First Law

$Q = \Delta U + W \quad [ML^2T^{-2}]\ [\text{SI: J}]$

Isothermal Process (ideal gas)

$W = Q = nRT\ln\!\left(\frac{V_2}{V_1}\right) \quad [ML^2T^{-2}]\ [\text{SI: J}]$

$\Delta U = 0; \quad PV = \text{const}$

Adiabatic Process

$Q = 0; \quad PV^\gamma = \text{const}; \quad TV^{\gamma-1} = \text{const}$

$W = -\Delta U = nC_v(T_1 - T_2) = \frac{P_1V_1 - P_2V_2}{\gamma - 1} \quad [ML^2T^{-2}]\ [\text{SI: J}]$

Isochoric Process

$W = 0; \quad Q = \Delta U = nC_v\Delta T \quad [ML^2T^{-2}]\ [\text{SI: J}]$

Isobaric Process

$W = P\Delta V = nR\Delta T \quad [ML^2T^{-2}]\ [\text{SI: J}]$

$Q = nC_p\Delta T \quad [ML^2T^{-2}]\ [\text{SI: J}]$

Mayer's Relation

$C_p - C_v = R \quad [ML^2T^{-2}K^{-1}mol^{-1}]\ [\text{SI: J mol}^{-1}\text{K}^{-1}]$

Carnot Efficiency

$\eta = 1 - \frac{T_2}{T_1} = \frac{W}{Q_1} = \frac{Q_1 - Q_2}{Q_1} \quad [\text{dimensionless}]$

Refrigerator COP

$\text{COP} = \frac{Q_2}{W} = \frac{T_2}{T_1 - T_2} \quad [\text{dimensionless}]$

Kinetic Theory — Pressure

$P = \frac{1}{3}\rho v_{rms}^2 \quad [ML^{-1}T^{-2}]\ [\text{SI: Pa}]$

Molecular Speeds

$v_{rms} = \sqrt{\frac{3RT}{M}} \quad [LT^{-1}]\ [\text{SI: m/s}]$

$v_{avg} = \sqrt{\frac{8RT}{\pi M}} \quad [LT^{-1}]\ [\text{SI: m/s}]$

$v_{mp} = \sqrt{\frac{2RT}{M}} \quad [LT^{-1}]\ [\text{SI: m/s}]$

Internal Energy

$U = \frac{f}{2}nRT \quad [ML^2T^{-2}]\ [\text{SI: J}]$

Specific Heats (Equipartition)

$C_v = \frac{f}{2}R; \quad C_p = \frac{f+2}{2}R; \quad \gamma = \frac{C_p}{C_v} = \frac{f+2}{f}$

KE Per Molecule (Translational)

$KE_{trans} = \frac{3}{2}k_BT \quad [ML^2T^{-2}]\ [\text{SI: J}]$

Ideal Gas

$PV = nRT = Nk_BT \quad [ML^2T^{-2}]\ [\text{SI: J}]$