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Math, physics, perl, and programming obscurity.

Thermodynamic identities

Posted by peeterjoot on March 7, 2013

[Click here for a PDF of this post with nicer formatting (especially if my latex to wordpress script has left FORMULA DOES NOT PARSE errors.)]

Impressed with the clarity of Baez’s entropic force discussion on differential forms [1], let’s use that methodology to find all the possible identities that we can get from the thermodynamic identity (for now assuming N is fixed, ignoring the chemical potential.)

This isn’t actually that much work to do, since a bit of editor regular expression magic can do most of the work.

Our starting point is the thermodynamic identity

\begin{aligned}dU = d Q + d W = T dS - P dV,\end{aligned} \hspace{\stretch{1}}(1.0.1)

or

\begin{aligned}0 = dU - T dS + P dV.\end{aligned} \hspace{\stretch{1}}(1.0.2)

It’s quite likely that many of the identities that can be obtained will be useful, but this should at least provide a handy reference of possible conversions.

Differentials in P, V

This first case illustrates the method.

\begin{aligned}0 &= dU - T dS + P dV \\ &= \left( \frac{\partial {U}}{\partial {P}} \right)_{V} dP +\left( \frac{\partial {U}}{\partial {V}} \right)_{P} dV- T\left(  \left( \frac{\partial {S}}{\partial {P}} \right)_{V} dP + \left( \frac{\partial {S}}{\partial {V}} \right)_{P} dV  \right)+ P dV \\ &= dP \left(  \left( \frac{\partial {U}}{\partial {P}} \right)_{V} - T \left( \frac{\partial {S}}{\partial {P}} \right)_{V}  \right)+dV \left(  \left( \frac{\partial {U}}{\partial {V}} \right)_{P} - T \left( \frac{\partial {S}}{\partial {V}} \right)_{P} + P  \right).\end{aligned} \hspace{\stretch{1}}(1.0.3)

Taking wedge products with dV and dP respectively, we form two two forms

\begin{aligned}0 = dP \wedge dV \left(  \left( \frac{\partial {U}}{\partial {P}} \right)_{V} - T \left( \frac{\partial {S}}{\partial {P}} \right)_{V}  \right)\end{aligned} \hspace{\stretch{1}}(1.0.4a)

\begin{aligned}0 = dV \wedge dP \left(  \left( \frac{\partial {U}}{\partial {V}} \right)_{P} - T \left( \frac{\partial {S}}{\partial {V}} \right)_{P} + P  \right).\end{aligned} \hspace{\stretch{1}}(1.0.4b)

Since these must both be zero we find

\begin{aligned}\left( \frac{\partial {U}}{\partial {P}} \right)_{V} = T \left( \frac{\partial {S}}{\partial {P}} \right)_{V}\end{aligned} \hspace{\stretch{1}}(1.0.5a)

\begin{aligned}P =-\left( \frac{\partial {U}}{\partial {V}} \right)_{P}- T \left( \frac{\partial {S}}{\partial {V}} \right)_{P}.\end{aligned} \hspace{\stretch{1}}(1.0.5b)

Differentials in P, T

\begin{aligned}0 &= dU - T dS + P dV \\ &= \left( \frac{\partial {U}}{\partial {P}} \right)_{T} dP + \left( \frac{\partial {U}}{\partial {T}} \right)_{P} dT-T \left(  \left( \frac{\partial {S}}{\partial {P}} \right)_{T} dP + \left( \frac{\partial {S}}{\partial {T}} \right)_{P} dT  \right)+\left( \frac{\partial {V}}{\partial {P}} \right)_{T} dP + \left( \frac{\partial {V}}{\partial {T}} \right)_{P} dT,\end{aligned} \hspace{\stretch{1}}(1.0.6)

or

\begin{aligned}0 = \left( \frac{\partial {U}}{\partial {P}} \right)_{T} -T \left( \frac{\partial {S}}{\partial {P}} \right)_{T} + \left( \frac{\partial {V}}{\partial {P}} \right)_{T}\end{aligned} \hspace{\stretch{1}}(1.0.7a)

\begin{aligned}0 = \left( \frac{\partial {U}}{\partial {T}} \right)_{P} -T \left( \frac{\partial {S}}{\partial {T}} \right)_{P} + \left( \frac{\partial {V}}{\partial {T}} \right)_{P}.\end{aligned} \hspace{\stretch{1}}(1.0.7b)

Differentials in P, S

\begin{aligned}0 &= dU - T dS + P dV \\ &= \left( \frac{\partial {U}}{\partial {P}} \right)_{S} dP + \left( \frac{\partial {U}}{\partial {S}} \right)_{P} dS- T dS+ P \left(  \left( \frac{\partial {V}}{\partial {P}} \right)_{S} dP + \left( \frac{\partial {V}}{\partial {S}} \right)_{P} dS  \right),\end{aligned} \hspace{\stretch{1}}(1.0.8)

or

\begin{aligned}\left( \frac{\partial {U}}{\partial {P}} \right)_{S} = -P \left( \frac{\partial {V}}{\partial {P}} \right)_{S}\end{aligned} \hspace{\stretch{1}}(1.0.9a)

\begin{aligned}T = \left( \frac{\partial {U}}{\partial {S}} \right)_{P} + P \left( \frac{\partial {V}}{\partial {S}} \right)_{P}.\end{aligned} \hspace{\stretch{1}}(1.0.9b)

Differentials in P, U

\begin{aligned}0 &= dU - T dS + P dV \\ &= dU - T \left(  \left( \frac{\partial {S}}{\partial {P}} \right)_{U} dP + \left( \frac{\partial {S}}{\partial {U}} \right)_{P} dU  \right)+ P\left(  \left( \frac{\partial {V}}{\partial {P}} \right)_{U} dP + \left( \frac{\partial {V}}{\partial {U}} \right)_{P} dU  \right),\end{aligned} \hspace{\stretch{1}}(1.0.10)

or

\begin{aligned}0 = 1 - T \left( \frac{\partial {S}}{\partial {U}} \right)_{P} + P \left( \frac{\partial {V}}{\partial {U}} \right)_{P} \end{aligned} \hspace{\stretch{1}}(1.0.11a)

\begin{aligned}T \left( \frac{\partial {S}}{\partial {P}} \right)_{U} = P \left( \frac{\partial {V}}{\partial {P}} \right)_{U}.\end{aligned} \hspace{\stretch{1}}(1.0.11b)

Differentials in V, T

\begin{aligned}0 &= dU - T dS + P dV \\ &= \left( \frac{\partial {U}}{\partial {V}} \right)_{T} dV + \left( \frac{\partial {U}}{\partial {T}} \right)_{V} dT - T \left(  \left( \frac{\partial {S}}{\partial {V}} \right)_{T} dV + \left( \frac{\partial {S}}{\partial {T}} \right)_{V} dT  \right)+ P dV,\end{aligned} \hspace{\stretch{1}}(1.0.12)

or

\begin{aligned}0 = \left( \frac{\partial {U}}{\partial {V}} \right)_{T} - T \left( \frac{\partial {S}}{\partial {V}} \right)_{T} + P \end{aligned} \hspace{\stretch{1}}(1.0.13a)

\begin{aligned}\left( \frac{\partial {U}}{\partial {T}} \right)_{V} = T \left( \frac{\partial {S}}{\partial {T}} \right)_{V}.\end{aligned} \hspace{\stretch{1}}(1.0.13b)

Differentials in V, S

\begin{aligned}0 &= dU - T dS + P dV \\ &= \left( \frac{\partial {U}}{\partial {V}} \right)_{S} dV + \left( \frac{\partial {U}}{\partial {S}} \right)_{V} dS - T dS+ P dV,\end{aligned} \hspace{\stretch{1}}(1.0.14)

or

\begin{aligned}P = -\left( \frac{\partial {U}}{\partial {V}} \right)_{S}\end{aligned} \hspace{\stretch{1}}(1.0.15a)

\begin{aligned}T = \left( \frac{\partial {U}}{\partial {S}} \right)_{V} .\end{aligned} \hspace{\stretch{1}}(1.0.15b)

Differentials in V, U

\begin{aligned}0 &= dU - T dS + P dV \\ &= dU- T \left(  \left( \frac{\partial {S}}{\partial {V}} \right)_{U} dV + \left( \frac{\partial {S}}{\partial {U}} \right)_{V} dU  \right)+ P \left(  \left( \frac{\partial {V}}{\partial {V}} \right)_{U} dV + \left( \frac{\partial {V}}{\partial {U}} \right)_{V} dU  \right)\end{aligned} \hspace{\stretch{1}}(1.0.16)

or

\begin{aligned}0 = 1 - T \left( \frac{\partial {S}}{\partial {U}} \right)_{V} + P \left( \frac{\partial {V}}{\partial {U}} \right)_{V} \end{aligned} \hspace{\stretch{1}}(1.0.17a)

\begin{aligned}T \left( \frac{\partial {S}}{\partial {V}} \right)_{U} = P \left( \frac{\partial {V}}{\partial {V}} \right)_{U}.\end{aligned} \hspace{\stretch{1}}(1.0.17b)

Differentials in S, T

\begin{aligned}0 &= dU - T dS + P dV \\ &= \left(  \left( \frac{\partial {U}}{\partial {S}} \right)_{T} dS + \left( \frac{\partial {U}}{\partial {T}} \right)_{S} dT  \right)- T dS+ P \left(  \left( \frac{\partial {V}}{\partial {S}} \right)_{T} dS + \left( \frac{\partial {V}}{\partial {T}} \right)_{S} dT  \right),\end{aligned} \hspace{\stretch{1}}(1.0.18)

or

\begin{aligned}0 = \left( \frac{\partial {U}}{\partial {S}} \right)_{T} - T + P \left( \frac{\partial {V}}{\partial {S}} \right)_{T} \end{aligned} \hspace{\stretch{1}}(1.0.19a)

\begin{aligned}0 = \left( \frac{\partial {U}}{\partial {T}} \right)_{S} + P \left( \frac{\partial {V}}{\partial {T}} \right)_{S}.\end{aligned} \hspace{\stretch{1}}(1.0.19b)

Differentials in S, U

\begin{aligned}0 &= dU - T dS + P dV \\ &= dU - T dS+ P \left(  \left( \frac{\partial {V}}{\partial {S}} \right)_{U} dS + \left( \frac{\partial {V}}{\partial {U}} \right)_{S} dU  \right)\end{aligned} \hspace{\stretch{1}}(1.0.20)

or

\begin{aligned}\frac{1}{{P}} = - \left( \frac{\partial {V}}{\partial {U}} \right)_{S} \end{aligned} \hspace{\stretch{1}}(1.0.21a)

\begin{aligned}T = P \left( \frac{\partial {V}}{\partial {S}} \right)_{U}.\end{aligned} \hspace{\stretch{1}}(1.0.21b)

Differentials in T, U

\begin{aligned}0 &= dU - T dS + P dV \\ &= dU - T \left(  \left( \frac{\partial {S}}{\partial {T}} \right)_{U} dT + \left( \frac{\partial {S}}{\partial {U}} \right)_{T} dU  \right)+ P\left(  \left( \frac{\partial {V}}{\partial {T}} \right)_{U} dT + \left( \frac{\partial {V}}{\partial {U}} \right)_{T} dU  \right),\end{aligned} \hspace{\stretch{1}}(1.0.22)

or

\begin{aligned}0 = 1 - T \left( \frac{\partial {S}}{\partial {U}} \right)_{T} + P \left( \frac{\partial {V}}{\partial {U}} \right)_{T} \end{aligned} \hspace{\stretch{1}}(1.0.23a)

\begin{aligned}T \left( \frac{\partial {S}}{\partial {T}} \right)_{U} = P \left( \frac{\partial {V}}{\partial {T}} \right)_{U}.\end{aligned} \hspace{\stretch{1}}(1.0.23b)

References

[1] John Baez. Entropic forces, 2012. URL http://johncarlosbaez.wordpress.com/2012/02/01/entropic-forces/. [Online; accessed 07-March-2013].

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