Cheat sheets/Thermodynamics in differential form

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Heat: dQ
Work: dW
Entropy: dS = dQ / T

Internal energy

$U=U(T,V)$
$dU=\left({\frac {\partial U}{\partial T}}\right)_{V}dT+\left({\frac {\partial U}{\partial V}}\right)_{T}dV$
$dU=dQ+dW$
$dU=C_{V}dT+\left({\frac {\partial U}{\partial V}}\right)_{T}dV$

Enthalpy

$H=H(T,P)$
$dH=\left({\frac {\partial H}{\partial T}}\right)_{P}dT+\left({\frac {\partial H}{\partial P}}\right)_{T}dP$
$dH=C_{P}dT+\left({\frac {\partial H}{\partial P}}\right)_{T}dP$

Combined first and second law

$dU=TdS-PdV$

Other thermodynamic potentials

$dU=TdS-PdV$
$dH=TdS+VdP$
$dA=-SdT-PdV$
$dG=-SdT+VdP$

Maxwell relations

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

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