Bridgman's thermodynamic equations

In Thermodynamics, Bridgman's Thermodynamic equations is actually a method of generating a large number of thermodynamic identities involving a number of thermodynamic quantities. Some of the most common thermodynamic quantites are:

Internal energy	U
Helmholtz free energy	F
Gibbs free energy	G
Enthalpy	H
Particle number	N
Pressure	P
Density	ρ
Entropy	S
Temperature	T
Specific heat (constant volume)	C_V
Specific heat (constant pressure)	C_P
Volume	V

Many thermodynamic equations are expressed in terms of partial derivatives. For example, the expression for the specific heat at constant volume is:

$C_P=\left(\frac{\partial H}{\partial T}\right)_P$

which is the partial derivative of the internal energy with respect to temperature while holding volume constant. We may write this equation as:

$C_P=\frac{(\partial H)_P}{(\partial T)_P}$

This method of rewriting the partial derivative was described by Bridgman (and also Lewis &; Randall), and allows the use of the following collection of expressions to express many thermodynamic equations. For example in the equations below we have:

$(\partial H)_P=C_P$

and

$(\partial T)_P=1$

Dividing, we recover the proper expression for C_P.

The following summary restates various partials in terms of S, T, P, and the following three derivatives which are easily measured experimentally.

$\left(\frac{\partial V}{\partial T}\right)_P,~~~ \left(\frac{\partial V}{\partial P}\right)_T,~~~ \left(\frac{\partial H}{\partial T}\right)_P (=C_P)$

Bridgman's thermodynamic equations

$(\partial T)_P=-(\partial P)_T=1$

$(\partial V)_P=-(\partial P)_V=\left(\frac{\partial V}{\partial T}\right)_P$

$(\partial S)_P=-(\partial P)_S=\frac{C_p}{T}$

$(\partial U)_P=-(\partial P)_U=C_P-P\left(\frac{\partial V}{\partial T}\right)_P$

$(\partial H)_P=-(\partial P)_H=C_P$

$(\partial G)_P=-(\partial P)_G=-S$

$(\partial F)_P=-(\partial P)_F=-S-P\left(\frac{\partial V}{\partial T}\right)_P$

$(\partial V)_T=-(\partial T)_V=-\left(\frac{\partial V}{\partial P}\right)_T$

$(\partial S)_T=-(\partial T)_S=\left(\frac{\partial V}{\partial T}\right)_P$

$(\partial U)_T=-(\partial T)_U=T\left(\frac{\partial V}{\partial T}\right)_P+P\left(\frac{\partial V}{\partial P}\right)_T$

$(\partial H)_T=-(\partial T)_H=-V+T\left(\frac{\partial V}{\partial }\right)_P$

$(\partial G)_T=-(\partial T)_G=-V$

$(\partial F)_T=-(\partial T)_F=P\left(\frac{\partial V}{\partial P}\right)_T$

$(\partial S)_V=-(\partial V)_S=\frac{C_P}{T}\left(\frac{\partial V}{\partial P}\right)_T+\left(\frac{\partial V}{\partial T}\right)_P^2$

$(\partial U)_V=-(\partial V)_U=C_P\left(\frac{\partial V}{\partial P}\right)_T+T\left(\frac{\partial V}{\partial T}\right)_P^2$

$(\partial H)_V=-(\partial V)_H=C_P\left(\frac{\partial V}{\partial P}\right)_T+T\left(\frac{\partial V}{\partial T}\right)_P^2-V\left(\frac{\partial V}{\partial T}\right)_P$

$(\partial G)_V=-(\partial V)_G=-V\left(\frac{\partial V}{\partial T}\right)_P-S\left(\frac{\partial V}{\partial P}\right)_T$

$(\partial F)_V=-(\partial V)_F=-S\left(\frac{\partial V}{\partial P}\right)_T$

$(\partial U)_S=-(\partial S)_U=\frac{PC_P}{T}\left(\frac{\partial V}{\partial P}\right)_T+P\left(\frac{\partial V}{\partial T}\right)_P^2$

$(\partial H)_S=-(\partial S)_H=-\frac{VC_P}{T}$

$(\partial G)_S=-(\partial S)_G=-\frac{VC_P}{T}+S\left(\frac{\partial V}{\partial T}\right)_P$

$(\partial F)_S=-(\partial S)_F=\frac{PC_P}{T}\left(\frac{\partial V}{\partial P}\right)_T+P\left(\frac{\partial V}{\partial T}\right)_P^2+S\left(\frac{\partial V}{\partial T}\right)_P$

$(\partial H)_U=-(\partial U)_H=-VC_P+PV\left(\frac{\partial V}{\partial T}\right)_P-PC_P\left(\frac{\partial V}{\partial P}\right)_T-PT\left(\frac{\partial V}{\partial T}\right)_P^2$

$(\partial G)_U=-(\partial U)_G=-VC_P+PV\left(\frac{\partial V}{\partial T}\right)_P+ST\left(\frac{\partial V}{\partial T}\right)_P+SP\left(\frac{\partial V}{\partial P}\right)_T$

$(\partial F)_U=-(\partial U)_F=-P(C_P+S)\left(\frac{\partial V}{\partial P}\right)_T+PT\left(\frac{\partial V}{\partial T}\right)_P^2+ST\left(\frac{\partial V}{\partial T}\right)_P$

$(\partial G)_H=-(\partial H)_G=-V(C_P+S)+TS\left(\frac{\partial V}{\partial T}\right)_P$

$(\partial F)_H=-(\partial H)_F=\left[S+P\left(\frac{\partial V}{\partial T}\right)_P\right]\left[V-T\left(\frac{\partial V}{\partial T}\right)_P\right]+PC_P\left(\frac{\partial V}{\partial P}\right)_T$

$(\partial F)_G=-(\partial G)_F=-SV-_P\left(\frac{\partial V}{\partial P}\right)_T-PV\left(\frac{\partial V}{\partial T}\right)_P$

References

Bridgman, P.W., Phys. Rev., 3, 273 (1914).

Lewis, G.N., and Randall, M., "Thermodynamics", 2nd Edition, McGraw-Hill Book Company, New York, 1961.

Categories: Thermodynamics