12.7: The Thermodynamic Functions for an Ideal Gas
- Page ID
- 8634
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)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.