27 Ideal Gas Properties

Ideal Gas Law

Relates pressure ([latex]P[/latex]), volume ([latex]V[/latex]), temperature ([latex]T[/latex]) and the number of moles ([latex]n[/latex]) of an ideal gas species using the ideal gas constant ([latex]R[/latex]):

Can also relate pressure, molar volume ([latex]\hat{V}[/latex]) and temperature:

The ideal gas law is an approximation that works well under some conditions:

[latex]\hat{V}\; or\; V_{m}=\frac{V}{n} \text{, with units of } \frac{volume}{mol}[/latex]

It is known experimentally that for gases at low density (such that their molecules occupy a negligible fraction of the total volume) and at temperatures well above the boiling point, these proportionalities hold to a good approximation. [latex]^{[1]}[/latex]

There are different ways to estimate how well the ideal gas approximation applies to a system. We will not go over these estimates explicitly in this course. Generally for this course, unless otherwise noted, we will assume the ideal gas law applies. However, in future thermodynamic courses, you will see when the ideal gas approximation may not be appropriate and will see other methods of relating gas properties in non-ideal scenarios.

Ideal Gas Mixtures

Dalton’s Law

If two or more gases are mixed, they will come to a thermal equilibrium as a result of collisions between molecules. When the gases have the same temperature, their molecules have the same average kinetic energy. Thus, each gas obeys the ideal gas law separately and exerts the same pressure on the walls of a container that it would if it were alone.

Therefore, in a mixture of gases, the total pressure is the sum of partial pressures of the component gases, assuming ideal gas behavior and no chemical reactions between the components. [latex]^{[2]}[/latex]

Using Dalton’s law, we can calculate the partial pressure of a gas component, which is defined by the pressure that individual component in the gas mixture exerts on the wall if it were alone.

where

[latex]y_{A}=[/latex] the mole fraction of the gas component [latex]=\frac{\text{moles of the component}}{\text{total moles of gas in the container}}[/latex]

Amgat’s Law

Amgat’s Law is analogous to Dalton’s Law, but is applied to the volume of the gas. The partial volume that each gas occupies will add up to the total system volume.

These partial volumes are proportional to the molar fraction of each gas in a system.

Standard Conditions

A word of caution that standard conditions may not always be the same in different industries or applications. This is shown below with two examples of standard conditions for gases from IUPAC (the International Union of Pure and Applied Chemistry) and NIST (the National Institute of Standards and Technology).

Using the IUPAC numbers we can calculate a standard volume from the ideal gas law. This is the amount of volume a mole of gas would occupy at these standard conditions:

[latex]\hat{V}_{s}=\frac{RT_{s}}{P_{s}}=\frac{8.314\frac{m^3Pa}{Kmol}×273.15K}{1×10^5Pa}×\frac{1000mol}{1kmol}=22.7\frac{m^3(ST\!P)}{kmol}=22.7\frac{L(ST\!P)}{mol}[/latex]

In some applications, gas flows may be expressed in standard cubic metres (SCM) or feet (SCF). A flow of 18.2 SCMH means 18.2 [latex]m^3/h[/latex] at standard conditions (whatever T/P these are defined to be).

We can calculate the temperature, pressure, and volume by comparing fluids at standard and non-standard conditions using the fact that [latex]R[/latex] is constant. We usually use it to compare the same system under different conditions, which means the number of gas molecules (n) may be constant when we perform the calculation.

[latex]R=\frac{V_{s}×P_{s}}{n×T_{s}}=\frac{V×P}{n×T}[/latex]