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# Thermodynamics Cheat Sheet ## Laws of Thermodynamics ### Zeroth Law If two thermodynamic systems are each in thermal equilibrium with a third, then they are in thermal equilibrium with each other. ### First Law The change in internal energy of a system is equal to the heat added to the system m...

# Thermodynamics Cheat Sheet ## Laws of Thermodynamics ### Zeroth Law If two thermodynamic systems are each in thermal equilibrium with a third, then they are in thermal equilibrium with each other. ### First Law The change in internal energy of a system is equal to the heat added to the system minus the work done by the system: $\qquad \Delta U = Q - W$ ### Second Law The total entropy of an isolated system can never decrease over time, and is constant if and only if all processes are reversible. Isolated systems spontaneously evolve towards thermal equilibrium—the state of maximum entropy. ### Third Law The entropy of a system approaches a constant value as the temperature approaches absolute zero. ## Thermodynamic Potentials | Potential | Definition | Differential Form | Natural Variables | | :-------- | :---------- | :---------------- | :---------------- | | Internal Energy $(U)$ | $U = TS - PV + \sum_i \mu_i N_i$ | $dU = TdS - PdV + \sum_i \mu_i dN_i$ | $S, V, \{N_i\}$ | | Enthalpy $(H)$ | $H = U + PV$ | $dH = TdS + VdP + \sum_i \mu_i dN_i$ | $S, P, \{N_i\}$ | | Helmholtz Free Energy $(F)$ | $F = U - TS$ | $dF = -SdT - PdV + \sum_i \mu_i dN_i$ | $T, V, \{N_i\}$ | | Gibbs Free Energy $(G)$ | $G = U + PV - TS$ | $dG = -SdT + VdP + \sum_i \mu_i dN_i$ | $T, P, \{N_i\}$ | ## Maxwell Relations Derived from the thermodynamic potentials: * $\left(\frac{\partial T}{\partial V}\right)_S = -\left(\frac{\partial P}{\partial S}\right)_V$ * $\left(\frac{\partial T}{\partial P}\right)_S = \left(\frac{\partial V}{\partial S}\right)_P$ * $\left(\frac{\partial S}{\partial V}\right)_T = \left(\frac{\partial P}{\partial T}\right)_V$ * $\left(\frac{\partial S}{\partial P}\right)_T = -\left(\frac{\partial V}{\partial T}\right)_P$ ## Statistical Mechanics ### Partition Function $Z = \sum_i e^{-\beta E_i}$, where $\beta = \frac{1}{k_B T}$ ### Average Energy $\langle E \rangle = -\frac{\partial}{\partial \beta} \ln Z$ ### Entropy $S = k_B (\ln Z + \beta \langle E \rangle)$ ### Boltzmann Distribution $P(E_i) = \frac{e^{-\beta E_i}}{Z}$ ## Ideal Gas Law $PV = nRT$, where $R$ is the ideal gas constant ## Heat Capacity $C = \frac{dQ}{dT}$ * Constant Volume: $C_V = \left(\frac{\partial U}{\partial T}\right)_V$ * Constant Pressure: $C_P = \left(\frac{\partial H}{\partial T}\right)_P$ ## Important Constants * Boltzmann Constant: $k_B \approx 1.38 \times 10^{-23} \, \text{J/K}$ * Ideal Gas Constant: $R \approx 8.314 \, \text{J/(mol⋅K)}$