Cheat sheets/Thermodynamics in differential form

• 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

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