{"id":882,"date":"2021-07-23T09:20:59","date_gmt":"2021-07-23T13:20:59","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/aperrott\/chapter\/potential-free-energy-and-equilibrium\/"},"modified":"2022-06-23T09:25:18","modified_gmt":"2022-06-23T13:25:18","slug":"potential-free-energy-and-equilibrium","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/aperrott\/chapter\/potential-free-energy-and-equilibrium\/","title":{"raw":"17.4 Potential, Free Energy, and Equilibrium","rendered":"17.4 Potential, Free Energy, and Equilibrium"},"content":{"raw":"&nbsp;\r\n<div class=\"textbox textbox--learning-objectives\">\r\n<h3><strong>Learning Objectives<\/strong><\/h3>\r\nBy the end of this section, you will be able to:\r\n<ul>\r\n \t<li>Explain the relations between potential, free energy change, and equilibrium constants<\/li>\r\n \t<li>Perform calculations involving the relations between cell potentials, free energy changes, and equilibrium<\/li>\r\n \t<li>Use the Nernst equation to determine cell potentials under nonstandard conditions<\/li>\r\n<\/ul>\r\n<\/div>\r\n<p id=\"fs-idm222137520\">So far in this chapter, the relationship between the cell potential and reaction <em data-effect=\"italics\">spontaneity<\/em> has been described, suggesting a link to the free energy change for the reaction (see chapter on thermodynamics). The interpretation of potentials as measures of oxidant <em data-effect=\"italics\">strength<\/em> was presented, bringing to mind similar measures of acid-base strength as reflected in equilibrium constants (see the chapter on acid-base equilibria). This section provides a summary of the relationships between potential and the related thermodynamic properties \u0394<em>G<\/em> and <em>K<\/em>.<\/p>\r\n\r\n<div id=\"fs-idm248064976\" class=\"bc-section section\" data-depth=\"1\">\r\n<h3 data-type=\"title\"><strong>E\u00b0 and \u0394G\u00b0<\/strong><\/h3>\r\n<p id=\"fs-idm215301376\">The standard free energy change of a process, \u0394<em data-effect=\"italics\">G<\/em>\u00b0, was defined in a previous chapter as the maximum work that could be performed by a system, <em data-effect=\"italics\">w<\/em><sub>max<\/sub>. In the case of a redox reaction taking place within a galvanic cell under standard state conditions, essentially all the work is associated with transferring the electrons from reductant-to-oxidant, <em data-effect=\"italics\">w<\/em><sub>elec<\/sub>:<\/p>\r\n\r\n<div id=\"fs-idm215765968\" style=\"padding-left: 40px\" data-type=\"equation\">\u0394<em>G<\/em>\u00b0 = <em>w<\/em><sub>max<\/sub> = <em>w<\/em><sub>elec<\/sub><\/div>\r\n<p id=\"fs-idm233319952\">The work associated with transferring electrons is determined by the total amount of charge (coulombs) transferred and the cell potential:<\/p>\r\n\r\n<div id=\"fs-idm655964560\" style=\"padding-left: 40px\" data-type=\"equation\">\u0394<em>G<\/em>\u00b0 = w<sub>elec<\/sub> = \u2212<em>nFE<\/em>\u00b0<sub>cell<\/sub><\/div>\r\n<div style=\"padding-left: 40px\" data-type=\"equation\">\u0394G\u00b0 = \u2212<em>nFE<\/em>\u00b0<sub>cell<\/sub><\/div>\r\n<p id=\"fs-idm648263408\">where <em data-effect=\"italics\">n<\/em> is the number of moles of electrons transferred, <em data-effect=\"italics\">F<\/em> is <strong data-effect=\"bold\">Faraday\u2019s constant<\/strong>, and <em data-effect=\"italics\">E<\/em>\u00b0<sub>cell<\/sub> is the standard cell potential. The relation between free energy change and standard cell potential confirms the sign conventions and spontaneity criteria previously discussed for both of these properties: spontaneous redox reactions exhibit positive potentials and negative free energy changes.<\/p>\r\n\r\n<\/div>\r\n<div class=\"bc-section section\" data-depth=\"1\">\r\n<h3 data-type=\"title\"><strong>E\u00b0 and K<\/strong><\/h3>\r\n<p id=\"fs-idm242158160\">Combining a previously derived relation between \u0394G\u00b0 and K (see the chapter on thermodynamics) and the equation above relating \u0394G\u00b0 and <em data-effect=\"italics\">E<\/em>\u00b0<sub>cell<\/sub> yields the following:<\/p>\r\n\r\n<div id=\"fs-idm205421792\" style=\"padding-left: 40px\" data-type=\"equation\"><img class=\"alignnone wp-image-2052\" src=\"https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/17.4a-300x62.png\" alt=\"\" width=\"257\" height=\"53\" \/><\/div>\r\n<p id=\"fs-idm242471888\">This equation indicates redox reactions with large (positive) standard cell potentials will proceed far towards completion, reaching equilibrium when the majority of reactant has been converted to product. A summary of the relations between <em data-effect=\"italics\">E<\/em>\u00b0, \u0394<em data-effect=\"italics\">G<\/em>\u00b0 and <em data-effect=\"italics\">K<\/em> is depicted in <a class=\"autogenerated-content\" href=\"#CNX_Chem_17_04_Relation\">(Figure)<\/a>, and a table correlating reaction spontaneity to values of these properties is provided in <a class=\"autogenerated-content\" href=\"#fs-idm241340256\">(Figure)<\/a>.<\/p>\r\n&nbsp;\r\n<div id=\"CNX_Chem_17_04_Relation\" class=\"scaled-down\">\r\n<div class=\"bc-figcaption figcaption\">Graphic depicting the relation between three important thermodynamic properties.<\/div>\r\n<span id=\"fs-idp78132768\" data-type=\"media\" data-alt=\"A diagram is shown that involves three double headed arrows positioned in the shape of an equilateral triangle. The vertices are labeled in red. The top vertex is labeled \u201cK.\u201c The vertex at the lower left is labeled \u201cdelta G superscript degree symbol.\u201d The vertex at the lower right is labeled \u201cE superscript degree symbol subscript cell.\u201d The right side of the triangle is labeled \u201cE superscript degree symbol subscript cell equals ( R T divided by n F ) l n K.\u201d The lower side of the triangle is labeled \u201cdelta G superscript degree symbol equals negative n F E superscript degree symbol subscript cell.\u201d The left side of the triangle is labeled \u201cdelta G superscript degree symbol equals negative R T l n K.\u201d\"><img src=\"https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/CNX_Chem_17_04_Relation.jpg\" alt=\"A diagram is shown that involves three double headed arrows positioned in the shape of an equilateral triangle. The vertices are labeled in red. The top vertex is labeled \u201cK.\u201c The vertex at the lower left is labeled \u201cdelta G superscript degree symbol.\u201d The vertex at the lower right is labeled \u201cE superscript degree symbol subscript cell.\u201d The right side of the triangle is labeled \u201cE superscript degree symbol subscript cell equals ( R T divided by n F ) l n K.\u201d The lower side of the triangle is labeled \u201cdelta G superscript degree symbol equals negative n F E superscript degree symbol subscript cell.\u201d The left side of the triangle is labeled \u201cdelta G superscript degree symbol equals negative R T l n K.\u201d\" data-media-type=\"image\/jpeg\" \/><\/span>\r\n\r\n<\/div>\r\n<table id=\"fs-idm241340256\" summary=\"No Summary\">\r\n<tbody>\r\n<tr valign=\"top\">\r\n<td style=\"width: 45px\" data-valign=\"top\" data-align=\"left\"><em data-effect=\"italics\">K<\/em><\/td>\r\n<td style=\"width: 46px\" data-valign=\"top\" data-align=\"left\">\u0394<em data-effect=\"italics\">G<\/em>\u00b0<\/td>\r\n<td style=\"width: 55px\" data-valign=\"top\" data-align=\"left\"><em data-effect=\"italics\">E<\/em>\u00b0<sub>cell<\/sub><\/td>\r\n<td style=\"width: 288px\" data-valign=\"top\" data-align=\"left\"><\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td style=\"width: 45px\" data-valign=\"top\" data-align=\"left\">&gt; 1<\/td>\r\n<td style=\"width: 46px\" data-valign=\"top\" data-align=\"left\">&lt; 0<\/td>\r\n<td style=\"width: 55px\" data-valign=\"top\" data-align=\"left\">&gt; 0<\/td>\r\n<td style=\"width: 288px\" data-valign=\"top\" data-align=\"left\">\r\n<p id=\"fs-idm239811792\">Reaction is spontaneous under standard conditions<\/p>\r\n<p id=\"fs-idm218962512\">Products more abundant at equilibrium<\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td style=\"width: 45px\" data-valign=\"top\" data-align=\"left\">&lt; 1<\/td>\r\n<td style=\"width: 46px\" data-valign=\"top\" data-align=\"left\">&gt; 0<\/td>\r\n<td style=\"width: 55px\" data-valign=\"top\" data-align=\"left\">&lt; 0<\/td>\r\n<td style=\"width: 288px\" data-valign=\"top\" data-align=\"left\">\r\n<p id=\"fs-idm248001280\">Reaction is non-spontaneous under standard conditions<\/p>\r\n<p id=\"fs-idm197346624\">Reactants more abundant at equilibrium<\/p>\r\n<\/td>\r\n<\/tr>\r\n<tr valign=\"top\">\r\n<td style=\"width: 45px\" data-valign=\"top\" data-align=\"left\">= 1<\/td>\r\n<td style=\"width: 46px\" data-valign=\"top\" data-align=\"left\">= 0<\/td>\r\n<td style=\"width: 55px\" data-valign=\"top\" data-align=\"left\">= 0<\/td>\r\n<td style=\"width: 288px\" data-valign=\"top\" data-align=\"left\">\r\n<p id=\"fs-idm249738032\">Reaction is at equilibrium under standard conditions<\/p>\r\n<p id=\"fs-idm197184048\">Reactants and products equally abundant<\/p>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div id=\"fs-idp16667568\" class=\"textbox textbox--examples\" data-type=\"example\">\r\n<p id=\"fs-idp58864784\"><strong>Equilibrium Constants, Standard Cell Potentials, and Standard Free Energy Changes <\/strong><\/p>\r\nUse data from Appendix L to calculate the standard cell potential, standard free energy change, and equilibrium constant for the following reaction at 25 \u00b0C. Comment on the spontaneity of the forward reaction and the composition of an equilibrium mixture of reactants and products.\r\n<div id=\"fs-idm89214592\" style=\"padding-left: 40px\" data-type=\"equation\">2Ag<sup>+<\/sup>(<em>aq<\/em>) + Fe(<em>s<\/em>) \u21cc 2Ag(<em>s<\/em>) + Fe<sup>2+<\/sup>(<em>aq<\/em>)<\/div>\r\n<div data-type=\"equation\"><\/div>\r\n<div data-type=\"equation\"><\/div>\r\n<p id=\"fs-idp148405856\"><strong>Solution:<\/strong><\/p>\r\nThe reaction involves an oxidation-reduction reaction, so the standard cell potential can be calculated using the data in Appendix L.\r\n<div id=\"fs-idm7297840\" style=\"padding-left: 40px\" data-type=\"equation\">anode (oxidation):\u00a0 Fe(<em>s<\/em>) \u27f6 Fe<sup>2+<\/sup>(<em>aq<\/em>) + 2e<sup>\u2212<\/sup>\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 E\u00b0<sub>Fe2+\/Fe<\/sub> = \u22120.447 V<\/div>\r\n<div style=\"padding-left: 40px\" data-type=\"equation\">cathode (reduction):\u00a0 2 \u00d7 (Ag<sup>+<\/sup>(<em>aq<\/em>) + e<sup>\u2212<\/sup> \u27f6 Ag(<em>s<\/em>))\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 E\u00b0<sub>Ag+\/Ag\u00b0<\/sub> = 0.7996 V<\/div>\r\n<div style=\"padding-left: 40px\" data-type=\"equation\">E\u00b0<sub>cell<\/sub> = E\u00b0<sub>cathode<\/sub> - E\u00b0<sub>anode<\/sub> = E\u00b0<sub>Ag<sup>+<\/sup>\/Ag<\/sub> - E\u00b0<sub>Fe<sup>2+<\/sup>\/Fe<\/sub> = +1.247 V<\/div>\r\n<div data-type=\"equation\"><\/div>\r\n<p id=\"fs-idp199399072\">With <em data-effect=\"italics\">n<\/em> = 2, the equilibrium constant is then<\/p>\r\n\r\n<div id=\"fs-idp2572928\" style=\"padding-left: 40px\" data-type=\"equation\"><img class=\"alignnone wp-image-2054\" src=\"https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/17.4b-300x190.png\" alt=\"\" width=\"219\" height=\"139\" \/><\/div>\r\n<div data-type=\"equation\"><\/div>\r\n<p id=\"fs-idp101330240\">The standard free energy is then<\/p>\r\n\r\n<div id=\"fs-idp126961280\" style=\"padding-left: 40px\" data-type=\"equation\">\u0394<em>G<\/em>\u00b0 = \u2212<em>nFE<\/em>\u00b0<sub>cell<\/sub><\/div>\r\n<div style=\"padding-left: 40px\" data-type=\"equation\">\u0394<em>G<\/em>\u00b0 = -2 \u00d7 96,485 C\/mol \u00d7 1.247 J\/C = -240.6 kJ\/mol<\/div>\r\n<div data-type=\"equation\"><\/div>\r\n<p id=\"fs-idm8503104\">The reaction is spontaneous, as indicated by a negative free energy change and a positive cell potential. The <em data-effect=\"italics\">K<\/em> value is very large, indicating the reaction proceeds to near completion to yield an equilibrium mixture containing mostly products.<\/p>\r\n<p id=\"fs-idp110872768\"><strong>Check Your Learning:<\/strong><\/p>\r\nWhat is the standard free energy change and the equilibrium constant for the following reaction at room temperature? Is the reaction spontaneous?\r\n<div id=\"fs-idp7040368\" style=\"padding-left: 40px\" data-type=\"equation\">Sn(<em>s<\/em>) + 2Cu<sup>2+<\/sup>(<em>aq<\/em>) \u21cc Sn<sup>2+<\/sup>(<em>aq<\/em>) + 2Cu<sup>+<\/sup>(<em>aq<\/em>)<\/div>\r\n<div data-type=\"equation\"><\/div>\r\n<div id=\"fs-idp195905712\" data-type=\"note\">\r\n<div data-type=\"title\"><strong>Answer:<\/strong><\/div>\r\n<p id=\"fs-idp101579952\">Spontaneous; <em data-effect=\"italics\">n<\/em> = 2; <em>E<\/em>\u00b0<sub>cell<\/sub> = +0.291 V; \u0394<em>G<\/em>\u00b0 =- 56.2 kJ\/mol; <em data-effect=\"italics\">K<\/em> = 6.8 \u00d7 10<sup>9<\/sup>.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-idm251393488\" class=\"bc-section section\" data-depth=\"1\">\r\n<h3 data-type=\"title\"><strong>Potentials at Nonstandard Conditions: The Nernst Equation<\/strong><\/h3>\r\n<p id=\"fs-idp76396192\">Most of the redox processes that interest science and society do not occur under standard state conditions, and so the potentials of these systems under nonstandard conditions are a property worthy of attention. Having established the relationship between potential and free energy change in this section, the previously discussed relation between free energy change and reaction mixture composition can be used for this purpose.<\/p>\r\n\r\n<div id=\"fs-idp76596960\" style=\"padding-left: 40px\" data-type=\"equation\">\u0394<em>G<\/em> = \u0394<em><span style=\"font-size: 1em\">G<\/span><\/em><span style=\"font-size: 1em\">\u00b0 + <em>RT<\/em> ln <em>Q<\/em><\/span><\/div>\r\n<p id=\"fs-idm983728\">Notice the reaction quotient, <em data-effect=\"italics\">Q<\/em>, appears in this equation, making the free energy change dependent upon the composition of the reaction mixture. Substituting the equation relating free energy change to cell potential yields the <span data-type=\"term\">Nernst equation<\/span>:<\/p>\r\n\r\n<div id=\"fs-idp17520176\" style=\"padding-left: 40px\" data-type=\"equation\">\u2212<em>nFE<\/em><sub>cell<\/sub> = -<span style=\"font-size: 1em\"><em>nFE<\/em>\u00b0<sub>cell<\/sub> + <em>RT<\/em> ln <em>Q<\/em><\/span><\/div>\r\n<div id=\"fs-idm642366960\" style=\"padding-left: 40px\" data-type=\"equation\"><img class=\"alignnone wp-image-2056\" src=\"https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/17.4c-300x75.png\" alt=\"\" width=\"172\" height=\"43\" \/><\/div>\r\n<p id=\"fs-idp24059424\">This equation describes how the potential of a redox system (such as a galvanic cell) varies from its standard state value, specifically, showing it to be a function of the number of electrons transferred, <em data-effect=\"italics\">n<\/em>, the temperature, <em data-effect=\"italics\">T<\/em>, and the reaction mixture composition as reflected in <em data-effect=\"italics\">Q<\/em>. A convenient form of the Nernst equation for most work is one in which values for the fundamental constants (R and F) and a factor converting from natural to base-10 logarithms have been included:<\/p>\r\n\r\n<div id=\"fs-idp195072512\" style=\"padding-left: 40px\" data-type=\"equation\"><img class=\"alignnone wp-image-2057\" src=\"https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/17.4d-300x55.png\" alt=\"\" width=\"218\" height=\"40\" \/><\/div>\r\n<div id=\"fs-idm240817536\" class=\"textbox textbox--examples\" data-type=\"example\">\r\n<p id=\"fs-idp23262432\"><strong>Predicting Redox Spontaneity Under Nonstandard Conditions <\/strong><\/p>\r\nUse the Nernst equation to predict the spontaneity of the redox reaction shown below.\r\n<div id=\"fs-idm365472\" style=\"padding-left: 40px\" data-type=\"equation\">Co(<em>s<\/em>) + Fe<sup>2+<\/sup>(<em>aq<\/em>, 1.94 M) \u27f6 Co<sup>2+<\/sup>(<em>aq<\/em>, 0.15 M) + Fe(<em>s<\/em>)<\/div>\r\n<div data-type=\"equation\"><\/div>\r\n<div data-type=\"equation\"><\/div>\r\n<p id=\"fs-idp102277360\"><strong>Solution:<\/strong><\/p>\r\nCollecting information from Appendix L and the problem,\r\n<div id=\"fs-idp61630144\" style=\"padding-left: 40px\" data-type=\"equation\">Anode (oxidation):\u00a0 Co(<em>s<\/em>) \u27f6 Co<sup>2+<\/sup>(aq) + 2e<sup>\u2212<\/sup>\u00a0 \u00a0 \u00a0E\u00b0<sub>Co<sup>2+<\/sup>\/Co<\/sub>\u00a0 = \u22120.28 V<\/div>\r\n<div style=\"padding-left: 40px\" data-type=\"equation\">Cathode (reduction):\u00a0 Fe<sup>2+<\/sup>(<em>aq<\/em>) + 2e<sup>\u2212<\/sup> \u27f6 Fe(<em>s<\/em>)\u00a0 \u00a0 \u00a0E\u00b0<sub>Fe<sup>2+<\/sup>\/Fe<\/sub> = <span style=\"font-size: 1em\">\u22120.447 V<\/span><\/div>\r\n<div style=\"padding-left: 40px\" data-type=\"equation\"><span style=\"font-size: 1em\">E\u00b0 = E\u00b0<sub>cathode<\/sub> - E\u00b0<sub>anode<\/sub> = \u22120.447 V - (\u22120.28 V) = \u22120.17 V<\/span><\/div>\r\n<div data-type=\"equation\"><\/div>\r\n<p id=\"fs-idm15620288\">Notice the negative value of the standard cell potential indicates the process is not spontaneous under standard conditions. Substitution of the Nernst equation terms for the nonstandard conditions yields:<\/p>\r\n\r\n<div data-type=\"equation\"><\/div>\r\n<div id=\"fs-idp99119696\" style=\"padding-left: 40px\" data-type=\"equation\"><img class=\"alignnone size-medium wp-image-2059\" src=\"https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/17.4e-300x122.png\" alt=\"\" width=\"300\" height=\"122\" \/><\/div>\r\n<p id=\"fs-idp160381408\">The cell potential remains negative (slightly) under the specified conditions, and so the reaction remains nonspontaneous.<\/p>\r\n<p id=\"fs-idp72968608\"><strong>Check Your Learning:<\/strong><\/p>\r\nFor the cell schematic below, identify values for <em data-effect=\"italics\">n<\/em> and <em data-effect=\"italics\">Q<\/em>, and calculate the cell potential, <em data-effect=\"italics\">E<\/em><sub>cell<\/sub>.\r\n<div id=\"fs-idp18135232\" style=\"padding-left: 40px\" data-type=\"equation\">Al(<em>s<\/em>)\u2502Al<sup>3+<\/sup>(<em>aq<\/em>, 0.15 M)\u2551Cu<sup>2+<\/sup>(<em>aq<\/em>, 0.025 M)\u2502Cu(<em>s<\/em>)<\/div>\r\n<div data-type=\"equation\"><\/div>\r\n<div id=\"fs-idp84594464\" data-type=\"note\">\r\n<div data-type=\"title\"><strong>Answer:<\/strong><\/div>\r\n<p id=\"fs-idp34686320\"><em data-effect=\"italics\">n<\/em> = 6; <em data-effect=\"italics\">Q<\/em> = 1440; <em data-effect=\"italics\">E<\/em><sub>cell<\/sub> = +1.97 V, spontaneous.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-idp264674608\">A <strong>concentration cell <\/strong>is constructed by connecting two nearly identical half-cells, each based on the same half-reaction and using the same electrode, varying only in the concentration of one redox species. The potential of a concentration cell, therefore, is determined only by the difference in concentration of the chosen redox species. The example problem below illustrates the use of the Nernst equation in calculations involving concentration cells.<\/p>\r\n\r\n<div id=\"fs-idp211103648\" class=\"textbox textbox--examples\" data-type=\"example\">\r\n<p id=\"fs-idp40117744\"><strong>Concentration Cells <\/strong><\/p>\r\nWhat is the cell potential of the concentration cell described by\r\n<div id=\"fs-idp85858736\" style=\"padding-left: 40px\" data-type=\"equation\">Zn(<em>s<\/em>)\u2502Zn<sup>2+<\/sup>(<em>aq<\/em>, 0.10 M)\u2551Zn<sup>2+<\/sup>(<em>aq<\/em>, 0.50 M)\u2502Zn(<em>s<\/em>)<\/div>\r\n<div data-type=\"equation\"><\/div>\r\n<p id=\"fs-idp142711536\"><strong>Solution:<\/strong><\/p>\r\nFrom the information given:\r\n<div id=\"fs-idp5275888\" style=\"padding-left: 40px\" data-type=\"equation\">Anode:\u00a0 Zn(<em>s<\/em>) \u27f6 Zn<sup>2+<\/sup>(<em>aq<\/em>, 0.10M) + 2e<sup>\u2212<\/sup>\u00a0 \u00a0 \u00a0E\u00b0<sub>anode<\/sub> = \u22120.7618 V<\/div>\r\n<div style=\"padding-left: 40px\" data-type=\"equation\">Cathode:\u00a0 Zn<sup>2+<\/sup>(<em>aq<\/em>, 0.50 M) + 2e<sup>\u2212<\/sup>\u27f6 Zn(<em>s<\/em>)\u00a0 \u00a0 \u00a0 E\u00b0<sub>cathode<\/sub> = \u22120.7618 V<\/div>\r\n<div style=\"padding-left: 40px\" data-type=\"equation\">Overall:\u00a0 Zn<sup>2+<\/sup>(<em>aq<\/em>, 0.50M) \u27f6 Zn<sup>2+<\/sup>(<em>aq<\/em>, 0.10M)\u00a0 \u00a0 \u00a0E\u00b0 = 0.000 V<\/div>\r\n<div data-type=\"equation\"><\/div>\r\n<p id=\"fs-idp2148576\">Substituting into the Nernst equation,<\/p>\r\n\r\n<div id=\"fs-idp235695968\" style=\"padding-left: 40px\" data-type=\"equation\"><img class=\"alignnone wp-image-2061\" src=\"https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/17.4f-300x37.png\" alt=\"\" width=\"349\" height=\"43\" \/><\/div>\r\n<p id=\"fs-idp47526144\">The positive value for cell potential indicates the overall cell reaction (see above) is spontaneous. This spontaneous reaction is one in which the zinc ion concentration in the cathode falls (it is reduced to elemental zinc) while that in the anode rises (it is produced by oxidation of the zinc anode). A greater driving force for zinc reduction is present in the cathode, where the zinc(II) ion concentration is greater (<em data-effect=\"italics\">E<\/em><sub>cathode<\/sub> &gt; <em data-effect=\"italics\">E<\/em><sub>anode<\/sub>).<\/p>\r\n<p id=\"fs-idm11024080\"><strong>Check Your Learning:<\/strong><\/p>\r\nThe concentration cell above was allowed to operate until the cell reaction reached equilibrium. What are the cell potential and the concentrations of zinc(II) in each half-cell for the cell now?\r\n\r\n&nbsp;\r\n<div id=\"fs-idm1444240\" data-type=\"note\">\r\n<div data-type=\"title\"><strong>Answer:<\/strong><\/div>\r\n<p id=\"fs-idp18466656\"><em data-effect=\"italics\">E<\/em><sub>cell<\/sub> = 0.000 V; [Zn<sup>2+<\/sup>]<sub>cathode<\/sub> = [Zn<sup>2+<\/sup>]<sub>anode<\/sub> = 0.30 <em data-effect=\"italics\">M<\/em><\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div id=\"fs-idp160224496\" class=\"summary\" data-depth=\"1\">\r\n<h3 data-type=\"title\"><strong>Key Concepts and Summary<\/strong><\/h3>\r\n<p id=\"fs-idp58835888\">Potential is a thermodynamic quantity reflecting the intrinsic driving force of a redox process, and it is directly related to the free energy change and equilibrium constant for the process. For redox processes taking place in electrochemical cells, the maximum (electrical) work done by the system is easily computed from the cell potential and the reaction stoichiometry and is equal to the free energy change for the process. The equilibrium constant for a redox reaction is logarithmically related to the reaction\u2019s cell potential, with larger (more positive) potentials indicating reactions with greater driving force that equilibrate when the reaction has proceeded far towards completion (large value of <em data-effect=\"italics\">K<\/em>). Finally, the potential of a redox process varies with the composition of the reaction mixture, being related to the reactions standard potential and the value of its reaction quotient, <em data-effect=\"italics\">Q<\/em>, as described by the Nernst equation.<\/p>\r\n\r\n<\/div>\r\n<div id=\"fs-idp39495232\" class=\"key-equations\" data-depth=\"1\">\r\n<h3 data-type=\"title\"><strong>Key Equations<\/strong><\/h3>\r\n<ul id=\"fs-idm17893664\" data-bullet-style=\"bullet\">\r\n \t<li><img class=\"alignnone wp-image-2062\" src=\"https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/17.4g-300x78.png\" alt=\"\" width=\"134\" height=\"35\" \/><\/li>\r\n \t<li><img class=\"alignnone wp-image-2063\" src=\"https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/17.4h-300x20.png\" alt=\"\" width=\"465\" height=\"31\" \/><\/li>\r\n \t<li><img class=\"alignnone wp-image-2064\" src=\"https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/17.4i-300x24.png\" alt=\"\" width=\"363\" height=\"29\" \/><\/li>\r\n \t<li><img class=\"alignnone wp-image-2065\" src=\"https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/17.4j-300x24.png\" alt=\"\" width=\"363\" height=\"29\" \/><\/li>\r\n \t<li>\u0394<em data-effect=\"italics\">G<\/em> = \u2212<em data-effect=\"italics\">nFE<\/em><sub>cell<\/sub><\/li>\r\n \t<li>\u0394<em>G<\/em>\u00b0 = \u2212<em>nFE<\/em>\u00b0<sub>cell<\/sub><\/li>\r\n \t<li>w<sub>elec<\/sub> = w<sub>max<\/sub> = \u2212<em>nFE<\/em><sub>cell<\/sub><\/li>\r\n<\/ul>\r\n<\/div>\r\n<div id=\"fs-idp39758576\" class=\"exercises\" data-depth=\"1\">\r\n<div id=\"fs-idp17461440\" data-type=\"exercise\">\r\n<div id=\"fs-idp58129232\" data-type=\"solution\">\r\n<p id=\"fs-idp161071024\"><\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox shaded\" data-type=\"glossary\">\r\n<h3 data-type=\"glossary-title\"><strong>Glossary<\/strong><\/h3>\r\n<dl id=\"fs-idp94537696\">\r\n \t<dt>concentration cell<\/dt>\r\n \t<dd id=\"fs-idp112973472\">galvanic cell comprising half-cells of identical composition but for the concentration of one redox reactant or product<\/dd>\r\n<\/dl>\r\n<dl id=\"fs-idp41925664\">\r\n \t<dt>Faraday\u2019s constant (F)<\/dt>\r\n \t<dd id=\"fs-idm11947792\">charge on 1 mol of electrons; <em data-effect=\"italics\">F<\/em> = 96,485 C\/mol e<sup>\u2212<\/sup><\/dd>\r\n<\/dl>\r\n<dl id=\"fs-idp211566960\">\r\n \t<dt>Nernst equation<\/dt>\r\n \t<dd id=\"fs-idp90500496\">relating the potential of a redox system to its composition<\/dd>\r\n<\/dl>\r\n<\/div>","rendered":"<p>&nbsp;<\/p>\n<div class=\"textbox textbox--learning-objectives\">\n<h3><strong>Learning Objectives<\/strong><\/h3>\n<p>By the end of this section, you will be able to:<\/p>\n<ul>\n<li>Explain the relations between potential, free energy change, and equilibrium constants<\/li>\n<li>Perform calculations involving the relations between cell potentials, free energy changes, and equilibrium<\/li>\n<li>Use the Nernst equation to determine cell potentials under nonstandard conditions<\/li>\n<\/ul>\n<\/div>\n<p id=\"fs-idm222137520\">So far in this chapter, the relationship between the cell potential and reaction <em data-effect=\"italics\">spontaneity<\/em> has been described, suggesting a link to the free energy change for the reaction (see chapter on thermodynamics). The interpretation of potentials as measures of oxidant <em data-effect=\"italics\">strength<\/em> was presented, bringing to mind similar measures of acid-base strength as reflected in equilibrium constants (see the chapter on acid-base equilibria). This section provides a summary of the relationships between potential and the related thermodynamic properties \u0394<em>G<\/em> and <em>K<\/em>.<\/p>\n<div id=\"fs-idm248064976\" class=\"bc-section section\" data-depth=\"1\">\n<h3 data-type=\"title\"><strong>E\u00b0 and \u0394G\u00b0<\/strong><\/h3>\n<p id=\"fs-idm215301376\">The standard free energy change of a process, \u0394<em data-effect=\"italics\">G<\/em>\u00b0, was defined in a previous chapter as the maximum work that could be performed by a system, <em data-effect=\"italics\">w<\/em><sub>max<\/sub>. In the case of a redox reaction taking place within a galvanic cell under standard state conditions, essentially all the work is associated with transferring the electrons from reductant-to-oxidant, <em data-effect=\"italics\">w<\/em><sub>elec<\/sub>:<\/p>\n<div id=\"fs-idm215765968\" style=\"padding-left: 40px\" data-type=\"equation\">\u0394<em>G<\/em>\u00b0 = <em>w<\/em><sub>max<\/sub> = <em>w<\/em><sub>elec<\/sub><\/div>\n<p id=\"fs-idm233319952\">The work associated with transferring electrons is determined by the total amount of charge (coulombs) transferred and the cell potential:<\/p>\n<div id=\"fs-idm655964560\" style=\"padding-left: 40px\" data-type=\"equation\">\u0394<em>G<\/em>\u00b0 = w<sub>elec<\/sub> = \u2212<em>nFE<\/em>\u00b0<sub>cell<\/sub><\/div>\n<div style=\"padding-left: 40px\" data-type=\"equation\">\u0394G\u00b0 = \u2212<em>nFE<\/em>\u00b0<sub>cell<\/sub><\/div>\n<p id=\"fs-idm648263408\">where <em data-effect=\"italics\">n<\/em> is the number of moles of electrons transferred, <em data-effect=\"italics\">F<\/em> is <strong data-effect=\"bold\">Faraday\u2019s constant<\/strong>, and <em data-effect=\"italics\">E<\/em>\u00b0<sub>cell<\/sub> is the standard cell potential. The relation between free energy change and standard cell potential confirms the sign conventions and spontaneity criteria previously discussed for both of these properties: spontaneous redox reactions exhibit positive potentials and negative free energy changes.<\/p>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\">\n<h3 data-type=\"title\"><strong>E\u00b0 and K<\/strong><\/h3>\n<p id=\"fs-idm242158160\">Combining a previously derived relation between \u0394G\u00b0 and K (see the chapter on thermodynamics) and the equation above relating \u0394G\u00b0 and <em data-effect=\"italics\">E<\/em>\u00b0<sub>cell<\/sub> yields the following:<\/p>\n<div id=\"fs-idm205421792\" style=\"padding-left: 40px\" data-type=\"equation\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-2052\" src=\"https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/17.4a-300x62.png\" alt=\"\" width=\"257\" height=\"53\" srcset=\"https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/17.4a-300x62.png 300w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/17.4a-65x13.png 65w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/17.4a-225x47.png 225w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/17.4a-350x72.png 350w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/17.4a.png 710w\" sizes=\"auto, (max-width: 257px) 100vw, 257px\" \/><\/div>\n<p id=\"fs-idm242471888\">This equation indicates redox reactions with large (positive) standard cell potentials will proceed far towards completion, reaching equilibrium when the majority of reactant has been converted to product. A summary of the relations between <em data-effect=\"italics\">E<\/em>\u00b0, \u0394<em data-effect=\"italics\">G<\/em>\u00b0 and <em data-effect=\"italics\">K<\/em> is depicted in <a class=\"autogenerated-content\" href=\"#CNX_Chem_17_04_Relation\">(Figure)<\/a>, and a table correlating reaction spontaneity to values of these properties is provided in <a class=\"autogenerated-content\" href=\"#fs-idm241340256\">(Figure)<\/a>.<\/p>\n<p>&nbsp;<\/p>\n<div id=\"CNX_Chem_17_04_Relation\" class=\"scaled-down\">\n<div class=\"bc-figcaption figcaption\">Graphic depicting the relation between three important thermodynamic properties.<\/div>\n<p><span id=\"fs-idp78132768\" data-type=\"media\" data-alt=\"A diagram is shown that involves three double headed arrows positioned in the shape of an equilateral triangle. The vertices are labeled in red. The top vertex is labeled \u201cK.\u201c The vertex at the lower left is labeled \u201cdelta G superscript degree symbol.\u201d The vertex at the lower right is labeled \u201cE superscript degree symbol subscript cell.\u201d The right side of the triangle is labeled \u201cE superscript degree symbol subscript cell equals ( R T divided by n F ) l n K.\u201d The lower side of the triangle is labeled \u201cdelta G superscript degree symbol equals negative n F E superscript degree symbol subscript cell.\u201d The left side of the triangle is labeled \u201cdelta G superscript degree symbol equals negative R T l n K.\u201d\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/CNX_Chem_17_04_Relation.jpg\" alt=\"A diagram is shown that involves three double headed arrows positioned in the shape of an equilateral triangle. The vertices are labeled in red. The top vertex is labeled \u201cK.\u201c The vertex at the lower left is labeled \u201cdelta G superscript degree symbol.\u201d The vertex at the lower right is labeled \u201cE superscript degree symbol subscript cell.\u201d The right side of the triangle is labeled \u201cE superscript degree symbol subscript cell equals ( R T divided by n F ) l n K.\u201d The lower side of the triangle is labeled \u201cdelta G superscript degree symbol equals negative n F E superscript degree symbol subscript cell.\u201d The left side of the triangle is labeled \u201cdelta G superscript degree symbol equals negative R T l n K.\u201d\" data-media-type=\"image\/jpeg\" \/><\/span><\/p>\n<\/div>\n<table id=\"fs-idm241340256\" summary=\"No Summary\">\n<tbody>\n<tr valign=\"top\">\n<td style=\"width: 45px\" data-valign=\"top\" data-align=\"left\"><em data-effect=\"italics\">K<\/em><\/td>\n<td style=\"width: 46px\" data-valign=\"top\" data-align=\"left\">\u0394<em data-effect=\"italics\">G<\/em>\u00b0<\/td>\n<td style=\"width: 55px\" data-valign=\"top\" data-align=\"left\"><em data-effect=\"italics\">E<\/em>\u00b0<sub>cell<\/sub><\/td>\n<td style=\"width: 288px\" data-valign=\"top\" data-align=\"left\"><\/td>\n<\/tr>\n<tr valign=\"top\">\n<td style=\"width: 45px\" data-valign=\"top\" data-align=\"left\">&gt; 1<\/td>\n<td style=\"width: 46px\" data-valign=\"top\" data-align=\"left\">&lt; 0<\/td>\n<td style=\"width: 55px\" data-valign=\"top\" data-align=\"left\">&gt; 0<\/td>\n<td style=\"width: 288px\" data-valign=\"top\" data-align=\"left\">\n<p id=\"fs-idm239811792\">Reaction is spontaneous under standard conditions<\/p>\n<p id=\"fs-idm218962512\">Products more abundant at equilibrium<\/p>\n<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td style=\"width: 45px\" data-valign=\"top\" data-align=\"left\">&lt; 1<\/td>\n<td style=\"width: 46px\" data-valign=\"top\" data-align=\"left\">&gt; 0<\/td>\n<td style=\"width: 55px\" data-valign=\"top\" data-align=\"left\">&lt; 0<\/td>\n<td style=\"width: 288px\" data-valign=\"top\" data-align=\"left\">\n<p id=\"fs-idm248001280\">Reaction is non-spontaneous under standard conditions<\/p>\n<p id=\"fs-idm197346624\">Reactants more abundant at equilibrium<\/p>\n<\/td>\n<\/tr>\n<tr valign=\"top\">\n<td style=\"width: 45px\" data-valign=\"top\" data-align=\"left\">= 1<\/td>\n<td style=\"width: 46px\" data-valign=\"top\" data-align=\"left\">= 0<\/td>\n<td style=\"width: 55px\" data-valign=\"top\" data-align=\"left\">= 0<\/td>\n<td style=\"width: 288px\" data-valign=\"top\" data-align=\"left\">\n<p id=\"fs-idm249738032\">Reaction is at equilibrium under standard conditions<\/p>\n<p id=\"fs-idm197184048\">Reactants and products equally abundant<\/p>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div id=\"fs-idp16667568\" class=\"textbox textbox--examples\" data-type=\"example\">\n<p id=\"fs-idp58864784\"><strong>Equilibrium Constants, Standard Cell Potentials, and Standard Free Energy Changes <\/strong><\/p>\n<p>Use data from Appendix L to calculate the standard cell potential, standard free energy change, and equilibrium constant for the following reaction at 25 \u00b0C. Comment on the spontaneity of the forward reaction and the composition of an equilibrium mixture of reactants and products.<\/p>\n<div id=\"fs-idm89214592\" style=\"padding-left: 40px\" data-type=\"equation\">2Ag<sup>+<\/sup>(<em>aq<\/em>) + Fe(<em>s<\/em>) \u21cc 2Ag(<em>s<\/em>) + Fe<sup>2+<\/sup>(<em>aq<\/em>)<\/div>\n<div data-type=\"equation\"><\/div>\n<div data-type=\"equation\"><\/div>\n<p id=\"fs-idp148405856\"><strong>Solution:<\/strong><\/p>\n<p>The reaction involves an oxidation-reduction reaction, so the standard cell potential can be calculated using the data in Appendix L.<\/p>\n<div id=\"fs-idm7297840\" style=\"padding-left: 40px\" data-type=\"equation\">anode (oxidation):\u00a0 Fe(<em>s<\/em>) \u27f6 Fe<sup>2+<\/sup>(<em>aq<\/em>) + 2e<sup>\u2212<\/sup>\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 E\u00b0<sub>Fe2+\/Fe<\/sub> = \u22120.447 V<\/div>\n<div style=\"padding-left: 40px\" data-type=\"equation\">cathode (reduction):\u00a0 2 \u00d7 (Ag<sup>+<\/sup>(<em>aq<\/em>) + e<sup>\u2212<\/sup> \u27f6 Ag(<em>s<\/em>))\u00a0 \u00a0 \u00a0 \u00a0 \u00a0 E\u00b0<sub>Ag+\/Ag\u00b0<\/sub> = 0.7996 V<\/div>\n<div style=\"padding-left: 40px\" data-type=\"equation\">E\u00b0<sub>cell<\/sub> = E\u00b0<sub>cathode<\/sub> &#8211; E\u00b0<sub>anode<\/sub> = E\u00b0<sub>Ag<sup>+<\/sup>\/Ag<\/sub> &#8211; E\u00b0<sub>Fe<sup>2+<\/sup>\/Fe<\/sub> = +1.247 V<\/div>\n<div data-type=\"equation\"><\/div>\n<p id=\"fs-idp199399072\">With <em data-effect=\"italics\">n<\/em> = 2, the equilibrium constant is then<\/p>\n<div id=\"fs-idp2572928\" style=\"padding-left: 40px\" data-type=\"equation\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-2054\" src=\"https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/17.4b-300x190.png\" alt=\"\" width=\"219\" height=\"139\" srcset=\"https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/17.4b-300x190.png 300w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/17.4b-65x41.png 65w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/17.4b-225x143.png 225w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/17.4b-350x222.png 350w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/17.4b.png 584w\" sizes=\"auto, (max-width: 219px) 100vw, 219px\" \/><\/div>\n<div data-type=\"equation\"><\/div>\n<p id=\"fs-idp101330240\">The standard free energy is then<\/p>\n<div id=\"fs-idp126961280\" style=\"padding-left: 40px\" data-type=\"equation\">\u0394<em>G<\/em>\u00b0 = \u2212<em>nFE<\/em>\u00b0<sub>cell<\/sub><\/div>\n<div style=\"padding-left: 40px\" data-type=\"equation\">\u0394<em>G<\/em>\u00b0 = -2 \u00d7 96,485 C\/mol \u00d7 1.247 J\/C = -240.6 kJ\/mol<\/div>\n<div data-type=\"equation\"><\/div>\n<p id=\"fs-idm8503104\">The reaction is spontaneous, as indicated by a negative free energy change and a positive cell potential. The <em data-effect=\"italics\">K<\/em> value is very large, indicating the reaction proceeds to near completion to yield an equilibrium mixture containing mostly products.<\/p>\n<p id=\"fs-idp110872768\"><strong>Check Your Learning:<\/strong><\/p>\n<p>What is the standard free energy change and the equilibrium constant for the following reaction at room temperature? Is the reaction spontaneous?<\/p>\n<div id=\"fs-idp7040368\" style=\"padding-left: 40px\" data-type=\"equation\">Sn(<em>s<\/em>) + 2Cu<sup>2+<\/sup>(<em>aq<\/em>) \u21cc Sn<sup>2+<\/sup>(<em>aq<\/em>) + 2Cu<sup>+<\/sup>(<em>aq<\/em>)<\/div>\n<div data-type=\"equation\"><\/div>\n<div id=\"fs-idp195905712\" data-type=\"note\">\n<div data-type=\"title\"><strong>Answer:<\/strong><\/div>\n<p id=\"fs-idp101579952\">Spontaneous; <em data-effect=\"italics\">n<\/em> = 2; <em>E<\/em>\u00b0<sub>cell<\/sub> = +0.291 V; \u0394<em>G<\/em>\u00b0 =- 56.2 kJ\/mol; <em data-effect=\"italics\">K<\/em> = 6.8 \u00d7 10<sup>9<\/sup>.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-idm251393488\" class=\"bc-section section\" data-depth=\"1\">\n<h3 data-type=\"title\"><strong>Potentials at Nonstandard Conditions: The Nernst Equation<\/strong><\/h3>\n<p id=\"fs-idp76396192\">Most of the redox processes that interest science and society do not occur under standard state conditions, and so the potentials of these systems under nonstandard conditions are a property worthy of attention. Having established the relationship between potential and free energy change in this section, the previously discussed relation between free energy change and reaction mixture composition can be used for this purpose.<\/p>\n<div id=\"fs-idp76596960\" style=\"padding-left: 40px\" data-type=\"equation\">\u0394<em>G<\/em> = \u0394<em><span style=\"font-size: 1em\">G<\/span><\/em><span style=\"font-size: 1em\">\u00b0 + <em>RT<\/em> ln <em>Q<\/em><\/span><\/div>\n<p id=\"fs-idm983728\">Notice the reaction quotient, <em data-effect=\"italics\">Q<\/em>, appears in this equation, making the free energy change dependent upon the composition of the reaction mixture. Substituting the equation relating free energy change to cell potential yields the <span data-type=\"term\">Nernst equation<\/span>:<\/p>\n<div id=\"fs-idp17520176\" style=\"padding-left: 40px\" data-type=\"equation\">\u2212<em>nFE<\/em><sub>cell<\/sub> = &#8211;<span style=\"font-size: 1em\"><em>nFE<\/em>\u00b0<sub>cell<\/sub> + <em>RT<\/em> ln <em>Q<\/em><\/span><\/div>\n<div id=\"fs-idm642366960\" style=\"padding-left: 40px\" data-type=\"equation\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-2056\" src=\"https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/17.4c-300x75.png\" alt=\"\" width=\"172\" height=\"43\" srcset=\"https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/17.4c-300x75.png 300w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/17.4c-65x16.png 65w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/17.4c-225x57.png 225w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/17.4c-350x88.png 350w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/17.4c.png 513w\" sizes=\"auto, (max-width: 172px) 100vw, 172px\" \/><\/div>\n<p id=\"fs-idp24059424\">This equation describes how the potential of a redox system (such as a galvanic cell) varies from its standard state value, specifically, showing it to be a function of the number of electrons transferred, <em data-effect=\"italics\">n<\/em>, the temperature, <em data-effect=\"italics\">T<\/em>, and the reaction mixture composition as reflected in <em data-effect=\"italics\">Q<\/em>. A convenient form of the Nernst equation for most work is one in which values for the fundamental constants (R and F) and a factor converting from natural to base-10 logarithms have been included:<\/p>\n<div id=\"fs-idp195072512\" style=\"padding-left: 40px\" data-type=\"equation\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-2057\" src=\"https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/17.4d-300x55.png\" alt=\"\" width=\"218\" height=\"40\" srcset=\"https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/17.4d-300x55.png 300w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/17.4d-65x12.png 65w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/17.4d-225x41.png 225w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/17.4d-350x64.png 350w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/17.4d.png 629w\" sizes=\"auto, (max-width: 218px) 100vw, 218px\" \/><\/div>\n<div id=\"fs-idm240817536\" class=\"textbox textbox--examples\" data-type=\"example\">\n<p id=\"fs-idp23262432\"><strong>Predicting Redox Spontaneity Under Nonstandard Conditions <\/strong><\/p>\n<p>Use the Nernst equation to predict the spontaneity of the redox reaction shown below.<\/p>\n<div id=\"fs-idm365472\" style=\"padding-left: 40px\" data-type=\"equation\">Co(<em>s<\/em>) + Fe<sup>2+<\/sup>(<em>aq<\/em>, 1.94 M) \u27f6 Co<sup>2+<\/sup>(<em>aq<\/em>, 0.15 M) + Fe(<em>s<\/em>)<\/div>\n<div data-type=\"equation\"><\/div>\n<div data-type=\"equation\"><\/div>\n<p id=\"fs-idp102277360\"><strong>Solution:<\/strong><\/p>\n<p>Collecting information from Appendix L and the problem,<\/p>\n<div id=\"fs-idp61630144\" style=\"padding-left: 40px\" data-type=\"equation\">Anode (oxidation):\u00a0 Co(<em>s<\/em>) \u27f6 Co<sup>2+<\/sup>(aq) + 2e<sup>\u2212<\/sup>\u00a0 \u00a0 \u00a0E\u00b0<sub>Co<sup>2+<\/sup>\/Co<\/sub>\u00a0 = \u22120.28 V<\/div>\n<div style=\"padding-left: 40px\" data-type=\"equation\">Cathode (reduction):\u00a0 Fe<sup>2+<\/sup>(<em>aq<\/em>) + 2e<sup>\u2212<\/sup> \u27f6 Fe(<em>s<\/em>)\u00a0 \u00a0 \u00a0E\u00b0<sub>Fe<sup>2+<\/sup>\/Fe<\/sub> = <span style=\"font-size: 1em\">\u22120.447 V<\/span><\/div>\n<div style=\"padding-left: 40px\" data-type=\"equation\"><span style=\"font-size: 1em\">E\u00b0 = E\u00b0<sub>cathode<\/sub> &#8211; E\u00b0<sub>anode<\/sub> = \u22120.447 V &#8211; (\u22120.28 V) = \u22120.17 V<\/span><\/div>\n<div data-type=\"equation\"><\/div>\n<p id=\"fs-idm15620288\">Notice the negative value of the standard cell potential indicates the process is not spontaneous under standard conditions. Substitution of the Nernst equation terms for the nonstandard conditions yields:<\/p>\n<div data-type=\"equation\"><\/div>\n<div id=\"fs-idp99119696\" style=\"padding-left: 40px\" data-type=\"equation\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-2059\" src=\"https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/17.4e-300x122.png\" alt=\"\" width=\"300\" height=\"122\" srcset=\"https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/17.4e-300x122.png 300w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/17.4e-768x312.png 768w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/17.4e-65x26.png 65w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/17.4e-225x91.png 225w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/17.4e-350x142.png 350w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/17.4e.png 832w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/div>\n<p id=\"fs-idp160381408\">The cell potential remains negative (slightly) under the specified conditions, and so the reaction remains nonspontaneous.<\/p>\n<p id=\"fs-idp72968608\"><strong>Check Your Learning:<\/strong><\/p>\n<p>For the cell schematic below, identify values for <em data-effect=\"italics\">n<\/em> and <em data-effect=\"italics\">Q<\/em>, and calculate the cell potential, <em data-effect=\"italics\">E<\/em><sub>cell<\/sub>.<\/p>\n<div id=\"fs-idp18135232\" style=\"padding-left: 40px\" data-type=\"equation\">Al(<em>s<\/em>)\u2502Al<sup>3+<\/sup>(<em>aq<\/em>, 0.15 M)\u2551Cu<sup>2+<\/sup>(<em>aq<\/em>, 0.025 M)\u2502Cu(<em>s<\/em>)<\/div>\n<div data-type=\"equation\"><\/div>\n<div id=\"fs-idp84594464\" data-type=\"note\">\n<div data-type=\"title\"><strong>Answer:<\/strong><\/div>\n<p id=\"fs-idp34686320\"><em data-effect=\"italics\">n<\/em> = 6; <em data-effect=\"italics\">Q<\/em> = 1440; <em data-effect=\"italics\">E<\/em><sub>cell<\/sub> = +1.97 V, spontaneous.<\/p>\n<\/div>\n<\/div>\n<p id=\"fs-idp264674608\">A <strong>concentration cell <\/strong>is constructed by connecting two nearly identical half-cells, each based on the same half-reaction and using the same electrode, varying only in the concentration of one redox species. The potential of a concentration cell, therefore, is determined only by the difference in concentration of the chosen redox species. The example problem below illustrates the use of the Nernst equation in calculations involving concentration cells.<\/p>\n<div id=\"fs-idp211103648\" class=\"textbox textbox--examples\" data-type=\"example\">\n<p id=\"fs-idp40117744\"><strong>Concentration Cells <\/strong><\/p>\n<p>What is the cell potential of the concentration cell described by<\/p>\n<div id=\"fs-idp85858736\" style=\"padding-left: 40px\" data-type=\"equation\">Zn(<em>s<\/em>)\u2502Zn<sup>2+<\/sup>(<em>aq<\/em>, 0.10 M)\u2551Zn<sup>2+<\/sup>(<em>aq<\/em>, 0.50 M)\u2502Zn(<em>s<\/em>)<\/div>\n<div data-type=\"equation\"><\/div>\n<p id=\"fs-idp142711536\"><strong>Solution:<\/strong><\/p>\n<p>From the information given:<\/p>\n<div id=\"fs-idp5275888\" style=\"padding-left: 40px\" data-type=\"equation\">Anode:\u00a0 Zn(<em>s<\/em>) \u27f6 Zn<sup>2+<\/sup>(<em>aq<\/em>, 0.10M) + 2e<sup>\u2212<\/sup>\u00a0 \u00a0 \u00a0E\u00b0<sub>anode<\/sub> = \u22120.7618 V<\/div>\n<div style=\"padding-left: 40px\" data-type=\"equation\">Cathode:\u00a0 Zn<sup>2+<\/sup>(<em>aq<\/em>, 0.50 M) + 2e<sup>\u2212<\/sup>\u27f6 Zn(<em>s<\/em>)\u00a0 \u00a0 \u00a0 E\u00b0<sub>cathode<\/sub> = \u22120.7618 V<\/div>\n<div style=\"padding-left: 40px\" data-type=\"equation\">Overall:\u00a0 Zn<sup>2+<\/sup>(<em>aq<\/em>, 0.50M) \u27f6 Zn<sup>2+<\/sup>(<em>aq<\/em>, 0.10M)\u00a0 \u00a0 \u00a0E\u00b0 = 0.000 V<\/div>\n<div data-type=\"equation\"><\/div>\n<p id=\"fs-idp2148576\">Substituting into the Nernst equation,<\/p>\n<div id=\"fs-idp235695968\" style=\"padding-left: 40px\" data-type=\"equation\"><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-2061\" src=\"https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/17.4f-300x37.png\" alt=\"\" width=\"349\" height=\"43\" srcset=\"https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/17.4f-300x37.png 300w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/17.4f-1024x126.png 1024w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/17.4f-768x94.png 768w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/17.4f-65x8.png 65w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/17.4f-225x28.png 225w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/17.4f-350x43.png 350w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/17.4f.png 1081w\" sizes=\"auto, (max-width: 349px) 100vw, 349px\" \/><\/div>\n<p id=\"fs-idp47526144\">The positive value for cell potential indicates the overall cell reaction (see above) is spontaneous. This spontaneous reaction is one in which the zinc ion concentration in the cathode falls (it is reduced to elemental zinc) while that in the anode rises (it is produced by oxidation of the zinc anode). A greater driving force for zinc reduction is present in the cathode, where the zinc(II) ion concentration is greater (<em data-effect=\"italics\">E<\/em><sub>cathode<\/sub> &gt; <em data-effect=\"italics\">E<\/em><sub>anode<\/sub>).<\/p>\n<p id=\"fs-idm11024080\"><strong>Check Your Learning:<\/strong><\/p>\n<p>The concentration cell above was allowed to operate until the cell reaction reached equilibrium. What are the cell potential and the concentrations of zinc(II) in each half-cell for the cell now?<\/p>\n<p>&nbsp;<\/p>\n<div id=\"fs-idm1444240\" data-type=\"note\">\n<div data-type=\"title\"><strong>Answer:<\/strong><\/div>\n<p id=\"fs-idp18466656\"><em data-effect=\"italics\">E<\/em><sub>cell<\/sub> = 0.000 V; [Zn<sup>2+<\/sup>]<sub>cathode<\/sub> = [Zn<sup>2+<\/sup>]<sub>anode<\/sub> = 0.30 <em data-effect=\"italics\">M<\/em><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div id=\"fs-idp160224496\" class=\"summary\" data-depth=\"1\">\n<h3 data-type=\"title\"><strong>Key Concepts and Summary<\/strong><\/h3>\n<p id=\"fs-idp58835888\">Potential is a thermodynamic quantity reflecting the intrinsic driving force of a redox process, and it is directly related to the free energy change and equilibrium constant for the process. For redox processes taking place in electrochemical cells, the maximum (electrical) work done by the system is easily computed from the cell potential and the reaction stoichiometry and is equal to the free energy change for the process. The equilibrium constant for a redox reaction is logarithmically related to the reaction\u2019s cell potential, with larger (more positive) potentials indicating reactions with greater driving force that equilibrate when the reaction has proceeded far towards completion (large value of <em data-effect=\"italics\">K<\/em>). Finally, the potential of a redox process varies with the composition of the reaction mixture, being related to the reactions standard potential and the value of its reaction quotient, <em data-effect=\"italics\">Q<\/em>, as described by the Nernst equation.<\/p>\n<\/div>\n<div id=\"fs-idp39495232\" class=\"key-equations\" data-depth=\"1\">\n<h3 data-type=\"title\"><strong>Key Equations<\/strong><\/h3>\n<ul id=\"fs-idm17893664\" data-bullet-style=\"bullet\">\n<li><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-2062\" src=\"https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/17.4g-300x78.png\" alt=\"\" width=\"134\" height=\"35\" srcset=\"https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/17.4g-300x78.png 300w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/17.4g-65x17.png 65w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/17.4g-225x59.png 225w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/17.4g-350x91.png 350w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/17.4g.png 372w\" sizes=\"auto, (max-width: 134px) 100vw, 134px\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-2063\" src=\"https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/17.4h-300x20.png\" alt=\"\" width=\"465\" height=\"31\" srcset=\"https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/17.4h-300x20.png 300w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/17.4h-1024x70.png 1024w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/17.4h-768x52.png 768w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/17.4h-65x4.png 65w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/17.4h-225x15.png 225w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/17.4h-350x24.png 350w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/17.4h.png 1288w\" sizes=\"auto, (max-width: 465px) 100vw, 465px\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-2064\" src=\"https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/17.4i-300x24.png\" alt=\"\" width=\"363\" height=\"29\" srcset=\"https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/17.4i-300x24.png 300w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/17.4i-1024x83.png 1024w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/17.4i-768x62.png 768w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/17.4i-65x5.png 65w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/17.4i-225x18.png 225w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/17.4i-350x28.png 350w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/17.4i.png 1107w\" sizes=\"auto, (max-width: 363px) 100vw, 363px\" \/><\/li>\n<li><img loading=\"lazy\" decoding=\"async\" class=\"alignnone wp-image-2065\" src=\"https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/17.4j-300x24.png\" alt=\"\" width=\"363\" height=\"29\" srcset=\"https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/17.4j-300x24.png 300w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/17.4j-1024x80.png 1024w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/17.4j-768x60.png 768w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/17.4j-65x5.png 65w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/17.4j-225x18.png 225w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/17.4j-350x28.png 350w, https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-content\/uploads\/sites\/1463\/2021\/07\/17.4j.png 1120w\" sizes=\"auto, (max-width: 363px) 100vw, 363px\" \/><\/li>\n<li>\u0394<em data-effect=\"italics\">G<\/em> = \u2212<em data-effect=\"italics\">nFE<\/em><sub>cell<\/sub><\/li>\n<li>\u0394<em>G<\/em>\u00b0 = \u2212<em>nFE<\/em>\u00b0<sub>cell<\/sub><\/li>\n<li>w<sub>elec<\/sub> = w<sub>max<\/sub> = \u2212<em>nFE<\/em><sub>cell<\/sub><\/li>\n<\/ul>\n<\/div>\n<div id=\"fs-idp39758576\" class=\"exercises\" data-depth=\"1\">\n<div id=\"fs-idp17461440\" data-type=\"exercise\">\n<div id=\"fs-idp58129232\" data-type=\"solution\">\n<p id=\"fs-idp161071024\">\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox shaded\" data-type=\"glossary\">\n<h3 data-type=\"glossary-title\"><strong>Glossary<\/strong><\/h3>\n<dl id=\"fs-idp94537696\">\n<dt>concentration cell<\/dt>\n<dd id=\"fs-idp112973472\">galvanic cell comprising half-cells of identical composition but for the concentration of one redox reactant or product<\/dd>\n<\/dl>\n<dl id=\"fs-idp41925664\">\n<dt>Faraday\u2019s constant (F)<\/dt>\n<dd id=\"fs-idm11947792\">charge on 1 mol of electrons; <em data-effect=\"italics\">F<\/em> = 96,485 C\/mol e<sup>\u2212<\/sup><\/dd>\n<\/dl>\n<dl id=\"fs-idp211566960\">\n<dt>Nernst equation<\/dt>\n<dd id=\"fs-idp90500496\">relating the potential of a redox system to its composition<\/dd>\n<\/dl>\n<\/div>\n","protected":false},"author":1392,"menu_order":5,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[48],"contributor":[],"license":[],"class_list":["post-882","chapter","type-chapter","status-publish","hentry","chapter-type-numberless"],"part":870,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-json\/pressbooks\/v2\/chapters\/882","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-json\/wp\/v2\/users\/1392"}],"version-history":[{"count":12,"href":"https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-json\/pressbooks\/v2\/chapters\/882\/revisions"}],"predecessor-version":[{"id":2187,"href":"https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-json\/pressbooks\/v2\/chapters\/882\/revisions\/2187"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-json\/pressbooks\/v2\/parts\/870"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-json\/pressbooks\/v2\/chapters\/882\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-json\/wp\/v2\/media?parent=882"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-json\/pressbooks\/v2\/chapter-type?post=882"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-json\/wp\/v2\/contributor?post=882"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/aperrott\/wp-json\/wp\/v2\/license?post=882"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}