First Law and Processes

$Q = \Delta U + W \qquad \text{Isothermal: } W = nRT\ln(V_2/V_1), \quad \Delta U = 0$

$\text{Adiabatic: } Q=0,\; PV^\gamma=\text{const},\; W = \frac{P_1V_1 - P_2V_2}{\gamma-1}$

$\text{Isochoric: } W=0,\; Q=\Delta U = nC_v\Delta T$

$\text{Isobaric: } W = P\Delta V = nR\Delta T,\; Q = nC_p\Delta T$

Specific Heats and γ

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

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

Carnot and COP

$\eta = 1 - \frac{T_2}{T_1} = \frac{W}{Q_1} \quad [\text{dimensionless, T in K}]$

$\text{COP}_{ref} = \frac{Q_2}{W} = \frac{T_2}{T_1 - T_2}$

Kinetic Theory

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

$v_{rms} = \sqrt{\frac{3RT}{M}},\quad v_{avg} = \sqrt{\frac{8RT}{\pi M}},\quad v_{mp} = \sqrt{\frac{2RT}{M}} \quad [LT^{-1}]$

Internal Energy and KE

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

$KE_{trans/molecule} = \frac{3}{2}k_BT,\quad k_B = 1.38\times10^{-23}\ \text{J/K}$

Ideal Gas

$PV = nRT = Nk_BT,\quad R = 8.314\ \text{J mol}^{-1}\text{K}^{-1}$