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12.7: The Thermodynamic Functions for an Ideal Gas

Last updated

Sep 10, 2020
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- 12.6: The Joule and Joule-Thomson Coefficients
- 12.8: The Thermodynamic Functions for Other Substances

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In this section I tabulate the changes in the thermodynamic functions for an ideal gas taken from one state to another.

One mole of an ideal gas going isothermally and reversibly from P₁V₁T to P₂V₂T or adiabatically and reversibly from P₁V₁T₁ to P₂V₂T₂.

	Isothermal	Adiabatic
Work done by gas	RT ln(V₂/V₁)*	$\frac{P_{1} V_{1}-P_{2} V_{2}}{\gamma-1}=\frac{R\left(T_{1}-T_{2}\right)}{\gamma-1}=C_{V}\left(T_{1}-T_{2}\right)$
U₂ − U₁	0	$-\frac{P_{1} V_{1}-P_{2} V_{2}}{\gamma-1}=-\frac{R\left(T_{1}-T_{2}\right)}{\gamma-1}=-C_{V}\left(T_{1}-T_{2}\right)$
Heat absorbed by gas	RT ln(V₂/V₁)	0
S₂ − S₁	R ln(V₂/V₁)	0
H₂ − H₁	0	$-\frac{P_{1} V_{1}-P_{2} V_{2}}{1-1 / \gamma}=-\frac{R\left(T_{1}-T_{2}\right)}{1-1 / \gamma}=-C_{P}\left(T_{1}-T_{2}\right)$
A₂ − A₁	−RT ln(V₂/V₁)	$-\frac{R\left(T_{1}-T_{2}\right)}{\gamma-1}-T_{2} S_{2}+T_{1} S_{1}$
G₂ − G₁	−RT ln(V₂/V₁)	$-\frac{R\left(T_{1}-T_{2}\right)}{1-1 / \gamma}-T_{2} S_{2}+T_{1} S_{1}$

*Note that for isothermal processes on an ideal gas, we can write (V₂/V₁) = (P₁/P₂).

A difficulty will be noted in the entries for the increase in the Helmholtz and Gibbs functions for an adiabatic process, in that, in order to calculate ∆A or ∆G, it is apparently necessary to know S₁ and S₂, and not merely their difference. For the time being this is a difficulty to note on one’s shirt-cuff, and perhaps return to it later.

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