# Bridgman thermodynamic formulas

Notation used (from Table I):

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

Bridgman thermodynamic formulas [1]

## Table II

#### pressure

${\displaystyle \left.\partial T\right\vert _{p}=-\left.\partial p\right\vert _{T}=1}$
${\displaystyle \left.\partial V\right\vert _{p}=-\left.\partial p\right\vert _{V}=\left.{\frac {\partial V}{\partial T}}\right\vert _{p}}$
${\displaystyle \left.\partial S\right\vert _{p}=-\left.\partial p\right\vert _{S}=C_{p}/T}$
${\displaystyle \left.\partial Q\right\vert _{p}=-\left.\partial p\right\vert _{Q}=C_{p}}$
${\displaystyle \left.\partial W\right\vert _{p}=-\left.\partial p\right\vert _{W}=p\left.{\frac {\partial V}{\partial T}}\right\vert _{p}}$
${\displaystyle \left.\partial U\right\vert _{p}=-\left.\partial p\right\vert _{U}=C_{p}-p\left.{\frac {\partial V}{\partial T}}\right\vert _{p}}$
${\displaystyle \left.\partial H\right\vert _{p}=-\left.\partial p\right\vert _{H}=C_{p}}$
${\displaystyle \left.\partial G\right\vert _{p}=-\left.\partial p\right\vert _{G}=-S}$
${\displaystyle \left.\partial A\right\vert _{p}=-\left.\partial p\right\vert _{A}=-\left(S+p\left.{\frac {\partial V}{\partial T}}\right\vert _{p}\right)}$

#### temperature

${\displaystyle \left.\partial V\right\vert _{T}=-\left.\partial T\right\vert _{V}=-\left.{\frac {\partial V}{\partial p}}\right\vert _{T}}$
${\displaystyle \left.\partial S\right\vert _{T}=-\left.\partial T\right\vert _{S}=\left.{\frac {\partial V}{\partial T}}\right\vert _{p}}$
${\displaystyle \left.\partial Q\right\vert _{T}=-\left.\partial T\right\vert _{Q}=T\left.{\frac {\partial V}{\partial T}}\right\vert _{p}}$
${\displaystyle \left.\partial W\right\vert _{T}=-\left.\partial T\right\vert _{W}=-p\left.{\frac {\partial V}{\partial p}}\right\vert _{T}}$
${\displaystyle \left.\partial U\right\vert _{T}=-\left.\partial T\right\vert _{U}=T\left.{\frac {\partial V}{\partial T}}\right\vert _{p}+p\left.{\frac {\partial V}{\partial p}}\right\vert _{T}}$
${\displaystyle \left.\partial H\right\vert _{T}=-\left.\partial T\right\vert _{H}=-V+T\left.{\frac {\partial V}{\partial T}}\right\vert _{p}}$
${\displaystyle \left.\partial G\right\vert _{T}=-\left.\partial T\right\vert _{G}=-V}$
${\displaystyle \left.\partial A\right\vert _{T}=-\left.\partial T\right\vert _{A}=p\left.{\frac {\partial V}{\partial p}}\right\vert _{T}}$

#### volume

${\displaystyle \left.\partial S\right\vert _{V}=-\left.\partial V\right\vert _{S}=1/T\left(C_{p}\left.{\frac {\partial V}{\partial p}}\right\vert _{T}+T\left.\left({\frac {\partial V}{\partial T}}\right)^{2}\right\vert _{p}\right)}$
${\displaystyle \left.\partial Q\right\vert _{V}=-\left.\partial V\right\vert _{Q}=C_{p}\left.{\frac {\partial V}{\partial p}}\right\vert _{T}+T\left.\left({\frac {\partial V}{\partial T}}\right)^{2}\right\vert _{p}}$
${\displaystyle \left.\partial W\right\vert _{V}=-\left.\partial V\right\vert _{W}=0}$
${\displaystyle \left.\partial U\right\vert _{V}=-\left.\partial V\right\vert _{U}=C_{p}\left.{\frac {\partial V}{\partial p}}\right\vert _{T}+T\left.\left({\frac {\partial V}{\partial T}}\right)^{2}\right\vert _{p}}$
${\displaystyle \left.\partial H\right\vert _{V}=-\left.\partial V\right\vert _{H}=C_{p}\left.{\frac {\partial V}{\partial p}}\right\vert _{T}+T\left.\left({\frac {\partial V}{\partial T}}\right)^{2}\right\vert _{p}-V\left.{\frac {\partial V}{\partial T}}\right\vert _{p}}$
${\displaystyle \left.\partial G\right\vert _{V}=-\left.\partial V\right\vert _{G}=-\left(V\left.{\frac {\partial V}{\partial T}}\right\vert _{p}+S\left.{\frac {\partial V}{\partial p}}\right\vert _{T}\right)}$
${\displaystyle \left.\partial A\right\vert _{V}=-\left.\partial V\right\vert _{A}=-S\left.{\frac {\partial V}{\partial p}}\right\vert _{T}}$

#### entropy

${\displaystyle \left.\partial Q\right\vert _{S}=-\left.\partial S\right\vert _{Q}=0}$
${\displaystyle \left.\partial W\right\vert _{S}=-\left.\partial S\right\vert _{W}=-(p/T)\left(C_{p}\left.{\frac {\partial V}{\partial p}}\right\vert _{T}+T\left.\left({\frac {\partial V}{\partial T}}\right)^{2}\right\vert _{p}\right)}$
${\displaystyle \left.\partial U\right\vert _{S}=-\left.\partial S\right\vert _{U}=(p/T)\left(C_{p}\left.{\frac {\partial V}{\partial p}}\right\vert _{T}+T\left.\left({\frac {\partial V}{\partial T}}\right)^{2}\right\vert _{p}\right)}$
${\displaystyle \left.\partial H\right\vert _{S}=-\left.\partial S\right\vert _{H}=-VC_{p}/T}$
${\displaystyle \left.\partial G\right\vert _{S}=-\left.\partial S\right\vert _{G}=-(1/T)\left(VC_{p}-ST\left.{\frac {\partial V}{\partial T}}\right\vert _{p}\right)}$
${\displaystyle \left.\partial A\right\vert _{S}=-\left.\partial S\right\vert _{A}=(1/T)\left(p\left(C_{p}\left.{\frac {\partial V}{\partial p}}\right\vert _{T}+T\left.\left({\frac {\partial V}{\partial T}}\right)^{2}\right\vert _{p}\right)+ST\left.{\frac {\partial V}{\partial T}}\right\vert _{p}\right)}$

#### heat

${\displaystyle \left.\partial W\right\vert _{Q}=-\left.\partial Q\right\vert _{W}=-p\left(C_{p}\left.{\frac {\partial V}{\partial p}}\right\vert _{T}+T\left.\left({\frac {\partial V}{\partial T}}\right)^{2}\right\vert _{p}\right)}$
${\displaystyle \left.\partial U\right\vert _{Q}=-\left.\partial Q\right\vert _{U}=p\left(C_{p}\left.{\frac {\partial V}{\partial p}}\right\vert _{T}+T\left.\left({\frac {\partial V}{\partial T}}\right)^{2}\right\vert _{p}\right)}$
${\displaystyle \left.\partial H\right\vert _{Q}=-\left.\partial Q\right\vert _{H}=-VC_{p}}$
${\displaystyle \left.\partial G\right\vert _{Q}=-\left.\partial Q\right\vert _{G}=-\left(ST\left.{\frac {\partial V}{\partial T}}\right\vert _{p}-VC_{p}\right)}$
${\displaystyle \left.\partial A\right\vert _{Q}=-\left.\partial Q\right\vert _{A}=p\left(C_{p}\left.{\frac {\partial V}{\partial p}}\right\vert _{T}+T\left.\left({\frac {\partial V}{\partial T}}\right)^{2}\right\vert _{p}\right)+ST\left.{\frac {\partial V}{\partial T}}\right\vert _{p}}$

#### work

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

#### internal energy

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

#### enthalpy

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

#### Gibbs energy function

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