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

 * $$\left ( \frac{\partial T}{\partial V} \right )_S = - \left ( \frac{\partial P}{\partial S}\right )_V$$
 * $$\left ( \frac{\partial T}{\partial P} \right )_S =  \left ( \frac{\partial V}{\partial S}\right )_P$$
 * $$\left ( \frac{\partial S}{\partial V} \right )_T =  \left ( \frac{\partial P}{\partial T}\right )_V$$
 * $$\left ( \frac{\partial S}{\partial P} \right )_T = - \left ( \frac{\partial V}{\partial T}\right )_P$$