Thermodynamic Equations

Entropy

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

Quasi-static process

  $~\; dQ=C\_v\; dT+l\_v\; d\_v$

Heat capacity at constant pressure

  $~\; C\_p=\backslash left\; (\; \{\backslash partial\; U\backslash over\; \backslash partial\; T\}\; \backslash right\; )\_p$

Heat capacity at constant volume

  $~\; C\_v=\backslash left\; (\; \{\backslash partial\; U\backslash over\; \backslash partial\; T\}\; \backslash right\; )\_v$

Helmholtz free energy

  $~\; F\; \backslash equiv\; U-TS\; =\; \backslash mu\; n\; -\; pv\; ~$ 

Gibbs free energy

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

Enthalpy

  $~\; H\; \backslash equiv\; U+pV\; =\; \backslash mu\; n\; +\; TS~$ 

Maxwell relations

  $~\; \backslash left\; (\; \{\backslash partial\; T\backslash over\; \backslash partial\; v\}\; \backslash right\; )\_\{S,n\}$

  $~\; \backslash left\; (\; \{\backslash partial\; T\backslash over\; \backslash partial\; p\}\; \backslash right\; )\_\{S,n\}$

  $~\; \backslash left\; (\; \{\backslash partial\; T\backslash over\; \backslash partial\; v\}\; \backslash right\; )\_\{p,n\}$

  $~\; \backslash left\; (\; \{\backslash partial\; T\backslash over\; \backslash partial\; p\}\; \backslash right\; )\_\{v,n\}$

Incremental processes

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

Compressibility at constant temperature

  $~\; K\_T\; =\; -\{\; 1\backslash over\; v\; \}\; \backslash left\; (\; \{\backslash partial\; v\backslash over\; \backslash partial\; p\}\; \backslash right\; )\_\{T,n\}\; ~$ 

More relations

  $~\; \backslash left\; (\; \{\backslash partial\; S\backslash over\; \backslash partial\; U\}\; \backslash right\; )\_\{v,n\}$

  $~\; \backslash left\; (\; \{\backslash partial\; S\backslash over\; \backslash partial\; v\}\; \backslash right\; )\_\{n,U\}$

  $~\; \backslash left\; (\; \{\backslash partial\; S\backslash over\; \backslash partial\; n\}\; \backslash right\; )\_\{v,U\}$

  $~\; \backslash left\; (\; \{\backslash partial\; T\backslash over\; \backslash partial\; S\}\; \backslash right\; )\_v$

  $~\; \backslash left\; (\; \{\backslash partial\; T\backslash over\; \backslash partial\; S\}\; \backslash right\; )\_p$

  $~\; -\backslash left\; (\; \{\backslash partial\; p\backslash over\; \backslash partial\; v\}\; \backslash right\; )\_T$

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