A Carnot engine operates with T_H = 800 K and T_C = 400 K. What is its efficiency?

E_Carnot = 1 − T_C/T_H. Substitute values.

Why is the actual efficiency of a gasoline engine always less than the Carnot efficiency for the same temperature range?

You can work it out this way: consider friction, heat loss through cylinder walls, and non-quasi-static compression. What do these have in common thermodynamically?

If the cold reservoir temperature is lowered from 300 K to 200 K while T_H stays at 800 K, how does efficiency change?

You can work it out this way: calculate e_Carnot for both T_C values and compare. What is the trend?

Pro 🔒~35 min

Heat Engines & Carnot Cycle

Discover the upper efficiency limit set by the second law of thermodynamics

How it works

The second law of thermodynamics places an absolute upper bound on engine efficiency: no heat engine operating between two reservoirs at T_H and T_C can be more efficient than a Carnot engine operating between the same reservoirs. The Carnot cycle consists of two reversible isothermal processes (heat exchange with reservoirs) and two reversible adiabatic processes (no heat exchange). Because all steps are reversible, no entropy is generated — this is the theoretical ideal. Real engines (Otto, Diesel, Rankine) suffer irreversibilities: friction, turbulence, finite temperature differences during heat transfer, and non-quasi-static compression. These generate entropy, increasing Q_C and reducing net work. The gap between actual and Carnot efficiency is a direct measure of irreversibility. Importantly, the Carnot efficiency depends only on the reservoir temperatures — higher T_H or lower T_C always improves it.

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Step-by-step

Set T_hot and T_cold (Pro).
The animated P-V diagram shows the cycle running with color-coded steps: isothermal expansion (red), adiabatic expansion (orange), isothermal compression (blue), adiabatic compression (cyan).
The efficiency meter shows actual vs. Carnot values.
Adjust Q_hot to scale the cycle.
Switch to Otto cycle (Max) to compare with the ideal gas engine used in cars — note the efficiency gap.
The Sankey diagram on the right shows heat flows at every cycle.

Key formulas

$e = \frac{W_{net}}{Q_H} = 1 - \frac{Q_C}{Q_H}$ Thermal efficiency: fraction of heat input converted to work
$e_{Carnot} = 1 - \frac{T_C}{T_H}$ Carnot efficiency: maximum possible for given reservoir temperatures
$W_{net} = Q_H - Q_C$ Net work equals heat absorbed minus heat rejected
$\text{COP} = \frac{Q_C}{W}$ Coefficient of performance for a refrigerator

Frequently asked questions

A Carnot engine operates with T_H = 800 K and T_C = 400 K. What is its efficiency?: E_Carnot = 1 − T_C/T_H. Substitute values.
Why is the actual efficiency of a gasoline engine always less than the Carnot efficiency for the same temperature range?: You can work it out this way: consider friction, heat loss through cylinder walls, and non-quasi-static compression. What do these have in common thermodynamically?
If the cold reservoir temperature is lowered from 300 K to 200 K while T_H stays at 800 K, how does efficiency change?: You can work it out this way: calculate e_Carnot for both T_C values and compare. What is the trend?