# Cheat sheets/Thermodynamics in differential form

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

## Internal energy

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

## Enthalpy

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

## Combined first and second law

• ${\displaystyle dU=TdS-PdV}$

### Other thermodynamic potentials

• ${\displaystyle dU=TdS-PdV}$
• ${\displaystyle dH=TdS+VdP}$
• ${\displaystyle dA=-SdT-PdV}$
• ${\displaystyle dG=-SdT+VdP}$

### Maxwell relations

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