Thermodynamics — Advanced Multi-Step Practice (Hard)

Quick Recap — Advanced Thermodynamics (Multi-step)

• Adiabatic work: $W=\dfrac{P_1V_1-P_2V_2}{\gamma-1}=\dfrac{nR(T_1-T_2)}{\gamma-1}$. Adiabatic ratios: $\dfrac{P_2}{P_1}=\left(\dfrac{V_1}{V_2}\right)^\gamma$, $\dfrac{T_2}{T_1}=\left(\dfrac{V_1}{V_2}\right)^{\gamma-1}$.
• Multi-step / cyclic: apply the first law leg by leg; net $W=$ enclosed area, net $Q=$ net $W$.
• Process molar heat capacity: for $PV^n=$ const, $C=C_V+\dfrac{R}{1-n}=\dfrac{R}{\gamma-1}+\dfrac{R}{1-n}$.
• Gas mixtures: $C_{V,mix}=\dfrac{n_1C_{V1}+n_2C_{V2}}{n_1+n_2}$, then $\gamma_{mix}=\dfrac{C_{P,mix}}{C_{V,mix}}$.
• Engines in series: the pair between $T_1$ and $T_3$ has the Carnot efficiency of that overall span.

Worked mini-example. A diatomic gas ($\gamma=1.4$) is compressed adiabatically to half its volume: $\dfrac{P_2}{P_1}=2^{1.4}=2.64$, so the pressure rises to about $2.64$ times its initial value while the temperature rises by $2^{0.4}=1.32$.