Carnot Cycle

No engine running between a hot source and a cold sink can beat one specific ideal: the Carnot cycle. It sets a ceiling on efficiency that depends only on the two reservoir temperatures, not on the working fluid or the cleverness of the design. That ceiling is the reason the Carnot cycle is worth memorizing.

The Formula And Its Symbols

For a reversible engine between a hot reservoir at $T_H$ and a cold reservoir at $T_C$, the maximum efficiency is

$$\eta = 1 - \frac{T_C}{T_H}$$

Here $\eta$ is the efficiency, $T_H$ is the hot-reservoir temperature, and $T_C$ is the cold-reservoir temperature. Both temperatures must be in Kelvin, because the formula is a ratio of absolute temperatures. Real engines fall short of this value, but the Carnot result tells you what the limit is.

The cycle itself has four stages: two isothermal stages, where heat is exchanged at constant temperature, and two reversible adiabatic stages, where no heat is exchanged and the temperature changes.

1. Isothermal expansion at $T_H$. The gas absorbs heat $Q_H$ from the hot reservoir and does work at the hot temperature.
2. Reversible adiabatic expansion. No heat enters or leaves. The gas keeps expanding, does work, and its temperature falls from $T_H$ to $T_C$.
3. Isothermal compression at $T_C$. The surroundings do work on the gas while it rejects heat $Q_C$ to the cold reservoir at the cold temperature.
4. Reversible adiabatic compression. No heat is exchanged. The gas is compressed until its temperature rises from $T_C$ back to $T_H$.

After stage four the system returns to its starting state, so the process repeats as a cycle.

Why The Formula Holds

The efficiency expression is not a definition pulled from nowhere; it follows from an entropy balance. In a reversible Carnot cycle, the entropy gained from the hot reservoir matches the entropy delivered to the cold reservoir in magnitude, so

$$\frac{Q_H}{T_H} = \frac{Q_C}{T_C}$$

which gives

$$\frac{Q_C}{Q_H} = \frac{T_C}{T_H}$$

and then

$$\eta = \frac{W}{Q_H} = 1 - \frac{Q_C}{Q_H} = 1 - \frac{T_C}{T_H}$$

That entropy balance only works with absolute temperature, which is exactly why Celsius and Fahrenheit values must be converted to Kelvin first.

Worked Example: Efficiency Between Two Temperatures

Suppose an ideal Carnot engine operates between $T_H = 600\ \mathrm{K}$ and $T_C = 300\ \mathrm{K}$ and absorbs $Q_H = 900\ \mathrm{J}$ from the hot reservoir each cycle.

Its efficiency is

$$\eta = 1 - \frac{300}{600} = 0.50$$

so the maximum possible efficiency is $50\%$.

The work done per cycle is

$$W = \eta Q_H = 0.50 \times 900 = 450\ \mathrm{J}$$

and the rejected heat is

$$Q_C = Q_H - W = 900 - 450 = 450\ \mathrm{J}$$

Once the reservoir temperatures are fixed, the maximum efficiency is fixed too. Better engineering can push a real engine closer to that limit, but never past it.

Your Turn, With A Check

Run the same procedure for $T_H = 500\ \mathrm{K}$ and $T_C = 350\ \mathrm{K}$. Compute the Carnot efficiency first, then pick a value of $Q_H$ and find the work and rejected heat. As a checkpoint, the efficiency should come out to $1 - 350/500 = 0.30$, or $30\%$; if you chose $Q_H = 900\ \mathrm{J}$, then $W = 270\ \mathrm{J}$ and $Q_C = 630\ \mathrm{J}$. As a final step, compare that ideal answer with a real engine that runs lower and explain why the gap appears.

Traps That Cost Points

Using Celsius in the formula. The ratio $T_C/T_H$ must use Kelvin.

Treating the Carnot cycle as a realistic everyday engine. It is an ideal reversible benchmark, not a description of how normal engines behave.

Memorizing the four stages without tracking heat flow. Heat enters during the hot isothermal expansion and leaves during the cold isothermal compression. The adiabatic stages have $Q = 0$.

Overreading the formula. A large $T_H$ does not automatically make a real engine efficient. Materials limits, irreversibility, and design constraints still matter.

Where The Carnot Cycle Is Used

The Carnot cycle connects entropy, reversibility, and engine efficiency in one clean model. It sets upper efficiency limits, lets you compare real engines with ideal ones, and builds intuition for refrigerators and heat pumps as well as heat engines. If you already know the second law of thermodynamics, the Carnot cycle is one of the clearest ways to watch that law turn into a quantitative limit.