Carnot Cycle & Heat Engines

The Carnot cycle is the most efficient possible heat engine operating between two fixed temperatures. It consists of four reversible steps:

1. Isothermal expansion at $T_H$: gas absorbs heat $Q_H$ from the hot reservoir and does work; temperature stays constant.
2. Adiabatic expansion: no heat exchange; gas does work, temperature falls from $T_H$ to $T_C$.
3. Isothermal compression at $T_C$: gas rejects heat $|Q_C|$ to the cold reservoir; work is done on the gas.
4. Adiabatic compression: no heat; work done on the gas raises its temperature back to $T_H$, completing the cycle.

The Second Law of Thermodynamics states that no engine can be more efficient than the Carnot engine — all real engines have irreversibilities (friction, turbulence, heat losses) that reduce efficiency below the Carnot limit. The enclosed area on a $P$-$V$ diagram equals the net work output per cycle.

$\eta_{\text{Carnot}} = 1 - \frac{T_C}{T_H} = \frac{W}{Q_H} = \frac{Q_H - |Q_C|}{Q_H}$

where $T_H$ and $T_C$ are in Kelvin. 100% efficiency would require $T_C = 0\,\text{K}$ — unattainable in practice.

A steam power plant operates between a boiler at $T_H = 600\,\text{K}$ and a condenser at $T_C = 300\,\text{K}$.

$\eta_{\text{Carnot}} = 1 - \frac{300}{600} = 0.50 = 50\%$

If the plant absorbs $Q_H = 1000\,\text{J}$ per cycle, maximum work output:

$W = \eta \times Q_H = 0.50 \times 1000 = 500\,\text{J}$

Heat rejected to cold reservoir: $|Q_C| = Q_H - W = 500\,\text{J}$.

A real plant with the same reservoirs might achieve only 35–40% due to irreversibilities.

All thermal power plants — coal, nuclear, gas — are bounded by Carnot efficiency. Engineers raise $T_H$ (supercritical steam at 600°C) and lower $T_C$ (river cooling) to push closer to the limit. Refrigerators and heat pumps run the Carnot cycle in reverse: they use work input to pump heat from cold to hot. Even electric vehicles are indirectly limited: the electricity often comes from thermal plants.

1. A Carnot engine operates between $T_H = 800\,\text{K}$ and $T_C = 400\,\text{K}$. What is its efficiency, and how much heat must it absorb to produce $600\,\text{J}$ of work?

2. Why can a real engine never reach Carnot efficiency, even in principle, at the same two reservoir temperatures?