# 12.7: The Thermodynamic Functions for an Ideal Gas

- Page ID
- 8634

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 P_{1}V_{1}T to P_{2}V_{2}T or adiabatically and reversibly from P_{1}V_{1}T_{1} to P_{2}V_{2}T_{2}.**

Isothermal | Adiabatic | |

Work done by gas |
RT ln(V_{2}/V_{1})* |
\(\frac{P_{1} V_{1}-P_{2} V_{2}}{\gamma-1}=\frac{R\left(T_{1}-T_{2}\right)}{\gamma-1}=C_{V}\left(T_{1}-T_{2}\right)\) |

U_{2} − U_{1} |
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(V_{2}/V_{1}) |
0 |

S_{2} − S_{1} |
R ln(V_{2}/V_{1}) |
0 |

H_{2} − H_{1} |
0 | \(-\frac{P_{1} V_{1}-P_{2} V_{2}}{1-1 / \gamma}=-\frac{R\left(T_{1}-T_{2}\right)}{1-1 / \gamma}=-C_{P}\left(T_{1}-T_{2}\right)\) |

A_{2} − A_{1} |
−RT ln(V_{2}/V_{1}) |
\(-\frac{R\left(T_{1}-T_{2}\right)}{\gamma-1}-T_{2} S_{2}+T_{1} S_{1}\) |

G_{2} − G_{1} |
−RT ln(V_{2}/V_{1}) |
\(-\frac{R\left(T_{1}-T_{2}\right)}{1-1 / \gamma}-T_{2} S_{2}+T_{1} S_{1}\) |

*Note that for isothermal processes on an ideal gas, we can write (*V*_{2}/*V*_{1}) = (*P*_{1}/*P*_{2}).

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 *S*_{1} and *S*_{2}, 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.