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

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

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

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

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

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

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

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 (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left. \partial H \right\vert_Q = - \left. \partial Q \right\vert_H = -VC_p }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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) }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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) }

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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left. \partial A \right\vert_W = - \left. \partial W \right\vert_A = -pS \left. \frac{\partial V}{\partial p} \right\vert_T }

internal energy

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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) }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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 (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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 (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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)