Check Your Understanding<\/div>\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1379188\">\n<p id=\"import-auto-id2763671\">The active ingredient in a Tylenol pill is 325 mg of acetaminophen <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-491aa33ae234eef8a962d09e52830a3c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#125;&#125;&#95;&#123;&#56;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#72;&#125;&#125;&#95;&#123;&#57;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#78;&#79;&#125;&#125;&#95;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"88\" style=\"vertical-align: -4px;\" \/>. Find the number of active molecules of acetaminophen in a single pill.<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1490941\">\n<p id=\"import-auto-id2763690\">We first need to calculate the molar mass (the mass of one mole) of acetaminophen. To do this, we need to multiply the number of atoms of each element by the element\u2019s atomic mass.<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id2919489\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-615d1009da975dd6b6f199f620beda98_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#56;&#32;&#109;&#111;&#108;&#101;&#115;&#32;&#111;&#102;&#32;&#99;&#97;&#114;&#98;&#111;&#110;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#50;&#32;&#103;&#114;&#97;&#109;&#115;&#47;&#109;&#111;&#108;&#101;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#57;&#32;&#109;&#111;&#108;&#101;&#115;&#32;&#104;&#121;&#100;&#114;&#111;&#103;&#101;&#110;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#32;&#103;&#114;&#97;&#109;&#47;&#109;&#111;&#108;&#101;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#92;&#32;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#32;&#109;&#111;&#108;&#101;&#32;&#110;&#105;&#116;&#114;&#111;&#103;&#101;&#110;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#52;&#32;&#103;&#114;&#97;&#109;&#115;&#47;&#109;&#111;&#108;&#101;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#43;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#32;&#109;&#111;&#108;&#101;&#115;&#32;&#111;&#120;&#121;&#103;&#101;&#110;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#54;&#32;&#103;&#114;&#97;&#109;&#115;&#47;&#109;&#111;&#108;&#101;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#53;&#49;&#32;&#103;&#125;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"40\" width=\"631\" style=\"vertical-align: -15px;\" \/><\/div>\n<p id=\"import-auto-id2763682\">Then we need to calculate the number of moles in 325 mg.<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id2386722\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-9d20f750618866be0fa95c884cf27512_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#50;&#53;&#32;&#109;&#103;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#53;&#49;&#32;&#103;&#114;&#97;&#109;&#115;&#47;&#109;&#111;&#108;&#101;&#125;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#102;&#114;&#97;&#99;&#123;&#49;&#32;&#103;&#114;&#97;&#109;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#48;&#48;&#32;&#109;&#103;&#125;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#50;&#46;&#49;&#53;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#51;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#111;&#108;&#101;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"33\" width=\"337\" style=\"vertical-align: -12px;\" \/><\/div>\n<p id=\"import-auto-id2386714\">Then use Avogadro\u2019s number to calculate the number of molecules.<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id2627386\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-214bd403ec955fcaf59669b8edab16b4_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#78;&#61;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#46;&#49;&#53;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#51;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#111;&#108;&#101;&#115;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#54;&#46;&#48;&#50;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#51;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#111;&#108;&#101;&#99;&#117;&#108;&#101;&#115;&#47;&#109;&#111;&#108;&#101;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#46;&#51;&#48;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#49;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#111;&#108;&#101;&#99;&#117;&#108;&#101;&#115;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"22\" width=\"545\" style=\"vertical-align: -7px;\" \/><\/div>\n<\/div>\n<\/div>\n<div data-type=\"example\" class=\"textbox examples\" id=\"fs-id2322275\">\n<div data-type=\"title\" class=\"title\">Calculating Moles per Cubic Meter and Liters per Mole<\/div>\n<p id=\"import-auto-id2870639\">Calculate: (a) the number of moles in <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-41734f66b0616f80a1e52449f84f43ee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#48;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#125;&#125;&#94;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"57\" style=\"vertical-align: -1px;\" \/> of gas at STP, and (b) the number of liters of gas per mole.<\/p>\n<p id=\"import-auto-id2870635\"><strong>Strategy and Solution<\/strong><\/p>\n<p id=\"import-auto-id2717734\">(a) We are asked to find the number of moles per cubic meter, and we know from <a href=\"#fs-id1843505\" class=\"autogenerated-content\">(Figure)<\/a> that the number of molecules per cubic meter at STP is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-1e6386caba7aa81b6f8debfe2a012cd8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#54;&#56;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#53;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"63\" style=\"vertical-align: -1px;\" \/>. The number of moles can be found by dividing the number of molecules by Avogadro\u2019s number. We let <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b170995d512c659d8668b4e42e1fef6b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\" \/> stand for the number of moles,<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id2564703\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-56a78d8ee4d53ab451463befae7e677d_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#111;&#108;&#47;&#109;&#125;&#125;&#94;&#123;&#51;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#78;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#111;&#108;&#101;&#99;&#117;&#108;&#101;&#115;&#47;&#109;&#125;&#125;&#94;&#123;&#51;&#125;&#125;&#123;&#54;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#50;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#51;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#111;&#108;&#101;&#99;&#117;&#108;&#101;&#115;&#47;&#109;&#111;&#108;&#125;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#50;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#54;&#56;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#53;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#111;&#108;&#101;&#99;&#117;&#108;&#101;&#115;&#47;&#109;&#125;&#125;&#94;&#123;&#51;&#125;&#125;&#123;&#54;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#50;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#51;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#111;&#108;&#101;&#99;&#117;&#108;&#101;&#115;&#47;&#109;&#111;&#108;&#125;&#125;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#52;&#52;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#53;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#111;&#108;&#47;&#109;&#125;&#125;&#94;&#123;&#51;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"30\" width=\"537\" style=\"vertical-align: -10px;\" \/><\/div>\n<p id=\"import-auto-id2595510\">(b) Using the value obtained for the number of moles in a cubic meter, and converting cubic meters to liters, we obtain<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id2919356\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-478f143abaf2850e99b4b52bc8896378_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#51;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#76;&#47;&#109;&#125;&#125;&#94;&#123;&#51;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#123;&#52;&#52;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#53;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#111;&#108;&#47;&#109;&#125;&#125;&#94;&#123;&#51;&#125;&#125;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#50;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#53;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#76;&#47;&#109;&#111;&#108;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"36\" width=\"190\" style=\"vertical-align: -12px;\" \/><\/div>\n<p id=\"import-auto-id2167136\"><strong>Discussion<\/strong><\/p>\n<p id=\"import-auto-id2883491\">This value is very close to the accepted value of 22.4 L\/mol. The slight difference is due to rounding errors caused by using three-digit input. Again this number is the same for all gases. In other words, it is independent of the gas.<\/p>\n<p id=\"import-auto-id2919351\">The (average) molar weight of air (approximately 80% <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ce901c2fc7f90b06af1a774799a4fa18_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#78;&#125;&#125;&#95;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"20\" style=\"vertical-align: -3px;\" \/> and 20% <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-0deb39be3c7640742283a1b426c549fd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#79;&#125;&#125;&#95;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"21\" style=\"vertical-align: -3px;\" \/> is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-717cf135d8edbcd28a2c4e5b6d455e01_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#77;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#56;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#56;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#103;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"92\" style=\"vertical-align: -3px;\" \/> Thus the mass of one cubic meter of air is 1.28 kg. If a living room has dimensions <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d2633632e36040a46b8469a90c0fde78_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#53;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#125;&times;&#92;&#116;&#101;&#120;&#116;&#123;&#53;&#32;&#109;&#125;&times;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#32;&#109;&#44;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"91\" style=\"vertical-align: -3px;\" \/> the mass of air inside the room is 96 kg, which is the typical mass of a human. <\/p>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2710286\" data-element-type=\"check-understanding\" data-label=\"\">\n<div data-type=\"title\">Check Your Understanding<\/div>\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2322677\">\n<p id=\"import-auto-id2017983\">The density of air at standard conditions <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-6d06f93ca65cde7158467c67a7a94099_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#80;&#61;&#49;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#116;&#109;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"88\" style=\"vertical-align: -4px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-85c906f6afcf9c08c6c5125f262d44fa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#84;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#48;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&ordm;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"66\" style=\"vertical-align: 0px;\" \/> is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-a582a754e1f52cc1fc9756855473731f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#56;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#107;&#103;&#47;&#109;&#125;&#125;&#94;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"84\" style=\"vertical-align: -4px;\" \/>. At what pressure is the density <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-1761bb6cd08ae568d662fbdfec71e344_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#54;&#52;&#32;&#107;&#103;&#47;&#109;&#125;&#125;&#94;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"86\" style=\"vertical-align: -4px;\" \/> if the temperature and number of molecules are kept constant?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1509649\">\n<p id=\"import-auto-id2183997\">The best way to approach this question is to think about what is happening. If the density drops to half its original value and no molecules are lost, then the volume must double. If we look at the equation <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-5d32b74a5df6abe7863a6f8bd9523f19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#80;&#86;&#125;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#78;&#107;&#84;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"84\" style=\"vertical-align: -1px;\" \/>, we see that when the temperature is constant, the pressure is inversely proportional to volume. Therefore, if the volume doubles, the pressure must drop to half its original value, and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-5d759f6bf0296667f699dfa7213f3532_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#80;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#102;&#125;&#125;&#61;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#53;&#48;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#116;&#109;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"111\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id2248571\">\n<h1 data-type=\"title\">The Ideal Gas Law Restated Using Moles<\/h1>\n<p id=\"import-auto-id2797949\">A very common expression of the ideal gas law uses the number of moles, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b170995d512c659d8668b4e42e1fef6b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\" \/>, rather than the number of atoms and molecules, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-5793832f979c2268e3694c246d53b1bb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#78;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"16\" style=\"vertical-align: 0px;\" \/>. We start from the ideal gas law,<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id2689659\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-8441e7f70b31985c36fddaca33fd14f6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#80;&#86;&#125;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#78;&#107;&#84;&#44;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"88\" style=\"vertical-align: -3px;\" \/><\/div>\n<p id=\"import-auto-id2889310\">and multiply and divide the equation by Avogadro\u2019s number <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-8c5d5641fa23b0f683a1641759a247aa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#78;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#65;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"24\" style=\"vertical-align: -4px;\" \/>. This gives<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id2889307\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-5c4374d72580462cbc7b535a43f1b542_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#80;&#86;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#78;&#125;&#123;&#123;&#78;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#65;&#125;&#125;&#125;&#123;&#78;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#65;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#107;&#84;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"25\" width=\"125\" style=\"vertical-align: -9px;\" \/><\/div>\n<p id=\"import-auto-id2084701\">Note that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-1d9c86cd91f47a285d5a73195c2e60c6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;&#61;&#78;&#47;&#123;&#78;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#65;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"81\" style=\"vertical-align: -5px;\" \/> is the number of moles. We define the universal gas constant <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-010e773bf698562a3b7141233995f7b8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;&#61;&#123;&#78;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#65;&#125;&#125;&#107;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"72\" style=\"vertical-align: -4px;\" \/>, and obtain the ideal gas law in terms of moles.<\/p>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id2798942\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">Ideal Gas Law (in terms of moles)<\/div>\n<p id=\"import-auto-id2386604\">The ideal gas law (in terms of moles) is<\/p>\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-369f3b062ff5c26a7eee188baae2ec25_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#80;&#86;&#125;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#110;&#82;&#84;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"87\" style=\"vertical-align: -1px;\" \/><\/div>\n<p id=\"import-auto-id2736663\">The numerical value of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-dae6bae3dcdac4629730754352c5e329_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/> in SI units is<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id3226853\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-01f5426fafef2cb68107d21bb1eecbe8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;&#61;&#123;&#78;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#65;&#125;&#125;&#107;&#61;&#92;&#108;&#101;&#102;&#116;&#40;&#54;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#50;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#51;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#111;&#108;&#125;&#125;&#94;&#123;&#45;&#49;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#56;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#51;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#74;&#47;&#75;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#56;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#49;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#74;&#125;&#47;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#111;&#108;&#125;&#92;&#99;&#100;&#111;&#116;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#75;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"23\" width=\"494\" style=\"vertical-align: -7px;\" \/><\/div>\n<p id=\"import-auto-id2919773\">In other units, <\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id2919790\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-29e437820f8cddb3cc1b46e750b72ba8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#108;&#108;&#108;&#125;&#82;&#38;&#32;&#61;&#38;&#32;&#49;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#57;&#57;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#97;&#108;&#47;&#109;&#111;&#108;&#125;&#92;&#99;&#100;&#111;&#116;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#75;&#125;&#92;&#92;&#32;&#82;&#38;&#32;&#61;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#56;&#50;&#49;&#32;&#76;&#125;&#92;&#99;&#100;&#111;&#116;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#116;&#109;&#47;&#109;&#111;&#108;&#125;&#92;&#99;&#100;&#111;&#116;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#75;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"40\" width=\"237\" style=\"vertical-align: -15px;\" \/><\/div>\n<p id=\"import-auto-id3227661\">You can use whichever value of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-dae6bae3dcdac4629730754352c5e329_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/> is most convenient for a particular problem. <\/p>\n<\/div>\n<div data-type=\"example\" class=\"textbox examples\" id=\"fs-id1444855\">\n<div data-type=\"title\" class=\"title\">Calculating Number of Moles: Gas in a Bike Tire<\/div>\n<p id=\"import-auto-id3224716\">How many moles of gas are in a bike tire with a volume of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f4a5e80d2c7bc9fec8886ae2173511d5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#48;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#51;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#125;&#125;&#94;&#123;&#51;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#48;&#32;&#76;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"167\" style=\"vertical-align: -4px;\" \/> a pressure of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-3607f165288a4edbdc16988644623465_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#48;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#53;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#80;&#97;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"82\" style=\"vertical-align: -1px;\" \/> (a gauge pressure of just under <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-282961fcce3340d63ebd8fe6d131dc30_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#57;&#48;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#48;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#98;&#47;&#105;&#110;&#125;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"81\" style=\"vertical-align: -4px;\" \/>), and at a temperature of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c272f96dee470023e23bc8f50037fede_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#56;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&ordm;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"43\" style=\"vertical-align: -1px;\" \/>?<\/p>\n<p id=\"import-auto-id2600714\"><strong>Strategy<\/strong><\/p>\n<p id=\"import-auto-id2396040\">Identify the knowns and unknowns, and choose an equation to solve for the unknown. In this case, we solve the ideal gas law, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-adab35e44758bbef6ddabc8ad1297fdd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#80;&#86;&#125;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#110;&#82;&#84;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"83\" style=\"vertical-align: -1px;\" \/>, for the number of moles <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b170995d512c659d8668b4e42e1fef6b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\" \/>. <\/p>\n<p id=\"import-auto-id2124325\"><strong>Solution<\/strong><\/p>\n<p id=\"import-auto-id2124328\">1. Identify the knowns.<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id2124330\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-1e43777f8fb9d9534d0eb13f964eee48_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#108;&#108;&#108;&#125;&#80;&#38;&#32;&#61;&#38;&#32;&#55;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#48;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#53;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#80;&#97;&#125;&#92;&#92;&#32;&#86;&#38;&#32;&#61;&#38;&#32;&#50;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#48;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#51;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#125;&#125;&#94;&#123;&#51;&#125;&#92;&#92;&#32;&#84;&#38;&#32;&#61;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#56;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&ordm;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#125;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#57;&#49;&#32;&#75;&#125;&#92;&#92;&#32;&#82;&#38;&#32;&#61;&#38;&#32;&#56;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#49;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#74;&#47;&#109;&#111;&#108;&#125;&#92;&#99;&#100;&#111;&#116;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#75;&#125;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"85\" width=\"175\" style=\"vertical-align: -37px;\" \/><\/div>\n<p id=\"import-auto-id2402615\">2. Rearrange the equation to solve for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b170995d512c659d8668b4e42e1fef6b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\" \/> and substitute known values.<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id2402631\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-8a3ebc6d5f4ea4ea479af20431141f59_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#108;&#108;&#108;&#125;&#110;&#38;&#32;&#61;&#38;&#32;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#80;&#86;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#82;&#84;&#125;&#125;&#61;&#92;&#102;&#114;&#97;&#99;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#55;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#48;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#53;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#80;&#97;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#50;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#48;&#48;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#51;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#125;&#125;&#94;&#123;&#51;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#123;&#92;&#108;&#101;&#102;&#116;&#40;&#56;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#49;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#74;&#47;&#109;&#111;&#108;&#125;&#92;&#99;&#100;&#111;&#116;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#75;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#57;&#49;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#75;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#125;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#125;&#38;&#32;&#61;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#53;&#55;&#57;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#111;&#108;&#125;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"49\" width=\"276\" style=\"vertical-align: -16px;\" \/><\/div>\n<p id=\"import-auto-id2862630\"><strong>Discussion<\/strong><\/p>\n<p id=\"import-auto-id2862614\">The most convenient choice for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-dae6bae3dcdac4629730754352c5e329_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/> in this case is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-9dd9e3408f9ad894008906986bbf4624_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#49;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#74;&#47;&#109;&#111;&#108;&#125;&#92;&#99;&#100;&#111;&#116;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#75;&#44;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"113\" style=\"vertical-align: -4px;\" \/> because our known quantities are in SI units. The pressure and temperature are obtained from the initial conditions in <a href=\"#fs-id1667893\" class=\"autogenerated-content\">(Figure)<\/a>, but we would get the same answer if we used the final values.<\/p>\n<\/div>\n<p id=\"import-auto-id2862600\">The ideal gas law can be considered to be another manifestation of the law of conservation of energy (see <a href=\"\/contents\/c5adb012-fbab-41a7-b5b2-4644984097fc@5\">Conservation of Energy<\/a>). Work done on a gas results in an increase in its energy, increasing pressure and\/or temperature, or decreasing volume. This increased energy can also be viewed as increased internal kinetic energy, given the gas\u2019s atoms and molecules.<\/p>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1542832\">\n<h1 data-type=\"title\">The Ideal Gas Law and Energy<\/h1>\n<p id=\"import-auto-id2717771\">Let us now examine the role of energy in the behavior of gases. When you inflate a bike tire by hand, you do work by repeatedly exerting a force through a distance. This energy goes into increasing the pressure of air inside the tire and increasing the temperature of the pump and the air.<\/p>\n<p id=\"import-auto-id2388350\">The ideal gas law is closely related to energy: the units on both sides are joules. The right-hand side of the ideal gas law in <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-5d32b74a5df6abe7863a6f8bd9523f19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#80;&#86;&#125;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#78;&#107;&#84;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"84\" style=\"vertical-align: -1px;\" \/> is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-8eaa29a6e9b2fa2ff4001dd1ba5cd3fc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#78;&#107;&#84;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"35\" style=\"vertical-align: -1px;\" \/>. This term is roughly the amount of translational kinetic energy of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-5793832f979c2268e3694c246d53b1bb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#78;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"16\" style=\"vertical-align: 0px;\" \/> atoms or molecules at an absolute temperature <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f9ed275b0bf1633b7ee83b78fcc28273_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#84;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"13\" style=\"vertical-align: 0px;\" \/>, as we shall see formally in <a href=\"\/contents\/3fffbab5-5b0d-4622-be83-eb65111f9c3e@5\">Kinetic Theory: Atomic and Molecular Explanation of Pressure and Temperature<\/a>. The left-hand side of the ideal gas law is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-bdc1ef9bc8ee074a1147649d94f3db01_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#80;&#86;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"25\" style=\"vertical-align: -1px;\" \/>, which also has the units of joules. We know from our study of fluids that pressure is one type of potential energy per unit volume, so pressure multiplied by volume is energy. The important point is that there is energy in a gas related to both its pressure and its volume. The energy can be changed when the gas is doing work as it expands\u2014something we explore in <a href=\"\/contents\/7892cddd-833e-42b8-bcaa-a1e94fb9eb9c@3\">Heat and Heat Transfer Methods<\/a>\u2014similar to what occurs in gasoline or steam engines and turbines.<\/p>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id2254366\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">Problem-Solving Strategy: The Ideal Gas Law<\/div>\n<p id=\"import-auto-id2564676\"><strong>Step 1<\/strong> Examine the situation to determine that an ideal gas is involved. Most gases are nearly ideal.<\/p>\n<p id=\"import-auto-id1747449\"><strong>Step 2<\/strong> Make a list of what quantities are given, or can be inferred from the problem as stated (identify the known quantities). Convert known values into proper SI units (K for temperature, Pa for pressure, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-e3b87847f2822cabdde235117b9f5dc8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#125;&#125;&#94;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"22\" style=\"vertical-align: 0px;\" \/> for volume, molecules for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-5793832f979c2268e3694c246d53b1bb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#78;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"16\" style=\"vertical-align: 0px;\" \/>, and moles for <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b170995d512c659d8668b4e42e1fef6b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\" \/>).<\/p>\n<p id=\"import-auto-id2731891\"><strong>Step 3<\/strong> Identify exactly what needs to be determined in the problem (identify the unknown quantities). A written list is useful.<\/p>\n<p id=\"import-auto-id3227648\"><strong>Step 4<\/strong> Determine whether the number of molecules or the number of moles is known, in order to decide which form of the ideal gas law to use. The first form is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-5d32b74a5df6abe7863a6f8bd9523f19_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#80;&#86;&#125;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#78;&#107;&#84;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"84\" style=\"vertical-align: -1px;\" \/> and involves <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-5793832f979c2268e3694c246d53b1bb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#78;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"16\" style=\"vertical-align: 0px;\" \/>, the number of atoms or molecules. The second form is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-adab35e44758bbef6ddabc8ad1297fdd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#80;&#86;&#125;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#110;&#82;&#84;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"83\" style=\"vertical-align: -1px;\" \/> and involves <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b170995d512c659d8668b4e42e1fef6b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\" \/>, the number of moles.<\/p>\n<p><strong>Step 5<\/strong> Solve the ideal gas law for the quantity to be determined (the unknown quantity). You may need to take a ratio of final states to initial states to eliminate the unknown quantities that are kept fixed.<\/p>\n<p id=\"import-auto-id2688944\"><strong>Step 6<\/strong> Substitute the known quantities, along with their units, into the appropriate equation, and obtain numerical solutions complete with units. Be certain to use absolute temperature and absolute pressure.<\/p>\n<p id=\"import-auto-id2688950\"><strong>Step 7<\/strong> Check the answer to see if it is reasonable: Does it make sense?<\/p>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1445894\" data-element-type=\"check-understanding\" data-label=\"\">\n<div data-type=\"title\">Check Your Understanding<\/div>\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1445900\">\n<p id=\"import-auto-id2688959\">Liquids and solids have densities about 1000 times greater than gases. Explain how this implies that the distances between atoms and molecules in gases are about 10 times greater than the size of their atoms and molecules. <\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1677733\">\n<p id=\"import-auto-id2688965\">Atoms and molecules are close together in solids and liquids. In gases they are separated by empty space. Thus gases have lower densities than liquids and solids. Density is mass per unit volume, and volume is related to the size of a body (such as a sphere) cubed. So if the distance between atoms and molecules increases by a factor of 10, then the volume occupied increases by a factor of 1000, and the density decreases by a factor of 1000.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"section-summary\" data-depth=\"1\" id=\"fs-id1698840\">\n<h1 data-type=\"title\">Section Summary<\/h1>\n<ul id=\"fs-id2398625\">\n<li id=\"import-auto-id739591\">The ideal gas law relates the pressure and volume of a gas to the number of gas molecules and the temperature of the gas.<\/li>\n<li id=\"import-auto-id739594\">The ideal gas law can be written in terms of the number of molecules of gas:\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id739597\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-589efb3659e66211d221a525587ee725_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#80;&#86;&#125;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#78;&#107;&#84;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"88\" style=\"vertical-align: -4px;\" \/><\/div>\n<p>where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-650eb7688af6737ac325425b5c9a5982_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#80;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/> is pressure, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-63ada879859a9e41fd935f035b7313bc_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#86;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/> is volume, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f9ed275b0bf1633b7ee83b78fcc28273_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#84;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"13\" style=\"vertical-align: 0px;\" \/> is temperature, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-5793832f979c2268e3694c246d53b1bb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#78;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"16\" style=\"vertical-align: 0px;\" \/> is number of molecules, and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-3422b6bb5c160593658b7c39425d9880_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#107;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"9\" style=\"vertical-align: 0px;\" \/> is the Boltzmann constant<\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id2564573\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b724ab40a5f93f0fa361148cd2570f07_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#107;&#61;&#49;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#56;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#51;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#74;&#47;&#75;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"148\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/li>\n<li id=\"import-auto-id2402831\">A mole is the number of atoms in a 12-g sample of carbon-12. <\/li>\n<li id=\"import-auto-id2402833\">The number of molecules in a mole is called Avogadro\u2019s number <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-8c5d5641fa23b0f683a1641759a247aa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#78;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#65;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"24\" style=\"vertical-align: -4px;\" \/>,\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id2402820\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-adfb42b92dbe90654406ff15b8cdb108_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#78;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#65;&#125;&#125;&#61;&#54;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#50;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#51;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#111;&#108;&#125;&#125;&#94;&#123;&#45;&#49;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"168\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/li>\n<li id=\"import-auto-id2599009\">A mole of any substance has a mass in grams equal to its molecular weight, which can be determined from the periodic table of elements.<\/li>\n<li id=\"import-auto-id2599036\">The ideal gas law can also be written and solved in terms of the number of moles of gas:\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id2599038\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c8292916e0e8ca73795580aad2ec5547_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#80;&#86;&#125;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#110;&#82;&#84;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"87\" style=\"vertical-align: -4px;\" \/><\/div>\n<p>where <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b170995d512c659d8668b4e42e1fef6b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#110;\" title=\"Rendered by QuickLaTeX.com\" height=\"8\" width=\"11\" style=\"vertical-align: 0px;\" \/> is number of moles and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-dae6bae3dcdac4629730754352c5e329_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/> is the universal gas constant, <\/p>\n<div data-type=\"equation\" class=\"equation\" id=\"import-auto-id2862692\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d660260381d09245280cd068287f0ecb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;&#61;&#56;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#49;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#74;&#47;&#109;&#111;&#108;&#125;&#92;&#99;&#100;&#111;&#116;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#75;&#125;&#46;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"150\" style=\"vertical-align: -4px;\" \/><\/div>\n<\/li>\n<li id=\"import-auto-id2883325\">The ideal gas law is generally valid at temperatures well above the boiling temperature.<\/li>\n<\/ul>\n<\/div>\n<div class=\"conceptual-questions\" data-depth=\"1\" id=\"fs-id2719827\" data-element-type=\"conceptual-questions\">\n<h1 data-type=\"title\">Conceptual Questions<\/h1>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2591367\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2591368\">\n<p id=\"import-auto-id2883312\">Find out the human population of Earth. Is there a mole of people inhabiting Earth? If the average mass of a person is 60 kg, calculate the mass of a mole of people. How does the mass of a mole of people compare with the mass of Earth?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2799079\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2799080\">\n<p id=\"import-auto-id2236451\">Under what circumstances would you expect a gas to behave significantly differently than predicted by the ideal gas law?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2378151\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2378152\">\n<p id=\"import-auto-id2236458\">A constant-volume gas thermometer contains a fixed amount of gas. What property of the gas is measured to indicate its temperature?<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"problems-exercises\" data-depth=\"1\" id=\"fs-id2682273\" data-element-type=\"problems-exercises\">\n<h1 data-type=\"title\">Problems &amp; Exercises<\/h1>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2682279\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2682280\">\n<p id=\"import-auto-id2236470\">The gauge pressure in your car tires is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-96b6931ee6d07929ded4f5614dd94367_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#53;&#48;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#53;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#78;&#47;&#109;&#125;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"105\" style=\"vertical-align: -4px;\" \/> at a temperature of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c143655cb1029e983b8af3ef07ba377f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#53;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&ordm;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"44\" style=\"vertical-align: 0px;\" \/> when you drive it onto a ferry boat to Alaska. What is their gauge pressure later, when their temperature has dropped to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-eebc21f5bc9da9da7c47a9bf94f17cfa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#116;&#101;&#120;&#116;&#123;&#52;&#48;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&ordm;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"56\" style=\"vertical-align: -1px;\" \/>?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id2684234\">\n<p id=\"import-auto-id2677607\">1.62 atm<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2798567\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2798568\">\n<p id=\"import-auto-id2677616\">Convert an absolute pressure of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-1808177dbdbcde3c7c036ac0959285c2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#48;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#53;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#78;&#47;&#109;&#125;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"105\" style=\"vertical-align: -4px;\" \/> to gauge pressure in <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-5f075a995fe83edc1e810546e3a1d75c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#108;&#98;&#47;&#105;&#110;&#125;&#125;&#94;&#123;&#50;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"50\" style=\"vertical-align: -4px;\" \/> (This value was stated to be just less than <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-6bc425e00535b0a1c3cfaafce12da6e2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#57;&#48;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#32;&#108;&#98;&#47;&#105;&#110;&#125;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"83\" style=\"vertical-align: -4px;\" \/> in <a href=\"#fs-id1444855\" class=\"autogenerated-content\">(Figure)<\/a>. Is it?) <\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2804147\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2804148\">\n<p id=\"import-auto-id2681625\">Suppose a gas-filled incandescent light bulb is manufactured so that the gas inside the bulb is at atmospheric pressure when the bulb has a temperature of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f205d73e3e16a867ce37b630c0c5cbaa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#48;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&ordm;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"44\" style=\"vertical-align: 0px;\" \/>. (a) Find the gauge pressure inside such a bulb when it is hot, assuming its average temperature is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f6abe5329592991c890609dc8940025a_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#54;&#48;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&ordm;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"44\" style=\"vertical-align: 0px;\" \/> (an approximation) and neglecting any change in volume due to thermal expansion or gas leaks. (b) The actual final pressure for the light bulb will be less than calculated in part (a) because the glass bulb will expand. What will the actual final pressure be, taking this into account? Is this a negligible difference?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id2736663\">\n<p id=\"import-auto-id2386001\">(a) 0.136 atm <\/p>\n<p id=\"import-auto-id2386017\">(b) 0.135 atm. The difference between this value and the value from part (a) is negligible.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2687223\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2687224\">\n<p id=\"import-auto-id2386026\">Large helium-filled balloons are used to lift scientific equipment to high altitudes. (a) What is the pressure inside such a balloon if it starts out at sea level with a temperature of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-11b3e9b994189112ed7be4890b58b10b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&ordm;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"43\" style=\"vertical-align: -1px;\" \/> and rises to an altitude where its volume is twenty times the original volume and its temperature is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-09a6f077ef71c1c739ba60e8464ef022_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#116;&#101;&#120;&#116;&#123;&#53;&#48;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&ordm;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"56\" style=\"vertical-align: 0px;\" \/>? (b) What is the gauge pressure? (Assume atmospheric pressure is constant.) <\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1746230\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1746231\">\n<p id=\"import-auto-id3224535\">Confirm that the units of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-a21130fc0cbf807fb8abd75cd5c4a135_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#110;&#82;&#84;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"34\" style=\"vertical-align: -1px;\" \/> are those of energy for each value of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-dae6bae3dcdac4629730754352c5e329_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#82;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"14\" style=\"vertical-align: 0px;\" \/>: (a) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-7f071d3e6a2a1b849aab32767ee4c1aa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#49;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#74;&#47;&#109;&#111;&#108;&#125;&#92;&#99;&#100;&#111;&#116;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#75;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"109\" style=\"vertical-align: -4px;\" \/>, (b) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-614971fb81799a472cd0d6047f2d0df9_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#57;&#57;&#32;&#99;&#97;&#108;&#47;&#109;&#111;&#108;&#125;&#92;&#99;&#100;&#111;&#116;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#75;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"122\" style=\"vertical-align: -4px;\" \/>, and (c) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d419285e357ac69c7c405a501a0e95d6_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#48;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#56;&#50;&#49;&#32;&#76;&#125;&#92;&#99;&#100;&#111;&#116;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#116;&#109;&#47;&#109;&#111;&#108;&#125;&#92;&#99;&#100;&#111;&#116;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#75;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"174\" style=\"vertical-align: -4px;\" \/>.<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id2404519\">\n<p id=\"import-auto-id2926437\">(a) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-53e3cb32f335b7bd710c28f64a28ef23_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#110;&#82;&#84;&#125;&#61;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#111;&#108;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#74;&#47;&#109;&#111;&#108;&#125;&#92;&#99;&#100;&#111;&#116;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#75;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#75;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#74;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"253\" style=\"vertical-align: -4px;\" \/><\/p>\n<p id=\"import-auto-id1563389\">(b) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-cc6fbc081e53cacba34991c834b53d67_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#110;&#82;&#84;&#125;&#61;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#111;&#108;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#97;&#108;&#47;&#109;&#111;&#108;&#125;&#92;&#99;&#100;&#111;&#116;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#75;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#75;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#97;&#108;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"279\" style=\"vertical-align: -4px;\" \/><\/p>\n<p id=\"import-auto-id2409504\">(c) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2022beeb1f3a34e22add3cf716375192_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#108;&#108;&#108;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#110;&#82;&#84;&#125;&#38;&#32;&#61;&#38;&#32;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#111;&#108;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#76;&#125;&#92;&#99;&#100;&#111;&#116;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#116;&#109;&#47;&#109;&#111;&#108;&#125;&#92;&#99;&#100;&#111;&#116;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#75;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#92;&#116;&#101;&#120;&#116;&#123;&#75;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#92;&#32;&#38;&#32;&#61;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#76;&#125;&#92;&#99;&#100;&#111;&#116;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#116;&#109;&#125;&#61;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#125;&#125;&#94;&#123;&#51;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#78;&#47;&#109;&#125;&#125;&#94;&#123;&#50;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#92;&#32;&#38;&#32;&#61;&#38;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#78;&#125;&#92;&#99;&#100;&#111;&#116;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#125;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#74;&#125;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"68\" width=\"288\" style=\"vertical-align: -27px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2094656\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2094657\">\n<p id=\"import-auto-id2637302\">In the text, it was shown that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-a4ad755b8fc0a78ffe3e55c00a3c9b25_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#78;&#47;&#86;&#61;&#50;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#54;&#56;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#53;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#125;&#125;&#94;&#123;&#45;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"163\" style=\"vertical-align: -5px;\" \/>  for gas at STP. (a) Show that this quantity is equivalent to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-12b8c8c217f46ab1a3fedaa8c2a30668_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#78;&#47;&#86;&#61;&#50;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#54;&#56;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#57;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#109;&#125;&#125;&#94;&#123;&#45;&#51;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"20\" width=\"175\" style=\"vertical-align: -5px;\" \/>  as stated. (b) About how many atoms are there in one <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-a9710cbd0b182c4b6e48565afb12719c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&mu;&#109;&#125;&#125;&#94;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"22\" style=\"vertical-align: 0px;\" \/> (a cubic micrometer) at STP? (c) What does your answer to part (b) imply about the separation of atoms and molecules? <\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2631403\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2631404\">\n<p id=\"import-auto-id2883481\">Calculate the number of moles in the 2.00-L volume of air in the lungs of the average person. Note that the air is at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-2f0ca0115a551bff9b106b13d0f34072_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#55;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&ordm;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"44\" style=\"vertical-align: 0px;\" \/> (body temperature).<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id2683493\">\n<p id=\"eip-id1318898\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-aa56fa061ddadeacfa5cc9c8e359a7ee_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#56;&#54;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#50;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#111;&#108;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"101\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2932069\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2932070\">\n<p id=\"import-auto-id3184957\">An airplane passenger has <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c64b89210c306062fe0a835395988fcb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#48;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#109;&#125;&#125;&#94;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"60\" style=\"vertical-align: -1px;\" \/> of air in his stomach just before the plane takes off from a sea-level airport. What volume will the air have at cruising altitude if cabin pressure drops to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-0515769e28c99be115a00b758d3c9a60_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#53;&#48;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#52;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#78;&#47;&#109;&#125;&#125;&#94;&#123;&#50;&#125;&#63;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"114\" style=\"vertical-align: -4px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2739655\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2739656\">\n<p id=\"import-auto-id2631677\">(a) What is the volume (in <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-234186468624fa4901d362177116eb7b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#107;&#109;&#125;&#125;&#94;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"31\" style=\"vertical-align: 0px;\" \/>) of Avogadro\u2019s number of sand grains if each grain is a cube and has sides that are 1.0 mm long? (b) How many kilometers of beaches in length would this cover if the beach averages 100 m in width and 10.0 m in depth? Neglect air spaces between grains.<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id2734849\">\n<p id=\"import-auto-id2393401\">(a) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b151d72cfb76f330185a1a8a2ecf7071_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#50;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#53;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#107;&#109;&#125;&#125;&#94;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"17\" width=\"92\" style=\"vertical-align: -1px;\" \/><\/p>\n<p id=\"import-auto-id2393424\">(b) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-04b6dbe81ce6122680551ea87f97e38e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#50;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#56;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#107;&#109;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"85\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1908858\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1908859\">\n<p id=\"import-auto-id2597680\">An expensive vacuum system can achieve a pressure as low as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-000cd41eaa5804cdc04ede0e4a005333_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#48;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#55;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#78;&#47;&#109;&#125;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"115\" style=\"vertical-align: -4px;\" \/> at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-a94a5528991784dff99f941113de9fe2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#48;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&ordm;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"30\" style=\"vertical-align: 0px;\" \/>. How many atoms are there in a cubic centimeter at this pressure and temperature? <\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2328150\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2328151\">\n<p id=\"import-auto-id3227463\">The number density of gas atoms at a certain location in the space above our planet is about <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-34fe85a07fa6718d81dee85c235598e5_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#48;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#49;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#125;&#125;&#94;&#123;&#45;&#51;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"104\" style=\"vertical-align: -4px;\" \/> and the pressure is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-95a40652d957ae8510b4f1ab89bf580f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#50;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#55;&#53;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#78;&#47;&#109;&#125;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"123\" style=\"vertical-align: -4px;\" \/> in this space. What is the temperature there?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id2631515\">\n<p id=\"eip-id3084073\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c8daf321a76946c62b13abf29bc84187_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#45;&#92;&#116;&#101;&#120;&#116;&#123;&#55;&#51;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#57;&#92;&#116;&#101;&#120;&#116;&#123;&ordm;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"56\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1804845\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1804846\">\n<p id=\"import-auto-id2688601\">A bicycle tire has a pressure of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-1808177dbdbcde3c7c036ac0959285c2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#48;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#53;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#78;&#47;&#109;&#125;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"105\" style=\"vertical-align: -4px;\" \/> at a temperature of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c272f96dee470023e23bc8f50037fede_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#56;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&ordm;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"43\" style=\"vertical-align: -1px;\" \/> and contains 2.00 L of gas. What will its pressure be if you let out an amount of air that has a volume of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c64b89210c306062fe0a835395988fcb_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#48;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#99;&#109;&#125;&#125;&#94;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"60\" style=\"vertical-align: -1px;\" \/> at atmospheric pressure? Assume tire temperature and volume remain constant. <\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2889733\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2409195\">\n<p id=\"import-auto-id2698879\">A high-pressure gas cylinder contains 50.0 L of toxic gas at a pressure of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-0c141d36557e4b7b36ddccf490fd170b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#49;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#52;&#48;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#55;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#78;&#47;&#109;&#125;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"104\" style=\"vertical-align: -4px;\" \/> and a temperature of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-cc820ae9a4e31f54215900ec9043cec0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#53;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&ordm;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"44\" style=\"vertical-align: 0px;\" \/>. Its valve leaks after the cylinder is dropped. The cylinder is cooled to dry ice temperature <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-1a0d073f12c7812c5323165cf66d44ef_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#108;&#101;&#102;&#116;&#40;&#45;&#92;&#116;&#101;&#120;&#116;&#123;&#55;&#56;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#53;&#92;&#116;&#101;&#120;&#116;&#123;&ordm;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"18\" width=\"70\" style=\"vertical-align: -4px;\" \/> to reduce the leak rate and pressure so that it can be safely repaired. (a) What is the final pressure in the tank, assuming a negligible amount of gas leaks while being cooled and that there is no phase change? (b) What is the final pressure if one-tenth of the gas escapes? (c) To what temperature must the tank be cooled to reduce the pressure to 1.00 atm (assuming the gas does not change phase and that there is no leakage during cooling)? (d) Does cooling the tank appear to be a practical solution? <\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id1695248\">\n<p id=\"import-auto-id2887240\">(a) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-87f07869ef8523e1df695ad4dfbe362c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#57;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#52;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#54;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#78;&#47;&#109;&#125;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"105\" style=\"vertical-align: -4px;\" \/><\/p>\n<p id=\"import-auto-id2385013\">(b) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-b2c7ada67f1a0471013e99d5bb0eba51_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#51;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#54;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#78;&#47;&#109;&#125;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"105\" style=\"vertical-align: -4px;\" \/><\/p>\n<p id=\"import-auto-id2692716\">(c) 2.16 K<\/p>\n<p id=\"import-auto-id2385055\">(d) No. The final temperature needed is much too low to be easily achieved for a large object.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2084983\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2084984\">\n<p id=\"import-auto-id2692718\">Find the number of moles in 2.00 L of gas at <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c143655cb1029e983b8af3ef07ba377f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#51;&#53;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&ordm;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"44\" style=\"vertical-align: 0px;\" \/> and under <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-29480217d4bbfb136eb519e89ae2ad06_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#55;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#52;&#49;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#55;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#78;&#47;&#109;&#125;&#125;&#94;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"105\" style=\"vertical-align: -4px;\" \/> of pressure. <\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2735156\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2735157\">\n<p id=\"import-auto-id2681492\">Calculate the depth to which Avogadro\u2019s number of table tennis balls would cover Earth. Each ball has a diameter of 3.75 cm. Assume the space between balls adds an extra 25.0% to their volume and assume they are not crushed by their own weight.<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id2735166\">\n<p id=\"import-auto-id2681473\">41 km<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1544154\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1544155\">\n<p id=\"import-auto-id2681482\">(a) What is the gauge pressure in a <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-cc820ae9a4e31f54215900ec9043cec0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#53;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&ordm;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"44\" style=\"vertical-align: 0px;\" \/> car tire containing 3.60 mol of gas in a 30.0 L volume? (b) What will its gauge pressure be if you add 1.00 L of gas originally at atmospheric pressure and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-cc820ae9a4e31f54215900ec9043cec0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#53;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&ordm;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"44\" style=\"vertical-align: 0px;\" \/>? Assume the temperature returns to <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-cc820ae9a4e31f54215900ec9043cec0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#53;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&ordm;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#67;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"44\" style=\"vertical-align: 0px;\" \/> and the volume remains constant. <\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2869966\" data-element-type=\"problems-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2869967\">\n<p id=\"import-auto-id2883532\">(a) In the deep space between galaxies, the density of atoms is as low as <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f2826a3b6de366a5bebf8381e92e6b4c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#54;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#97;&#116;&#111;&#109;&#115;&#47;&#109;&#125;&#125;&#94;&#123;&#51;&#125;&#44;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"111\" style=\"vertical-align: -4px;\" \/> and the temperature is a frigid 2.7 K. What is the pressure? (b) What volume (in <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-e3b87847f2822cabdde235117b9f5dc8_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#125;&#125;&#94;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"22\" style=\"vertical-align: 0px;\" \/>) is occupied by 1 mol of gas? (c) If this volume is a cube, what is the length of its sides in kilometers?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id2403810\">\n<p id=\"import-auto-id2393393\">(a) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-bd678d5a2d88eebd574f28e84297fc5f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#51;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#55;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#45;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#55;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#80;&#97;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"91\" style=\"vertical-align: -1px;\" \/><\/p>\n<p id=\"import-auto-id3177718\">(b) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-189f2e70bcd03164c13891c7321faf6c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#54;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#48;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#55;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#109;&#125;&#125;&#94;&#123;&#51;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"81\" style=\"vertical-align: -1px;\" \/><\/p>\n<p id=\"import-auto-id2677758\">(c) <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-eed622eeb804539dfcbd06a8614dbd9e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#56;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#52;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#50;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#107;&#109;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"77\" style=\"vertical-align: -1px;\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"glossary\" class=\"textbox shaded\">\n<h2 data-type=\"glossary-title\">Glossary<\/h2>\n<dl class=\"definition\" id=\"import-auto-id2744320\">\n<dt>ideal gas law<\/dt>\n<dd id=\"fs-id2084747\">the physical law that relates the pressure and volume of a gas to the number of gas molecules or number of moles of gas and the temperature of the gas<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id2744329\">\n<dt>Boltzmann constant<\/dt>\n<dd id=\"fs-id2699740\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-3422b6bb5c160593658b7c39425d9880_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#107;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"9\" style=\"vertical-align: 0px;\" \/> , a physical constant that relates energy to temperature; <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ce4cbb0cd268280d9e2990f1c1f5b91b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#107;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#46;&#51;&#56;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#45;&#50;&#51;&#125;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#74;&#47;&#75;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"138\" style=\"vertical-align: -4px;\" \/><\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id2626622\">\n<dt>Avogadro\u2019s number<\/dt>\n<dd id=\"fs-id2818566\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-8c5d5641fa23b0f683a1641759a247aa_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#78;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#65;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"24\" style=\"vertical-align: -4px;\" \/> , the number of molecules or atoms in one mole of a substance;<br \/>\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-479bb00824d5ad4bfe91808c7ac17ed2_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#78;&#125;&#95;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#65;&#125;&#125;&#61;&#54;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#50;&#125;&times;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#49;&#48;&#125;&#125;&#94;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#51;&#125;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"19\" width=\"112\" style=\"vertical-align: -4px;\" \/><br \/>\n     particles\/mole<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"fs-id1609952\">\n<dt>mole<\/dt>\n<dd id=\"fs-id1609957\">the quantity of a substance whose mass (in grams) is equal to its molecular mass<\/dd>\n<\/dl>\n<\/div>\n","protected":false},"author":211,"menu_order":1,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":"all-rights-reserved"},"chapter-type":[],"contributor":[],"license":[56],"class_list":["post-694","chapter","type-chapter","status-publish","hentry","license-all-rights-reserved"],"part":669,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters\/694","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/users\/211"}],"version-history":[{"count":1,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters\/694\/revisions"}],"predecessor-version":[{"id":695,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters\/694\/revisions\/695"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/parts\/669"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters\/694\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/media?parent=694"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapter-type?post=694"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/contributor?post=694"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/license?post=694"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}