Bridgman's thermodynamic equations

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In thermodynamics, Bridgman's thermodynamic equations are the basic set of thermodynamic equations, derived using a method of generating a large number of thermodynamic identities involving a number of thermodynamic quantities. The equations are named after the American physicist Percy Williams Bridgman. (See also the exact differential article for general differential relationships) Some of the most common thermodynamic quantities are:

Internal energy	U
Helmholtz free energy	A
Gibbs free energy	G
Enthalpy	H
Particle number	N
Pressure	P
Density	ρ
Entropy	S
Temperature	T
heat capacity (constant volume)	C_V
heat capacity (constant pressure)	C_P
Volume	V
Coefficient of thermal expansion	α
Isothermal compressibility	β_T

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

<math>C_P=\left(\frac{\partial H}{\partial T}\right)_P</math>

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

<math>C_P=\frac{(\partial H)_P}{(\partial T)_P}</math>

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 from the equations below we have:

<math>(\partial H)_P=C_P</math>

and

<math>(\partial T)_P=1</math>

Dividing, we recover the proper expression for C_P.

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

<math>\left(\frac{\partial V}{\partial T}\right)_P = \alpha V</math>

<math>\left(\frac{\partial V}{\partial P}\right)_T = -\beta_T V</math>

<math>\left(\frac{\partial H}{\partial T}\right)_P = C_P = c_P N</math>

[edit] Bridgman's thermodynamic equations

Note that Lewis and Randall use F and E for the Gibbs energy and internal energy, respectively, rather than G and U which are used in this article.

<math> (\partial T)_P=-(\partial P)_T=1</math>

<math> (\partial V)_P=-(\partial P)_V=\left(\frac{\partial V}{\partial T}\right)_P</math>

<math> (\partial S)_P=-(\partial P)_S=\frac{C_p}{T}</math>

<math> (\partial U)_P=-(\partial P)_U=C_P-P\left(\frac{\partial V}{\partial T}\right)_P</math>

<math> (\partial H)_P=-(\partial P)_H=C_P</math>

<math> (\partial G)_P=-(\partial P)_G=-S</math>

<math> (\partial A)_P=-(\partial P)_A=-S-P\left(\frac{\partial V}{\partial T}\right)_P</math>

<math> (\partial V)_T=-(\partial T)_V=-\left(\frac{\partial V}{\partial P}\right)_T</math>

<math> (\partial S)_T=-(\partial T)_S=\left(\frac{\partial V}{\partial T}\right)_P</math>

<math> (\partial U)_T=-(\partial T)_U=T\left(\frac{\partial V}{\partial T}\right)_P+P\left(\frac{\partial V}{\partial P}\right)_T</math>

<math> (\partial H)_T=-(\partial T)_H=-V+T\left(\frac{\partial V}{\partial T}\right)_P</math>

<math> (\partial G)_T=-(\partial T)_G=-V</math>

<math> (\partial A)_T=-(\partial T)_A=P\left(\frac{\partial V}{\partial P}\right)_T</math>

<math> (\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</math>

<math> (\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</math>

<math> (\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</math>

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

<math> (\partial A)_V=-(\partial V)_A=-S\left(\frac{\partial V}{\partial P}\right)_T</math>

<math> (\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</math>

<math> (\partial H)_S=-(\partial S)_H=-\frac{VC_P}{T}</math>

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

<math> (\partial A)_S=-(\partial S)_A=\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</math>

<math> (\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</math>

<math> (\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</math>

<math> (\partial A)_U=-(\partial U)_A=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</math>

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

<math> (\partial A)_H=-(\partial H)_A=-\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</math>

<math> (\partial A)_G=-(\partial G)_A=-S\left[V+P\left(\frac{\partial V}{\partial P}\right)_T\right]-PV\left(\frac{\partial V}{\partial T}\right)_P</math>