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

${\partial T|}_p=-{\partial p|}_T=1$

${\partial V|}_p=-{\partial p|}_V={\frac{\partial V}{\partial T}|}_p$

${\partial S|}_p=-{\partial p|}_S=C_p/T$

${\partial Q|}_p=-{\partial p|}_Q=C_p$

${\partial W|}_p=-{\partial p|}_W=p{\frac{\partial V}{\partial T}|}_p$

${\partial U|}_p=-{\partial p|}_U=C_p-p{\frac{\partial V}{\partial T}|}_p$

${\partial H|}_p=-{\partial p|}_H=C_p$

${\partial G|}_p=-{\partial p|}_G=-S$

${\partial A|}_p=-{\partial p|}_A=-(S+p{\frac{\partial V}{\partial T}|}_p)$

temperature

${\partial V|}_T=-{\partial T|}_V=-{\frac{\partial V}{\partial p}|}_T$

${\partial S|}_T=-{\partial T|}_S={\frac{\partial V}{\partial T}|}_p$

${\partial Q|}_T=-{\partial T|}_Q=T{\frac{\partial V}{\partial T}|}_p$

${\partial W|}_T=-{\partial T|}_W=-p{\frac{\partial V}{\partial p}|}_T$

${\partial U|}_T=-{\partial T|}_U=T{\frac{\partial V}{\partial T}|}_p+p{\frac{\partial V}{\partial p}|}_T$

${\partial H|}_T=-{\partial T|}_H=-V+T{\frac{\partial V}{\partial T}|}_p$

${\partial G|}_T=-{\partial T|}_G=-V$

${\partial A|}_T=-{\partial T|}_A=p{\frac{\partial V}{\partial p}|}_T$

volume

${\partial S|}_V=-{\partial V|}_S=1/T(C_p{\frac{\partial V}{\partial p}|}_T+T{{(\frac{\partial V}{\partial T})}^2|}_p)$

${\partial Q|}_V=-{\partial V|}_Q=C_p{\frac{\partial V}{\partial p}|}_T+T{{(\frac{\partial V}{\partial T})}^2|}_p$

${\partial W|}_V=-{\partial V|}_W=0$

${\partial U|}_V=-{\partial V|}_U=C_p{\frac{\partial V}{\partial p}|}_T+T{{(\frac{\partial V}{\partial T})}^2|}_p$

${\partial H|}_V=-{\partial V|}_H=C_p{\frac{\partial V}{\partial p}|}_T+T{{(\frac{\partial V}{\partial T})}^2|}_p-V{\frac{\partial V}{\partial T}|}_p$

${\partial G|}_V=-{\partial V|}_G=-(V{\frac{\partial V}{\partial T}|}_p+S{\frac{\partial V}{\partial p}|}_T)$

${\partial A|}_V=-{\partial V|}_A=-S{\frac{\partial V}{\partial p}|}_T$

entropy

${\partial Q|}_S=-{\partial S|}_Q=0$

${\partial W|}_S=-{\partial S|}_W=-(p/T)(C_p{\frac{\partial V}{\partial p}|}_T+T{{(\frac{\partial V}{\partial T})}^2|}_p)$

${\partial U|}_S=-{\partial S|}_U=(p/T)(C_p{\frac{\partial V}{\partial p}|}_T+T{{(\frac{\partial V}{\partial T})}^2|}_p)$

${\partial H|}_S=-{\partial S|}_H=-V{C}_p/T$

${\partial G|}_S=-{\partial S|}_G=-(1/T)(V{C}_p-ST{\frac{\partial V}{\partial T}|}_p)$

${\partial A|}_S=-{\partial S|}_A=(1/T)(p(C_p{\frac{\partial V}{\partial p}|}_T+T{{(\frac{\partial V}{\partial T})}^2|}_p)+ST{\frac{\partial V}{\partial T}|}_p)$

heat

${\partial W|}_Q=-{\partial Q|}_W=-p(C_p{\frac{\partial V}{\partial p}|}_T+T{{(\frac{\partial V}{\partial T})}^2|}_p)$

${\partial U|}_Q=-{\partial Q|}_U=p(C_p{\frac{\partial V}{\partial p}|}_T+T{{(\frac{\partial V}{\partial T})}^2|}_p)$

${\partial H|}_Q=-{\partial Q|}_H=-V{C}_p$

${\partial G|}_Q=-{\partial Q|}_G=-(ST{\frac{\partial V}{\partial T}|}_p-V{C}_p)$

${\partial A|}_Q=-{\partial Q|}_A=p(C_p{\frac{\partial V}{\partial p}|}_T+T{{(\frac{\partial V}{\partial T})}^2|}_p)+ST{\frac{\partial V}{\partial T}|}_p$

work

${\partial U|}_W=-{\partial W|}_U=p(C_p{\frac{\partial V}{\partial p}|}_T+T{{(\frac{\partial V}{\partial T})}^2|}_p)$

${\partial H|}_W=-{\partial W|}_H=p(C_p{\frac{\partial V}{\partial p}|}_T+T{{(\frac{\partial V}{\partial T})}^2|}_p-V{\frac{\partial V}{\partial T}|}_p)$

${\partial G|}_W=-{\partial W|}_G=-p(V{\frac{\partial V}{\partial p}|}_T+S{\frac{\partial V}{\partial p}|}_T)$

${\partial A|}_W=-{\partial W|}_A=-pS{\frac{\partial V}{\partial p}|}_T$

internal energy

${\partial H|}_U=-{\partial U|}_H=-V(C_p-p{\frac{\partial V}{\partial T}|}_p)-p(C_p{\frac{\partial V}{\partial p}|}_T+T{{(\frac{\partial V}{\partial T})}^2|}_p)$

${\partial G|}_U=-{\partial U|}_G=-V(C_p-p{\frac{\partial V}{\partial T}|}_p)+S(T{\frac{\partial V}{\partial T}|}_p+p{\frac{\partial V}{\partial p}|}_T)$

${\partial A|}_U=-{\partial U|}_A=p(C_p{\frac{\partial V}{\partial p}|}_T+T{{(\frac{\partial V}{\partial T})}^2|}_p)$

enthalpy

${\partial G|}_H=-{\partial H|}_G=-V(C_p+S)+TS{\frac{\partial V}{\partial T}|}_p$

${\partial A|}_H=-{\partial H|}_A=-(S+p{\frac{\partial V}{\partial T}|}_p)(V-T{\frac{\partial V}{\partial T}|}_p)+p{\frac{\partial V}{\partial p}|}_T$

Gibbs energy function

${\partial A|}_G=-{\partial G|}_A=-S(V+p{\frac{\partial V}{\partial p}|}_T)-pV{\frac{\partial V}{\partial T}|}_p$

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)