Bridgman thermodynamic formulas

Notation used (from Table I):

• $p$ is the pressure.
• $T$ is the temperature (in Kelvin).
• $V$ is the volume.
• $S$ is the entropy.
• $Q$ is the heat.
• $W$ is the work.
• $U$ is the internal energy.
• $H$ is the enthalpy
• $G$ is the Gibbs energy function
• $A$ is the Helmholtz energy function.

Bridgman thermodynamic formulas

Table II

pressure

$\left.\partial T\right\vert _{p}=-\left.\partial p\right\vert _{T}=1$

$\left.\partial V\right\vert _{p}=-\left.\partial p\right\vert _{V}=\left.{\frac {\partial V}{\partial T}}\right\vert _{p}$

$\left.\partial S\right\vert _{p}=-\left.\partial p\right\vert _{S}=C_{p}/T$

$\left.\partial Q\right\vert _{p}=-\left.\partial p\right\vert _{Q}=C_{p}$

$\left.\partial W\right\vert _{p}=-\left.\partial p\right\vert _{W}=p\left.{\frac {\partial V}{\partial T}}\right\vert _{p}$

$\left.\partial U\right\vert _{p}=-\left.\partial p\right\vert _{U}=C_{p}-p\left.{\frac {\partial V}{\partial T}}\right\vert _{p}$

$\left.\partial H\right\vert _{p}=-\left.\partial p\right\vert _{H}=C_{p}$

$\left.\partial G\right\vert _{p}=-\left.\partial p\right\vert _{G}=-S$

$\left.\partial A\right\vert _{p}=-\left.\partial p\right\vert _{A}=-\left(S+p\left.{\frac {\partial V}{\partial T}}\right\vert _{p}\right)$

temperature

$\left.\partial V\right\vert _{T}=-\left.\partial T\right\vert _{V}=-\left.{\frac {\partial V}{\partial p}}\right\vert _{T}$

$\left.\partial S\right\vert _{T}=-\left.\partial T\right\vert _{S}=\left.{\frac {\partial V}{\partial T}}\right\vert _{p}$

$\left.\partial Q\right\vert _{T}=-\left.\partial T\right\vert _{Q}=T\left.{\frac {\partial V}{\partial T}}\right\vert _{p}$

$\left.\partial W\right\vert _{T}=-\left.\partial T\right\vert _{W}=-p\left.{\frac {\partial V}{\partial p}}\right\vert _{T}$

$\left.\partial U\right\vert _{T}=-\left.\partial T\right\vert _{U}=T\left.{\frac {\partial V}{\partial T}}\right\vert _{p}+p\left.{\frac {\partial V}{\partial p}}\right\vert _{T}$

$\left.\partial H\right\vert _{T}=-\left.\partial T\right\vert _{H}=-V+T\left.{\frac {\partial V}{\partial T}}\right\vert _{p}$

$\left.\partial G\right\vert _{T}=-\left.\partial T\right\vert _{G}=-V$

$\left.\partial A\right\vert _{T}=-\left.\partial T\right\vert _{A}=p\left.{\frac {\partial V}{\partial p}}\right\vert _{T}$

volume

$\left.\partial S\right\vert _{V}=-\left.\partial V\right\vert _{S}=1/T\left(C_{p}\left.{\frac {\partial V}{\partial p}}\right\vert _{T}+T\left.\left({\frac {\partial V}{\partial T}}\right)^{2}\right\vert _{p}\right)$

$\left.\partial Q\right\vert _{V}=-\left.\partial V\right\vert _{Q}=C_{p}\left.{\frac {\partial V}{\partial p}}\right\vert _{T}+T\left.\left({\frac {\partial V}{\partial T}}\right)^{2}\right\vert _{p}$

$\left.\partial W\right\vert _{V}=-\left.\partial V\right\vert _{W}=0$

$\left.\partial U\right\vert _{V}=-\left.\partial V\right\vert _{U}=C_{p}\left.{\frac {\partial V}{\partial p}}\right\vert _{T}+T\left.\left({\frac {\partial V}{\partial T}}\right)^{2}\right\vert _{p}$

$\left.\partial H\right\vert _{V}=-\left.\partial V\right\vert _{H}=C_{p}\left.{\frac {\partial V}{\partial p}}\right\vert _{T}+T\left.\left({\frac {\partial V}{\partial T}}\right)^{2}\right\vert _{p}-V\left.{\frac {\partial V}{\partial T}}\right\vert _{p}$

$\left.\partial G\right\vert _{V}=-\left.\partial V\right\vert _{G}=-\left(V\left.{\frac {\partial V}{\partial T}}\right\vert _{p}+S\left.{\frac {\partial V}{\partial p}}\right\vert _{T}\right)$

$\left.\partial A\right\vert _{V}=-\left.\partial V\right\vert _{A}=-S\left.{\frac {\partial V}{\partial p}}\right\vert _{T}$

entropy

$\left.\partial Q\right\vert _{S}=-\left.\partial S\right\vert _{Q}=0$

$\left.\partial W\right\vert _{S}=-\left.\partial S\right\vert _{W}=-(p/T)\left(C_{p}\left.{\frac {\partial V}{\partial p}}\right\vert _{T}+T\left.\left({\frac {\partial V}{\partial T}}\right)^{2}\right\vert _{p}\right)$

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \left.\partial U\right\vert _{S}=-\left.\partial S\right\vert _{U}=(p/T)\left(C_{p}\left.{\frac {\partial V}{\partial p}}\right\vert _{T}+T\left.\left({\frac {\partial V}{\partial T}}\right)^{2}\right\vert _{p}\right)}

$\left.\partial H\right\vert _{S}=-\left.\partial S\right\vert _{H}=-VC_{p}/T$

$\left.\partial G\right\vert _{S}=-\left.\partial S\right\vert _{G}=-(1/T)\left(VC_{p}-ST\left.{\frac {\partial V}{\partial T}}\right\vert _{p}\right)$

$\left.\partial A\right\vert _{S}=-\left.\partial S\right\vert _{A}=(1/T)\left(p\left(C_{p}\left.{\frac {\partial V}{\partial p}}\right\vert _{T}+T\left.\left({\frac {\partial V}{\partial T}}\right)^{2}\right\vert _{p}\right)+ST\left.{\frac {\partial V}{\partial T}}\right\vert _{p}\right)$

heat

$\left.\partial W\right\vert _{Q}=-\left.\partial Q\right\vert _{W}=-p\left(C_{p}\left.{\frac {\partial V}{\partial p}}\right\vert _{T}+T\left.\left({\frac {\partial V}{\partial T}}\right)^{2}\right\vert _{p}\right)$

$\left.\partial U\right\vert _{Q}=-\left.\partial Q\right\vert _{U}=p\left(C_{p}\left.{\frac {\partial V}{\partial p}}\right\vert _{T}+T\left.\left({\frac {\partial V}{\partial T}}\right)^{2}\right\vert _{p}\right)$

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \left.\partial H\right\vert _{Q}=-\left.\partial Q\right\vert _{H}=-VC_{p}}

$\left.\partial G\right\vert _{Q}=-\left.\partial Q\right\vert _{G}=-\left(ST\left.{\frac {\partial V}{\partial T}}\right\vert _{p}-VC_{p}\right)$

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \left.\partial A\right\vert _{Q}=-\left.\partial Q\right\vert _{A}=p\left(C_{p}\left.{\frac {\partial V}{\partial p}}\right\vert _{T}+T\left.\left({\frac {\partial V}{\partial T}}\right)^{2}\right\vert _{p}\right)+ST\left.{\frac {\partial V}{\partial T}}\right\vert _{p}}

work

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \left.\partial U\right\vert _{W}=-\left.\partial W\right\vert _{U}=p\left(C_{p}\left.{\frac {\partial V}{\partial p}}\right\vert _{T}+T\left.\left({\frac {\partial V}{\partial T}}\right)^{2}\right\vert _{p}\right)}

$\left.\partial H\right\vert _{W}=-\left.\partial W\right\vert _{H}=p\left(C_{p}\left.{\frac {\partial V}{\partial p}}\right\vert _{T}+T\left.\left({\frac {\partial V}{\partial T}}\right)^{2}\right\vert _{p}-V\left.{\frac {\partial V}{\partial T}}\right\vert _{p}\right)$

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \left.\partial G\right\vert _{W}=-\left.\partial W\right\vert _{G}=-p\left(V\left.{\frac {\partial V}{\partial p}}\right\vert _{T}+S\left.{\frac {\partial V}{\partial p}}\right\vert _{T}\right)}

$\left.\partial A\right\vert _{W}=-\left.\partial W\right\vert _{A}=-pS\left.{\frac {\partial V}{\partial p}}\right\vert _{T}$

internal energy

$\left.\partial H\right\vert _{U}=-\left.\partial U\right\vert _{H}=-V\left(C_{p}-p\left.{\frac {\partial V}{\partial T}}\right\vert _{p}\right)-p\left(C_{p}\left.{\frac {\partial V}{\partial p}}\right\vert _{T}+T\left.\left({\frac {\partial V}{\partial T}}\right)^{2}\right\vert _{p}\right)$

$\left.\partial G\right\vert _{U}=-\left.\partial U\right\vert _{G}=-V\left(C_{p}-p\left.{\frac {\partial V}{\partial T}}\right\vert _{p}\right)+S\left(T\left.{\frac {\partial V}{\partial T}}\right\vert _{p}+p\left.{\frac {\partial V}{\partial p}}\right\vert _{T}\right)$

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \left.\partial A\right\vert _{U}=-\left.\partial U\right\vert _{A}=p\left(C_{p}\left.{\frac {\partial V}{\partial p}}\right\vert _{T}+T\left.\left({\frac {\partial V}{\partial T}}\right)^{2}\right\vert _{p}\right)}

enthalpy

$\left.\partial G\right\vert _{H}=-\left.\partial H\right\vert _{G}=-V(C_{p}+S)+TS\left.{\frac {\partial V}{\partial T}}\right\vert _{p}$

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \left.\partial A\right\vert _{H}=-\left.\partial H\right\vert _{A}=-\left(S+p\left.{\frac {\partial V}{\partial T}}\right\vert _{p}\right)\left(V-T\left.{\frac {\partial V}{\partial T}}\right\vert _{p}\right)+p\left.{\frac {\partial V}{\partial p}}\right\vert _{T}}

Gibbs energy function

Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle \left.\partial A\right\vert _{G}=-\left.\partial G\right\vert _{A}=-S\left(V+p\left.{\frac {\partial V}{\partial p}}\right\vert _{T}\right)-pV\left.{\frac {\partial V}{\partial T}}\right\vert _{p}}

See also

• Thermodynamic relations
• Maxwell's relations

References

Related reading

• James B. Cooper and T. Russell "On the Mathematics of Thermodynamics", arXiv:1102.1540v1 Tue, 8 Feb (2011)
• James B. Cooper "Thermodynamical identities - a systematic approach", arXiv:1108.4760v1 Wed, 24 Aug (2011)