Thermodynamics

First law and processes

First law
$Δ Q = Δ U + W,$ $Δ U = n C_{v} Δ T,$ $W = \int P d V$
Isothermal
$W = n R T ln \frac{V _{2}}{V _{1}},$ $Δ U = 0$
Adiabatic
$P V^{γ} =$ const $,$ $T V^{γ - 1} =$ const $,$ $W = \frac{P _{1} V _{1} - P _{2} V _{2}}{γ - 1}$
Isobaric / isochoric
$W = P Δ V, Q = n C_{p} Δ T (P$ const $);$ $W = 0, Q = n C_{v} Δ T (V$ const $)$
Polytropic
$P V^{x} =$ const $: W = \frac{n R Δ T}{1 - x},$ $C = C_{v} + \frac{R}{1 - x}$
Slopes on P–V
adiabatic slope $= γ \times$ isothermal slope

Efficiency
$η = \frac{W}{Q _{1}} = 1 - \frac{Q _{2}}{Q _{1}};$ $η_{Carnot} = 1 - \frac{T _{2}}{T _{1}}$
Refrigerator
$β = \frac{Q _{2}}{W} = \frac{T _{2}}{T _{1} - T _{2}}$ (Carnot)