Turning heat flow into useful work
A thermodynamic cycle returns a working substance to its starting state while transferring energy between reservoirs and the surroundings. The first law balances energy; the second law limits direction and efficiency.
Learning goals
- read work and direction from a pressure-volume diagram;
- apply $\Delta U=Q-W_{by}$ consistently;
- compare isochoric, isobaric, isothermal and adiabatic paths;
- calculate engine efficiency and refrigerator performance;
- use entropy and Carnot efficiency to distinguish possible from impossible claims.
1. States, paths and work
Pressure, volume, temperature and internal energy describe an equilibrium state. Heat $Q$ and work $W$ are transfers that depend on the path. Using the convention that $W_{by}$ is work done by the system, $\Delta U=Q-W_{by}$. Quasi-static boundary work is $W_{by}=\int P\,dV$: expansion gives positive work, compression negative work. On a $pV$ plot, the area under a path is work; the enclosed area of a cycle is net work.
2. Four useful processes
| Process | Constraint | Ideal-gas consequence |
|---|---|---|
| Isochoric | $V$ constant | $W=0$, so $Q=\Delta U$ |
| Isobaric | $P$ constant | $W=P\Delta V$ |
| Isothermal | $T$ constant | $\Delta U=0$ for an ideal gas, so $Q=W$ |
| Adiabatic | $Q=0$ | $PV^\gamma=constant$ for a reversible ideal-gas path |