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Thermodynamic equations

In thermodynamics, there are a large number of equations relating the various 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) CV
Specific heat (constant pressure) CP
Volume V

The first four are known as thermodynamic potentials, and their definitions and differential definitions are given in the thermodynamic potentials article. The following equations are classified by subject. See also Bridgman's equations for a technique for building a large number of thermodynamic identities.

Contents
1 Entropy
2 Quasi-static process
3 Heat capacity at constant pressure
4 Heat capacity at constant volume
5 Helmholtz free energy
6 Gibbs free energy
7 Enthalpy
8 Maxwell relations
9 Incremental processes
10 Compressibility at constant temperature
11 More relations

Entropy

~ S = k (\ln \Omega) ~
~ \Delta S = \frac{\Delta Q}{T} ~

Quasi-static process

~ dQ=C_v dT+l_v d_v  =dU+pdv =TdS~

Heat capacity at constant pressure

~ C_p=\left ( {\partial U\over \partial T} \right )_p  +p \left ( {\partial v\over \partial T} \right )_p  = \left ( {\partial H\over \partial T} \right )_p  = T \left ( {\partial S\over \partial T} \right )_p ~

Heat capacity at constant volume

~ C_v=\left ( {\partial U\over \partial T} \right )_v = T \left ( {\partial S\over \partial T} \right )_v ~

Helmholtz free energy

~ F \equiv U-TS = \mu n - pv ~

Gibbs free energy

~ G \equiv U-TS+pv = \mu n ~

Enthalpy

~ H \equiv U+pV = \mu n + TS~

Maxwell relations

~ \left ( {\partial T\over \partial v} \right )_{S,n}  = \left ( {\partial p\over \partial S} \right )_{v,n} ~

~ \left ( {\partial T\over \partial p} \right )_{S,n}  = \left ( {\partial v\over \partial S} \right )_{p,n} ~

~ \left ( {\partial T\over \partial v} \right )_{p,n}  = \left ( {\partial p\over \partial S} \right )_{T,n} ~

~ \left ( {\partial T\over \partial p} \right )_{v,n}  = \left ( {\partial v\over \partial S} \right )_{T,n} ~

Incremental processes

~ dU = T\,dS-p\,dv + \mu\,dn ~
~ dF = -S\,dT-p\,dv + \mu\,dn ~
~ dG = -S\,dT+v\,dp + \mu\,dn = \mu\,dn +n\,d\mu ~
~ dH = T\,dS+v\,dp + \mu\,dn ~

Compressibility at constant temperature

~ K_T = -{ 1\over v } \left ( {\partial v\over \partial p} \right )_{T,n} ~

More relations

~ \left ( {\partial S\over \partial U} \right )_{v,n}  =  { 1\over T } ~

~ \left ( {\partial S\over \partial v} \right )_{n,U}  =  { p\over T } ~

~ \left ( {\partial S\over \partial n} \right )_{v,U}  = - { \mu \over T } ~

~ \left ( {\partial T\over \partial S} \right )_v  = { T \over C_v } ~

~ \left ( {\partial T\over \partial S} \right )_p  = { T \over C_p } ~

~ -\left ( {\partial p\over \partial v} \right )_T  = { 1 \over {vK_T} } ~
01-04-2007 01:16:19
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