B.4.13 (HL)—Carnot efficiency limit

Carnot limit

For an ideal reversible engine operating between hot and cold reservoirs,

$$\eta_{\mathrm C}=1-\frac{T_C}{T_H}$$

Both reservoir temperatures must be in kelvin.

What the limit means

No real engine operating between the same reservoir temperatures can exceed $$\eta_{\mathrm C}$$. Real friction, finite temperature differences and other irreversibilities make actual efficiency lower.

Read temperature changes

At fixed $$T_H$$, lowering $$T_C$$ increases the Carnot efficiency. At fixed $$T_C$$, increasing $$T_H$$ also increases the limit.

Worked example from local Question Bank row 31493

A plant operates between reservoirs at $T_H=885\,\mathrm{K}$ and $T_C=622\,\mathrm{K}$.

$$\eta_C=1-\frac{T_C}{T_H}=1-\frac{622}{885}=0.297\approx29.7\%$$

This is the theoretical maximum. A real plant between the same temperatures must have lower efficiency because its processes are irreversible.

Common trap

Carnot efficiency is a maximum, not the efficiency every engine achieves. Never use Celsius in the temperature ratio.