Thermodynamics Equations

other useful identities

$\backslash Delta\; U\; =\; q\_\{by\}\; +\; w\_\{on\}\; =\; q\_\{by\}\; -\; \backslash int\; P\_\{ext\}dV\; =\; q\_\{by\}\; -\; P\_\{ext\}\backslash Delta\; V$

$H\; =\; U\; +\; PV\; \backslash ,\backslash !$

$A\; =\; U\; -\; TS\; \backslash ,\backslash !$

$G\; =\; H\; -\; TS\; =\; \backslash sum\_\{i\}\; \backslash mu\_\{i\}\; n\_\{i\}\; \backslash ,\backslash !$

$dU\backslash left(S,V,\{n\_\{i\}\}\backslash right)\; =\; TdS\; -\; PdV\; +\; \backslash sum\_\{i\}\; \backslash mu\_\{i\}\; dn\_i$

$dH\backslash left(S,P,n\_\{i\}\backslash right)\; =\; TdS\; +\; VdP\; +\; \backslash sum\_\{i\}\; \backslash mu\_\{i\}\; dn\_\{i\}$

$dA\backslash left(T,V,n\_\{i\}\backslash right)\; =\; -SdT\; -\; PdV\; +\; \backslash sum\_\{i\}\; \backslash mu\_\{i\}\; dn\_\{i\}$

$dG\backslash left(T,P,n\_\{i\}\backslash right)\; =\; -SdT\; +\; VdP\; +\; \backslash sum\_\{i\}\; \backslash mu\_\{i\}\; dn\_\{i\}$

$C\_v\; =\; \backslash left(\backslash frac\{\backslash partial\; U\}\{\backslash partial\; T\}\backslash right)\_V\; =\; T\backslash left(\backslash frac\{\backslash partial\; S\}\{\backslash partial\; T\}\backslash right)\_V$

$C\_p\; =\; \backslash left(\backslash frac\{\backslash partial\; H\}\{\backslash partial\; T\}\backslash right)\_P$

$\backslash mu\_\{JT\}\; =\; \backslash left(\backslash frac\{\backslash partial\; T\}\{\backslash partial\; P\}\backslash right)\_H$

$\backslash kappa\_\{T\}\; =\; -\backslash frac\{1\}\{V\}\backslash left(\backslash frac\{\backslash partial\; V\}\{\backslash partial\; P\}\backslash right)\_T$

$\backslash alpha\_\{P\}\; =\; \backslash frac\{1\}\{V\}\backslash left(\backslash frac\{\backslash partial\; V\}\{\backslash partial\; T\}\backslash right)\_P$

$\backslash left(\backslash frac\{\backslash partial\; H\}\{\backslash partial\; P\}\backslash right)\_T\; =\; V\; -\; T\backslash left(\backslash frac\{\backslash partial\; V\}\{\backslash partial\; T\}\backslash right)\_P$

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

$H\; =\; -T^2\backslash left(\backslash frac\{\backslash partial\; \backslash left(G/T\backslash right)\}\{\backslash partial\; T\}\backslash right)\_P$

$U\; =\; -T^2\backslash left(\backslash frac\{\backslash partial\; \backslash left(A/T\backslash right)\}\{\backslash partial\; T\}\backslash right)\_V$

Proof #1

    \left(\frac{\partial H}{\partial P}\right)_T 

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

   \left(\frac{\partial H}{\partial P}\right)_T 

    \left(\frac{\partial H}{\partial P}\right)_T 

Proof #2

References

• Peter Atkins and Julio de Paula, Physical Chemistry, 7th edition, W.H. Freeman and Company, 2002 0-7167-3539-3.
• Chapters 1 - 10, Part 1: Equilibrium.

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

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