Thermodynamics — Engines, Carnot & Second Law (Medium)

Quick Recap — Engines, Carnot & Second Law

• Heat engine: $\eta=1-\dfrac{Q_2}{Q_1}$; net work $W=Q_1-Q_2$. Carnot (reversible): $\eta=1-\dfrac{T_2}{T_1}$ (temperatures in kelvin) — the maximum possible, independent of the working substance.
• Refrigerator / heat pump: COP$_{fridge}=\dfrac{Q_2}{W}=\dfrac{T_2}{T_1-T_2}$; COP$_{pump}=\dfrac{Q_1}{W}=\dfrac{T_1}{T_1-T_2}$.
• Second law: no engine can convert heat wholly into work (Kelvin), and heat cannot flow unaided from cold to hot (Clausius); entropy of an isolated system never decreases.
• Process specific heats: adiabatic $C=0$ (since $Q=0$), isothermal $C\to\infty$ (since $\Delta T=0$). Adiabatic bulk modulus $=\gamma P$.

Worked mini-example. A Carnot engine between 400 K and 300 K has $\eta=1-\dfrac{300}{400}=0.25$. Absorbing 800 J, it does $W=0.25\times800=200$ J and rejects 600 J.