KAHIBARO
Table of Contents
The Ideal Heat Engine Cycle
The Carnot cycle is an ideal thermodynamic cycle that shows the greatest possible efficiency any heat engine can have when operating between two temperatures. It is not mainly important because real engines follow it exactly, they do not, but because it sets the upper limit for performance.
A heat engine takes in heat from a hot reservoir, converts part of that energy into work, and rejects the remaining heat to a cold reservoir. The Carnot cycle describes the most efficient way this can happen if all processes are reversible.
The cycle operates between two fixed temperatures, a hot temperature $T_H$ and a cold temperature $T_C$, with $T_H > T_C$.
The Four Stages of the Carnot Cycle
The Carnot cycle consists of four reversible processes. Two are isothermal, meaning temperature stays constant, and two are adiabatic, meaning no heat enters or leaves the system.
Stage 1, Reversible Isothermal Expansion
The gas is placed in contact with the hot reservoir at temperature $T_H$. It expands slowly and reversibly. Because the temperature remains constant, the gas absorbs heat $Q_H$ from the hot reservoir and does work on the surroundings.
For an ideal gas in an isothermal process, the internal energy does not change, so the absorbed heat equals the work done by the gas.
Stage 2, Reversible Adiabatic Expansion
The system is now thermally isolated. The gas continues to expand, but no heat enters or leaves, so $Q = 0$. Since the gas does work, its internal energy decreases and its temperature falls from $T_H$ to $T_C$.
Stage 3, Reversible Isothermal Compression
The gas is brought into contact with the cold reservoir at temperature $T_C$. It is compressed slowly and reversibly. Heat $Q_C$ leaves the gas and flows into the cold reservoir while work is done on the gas.
Stage 4, Reversible Adiabatic Compression
The system is again thermally isolated. The gas is compressed further with no heat exchange. Its temperature rises from $T_C$ back to $T_H$, returning the system to its initial state.
Visualizing the Cycle
On a pressure-volume diagram, the cycle forms a closed loop. The isothermal curves and adiabatic curves are different in shape, with adiabatic curves generally steeper than isothermal ones.
The area enclosed by the loop represents the net work done in one complete cycle.
Heat, Work, and Net Result
Because the system returns to its original state after one full cycle, the total change in internal energy over the cycle is zero.
So for one complete cycle,
$$
\Delta U = 0
$$
and therefore
$$
W_{\text{net}} = Q_H - Q_C
$$
where $Q_H$ is the heat absorbed from the hot reservoir and $Q_C$ is the heat rejected to the cold reservoir.
For any complete thermodynamic cycle,
$$
\Delta U = 0
$$
so the net work done by the engine is
$$
W_{\text{net}} = Q_{\text{in}} - Q_{\text{out}}
$$
Efficiency of a Carnot Engine
The efficiency of any heat engine is defined as
$$
e = \frac{W_{\text{net}}}{Q_H}
$$
Using $W_{\text{net}} = Q_H - Q_C$, we get
$$
e = \frac{Q_H - Q_C}{Q_H} = 1 - \frac{Q_C}{Q_H}
$$
For a Carnot engine, because the cycle is fully reversible, the heat exchanges are related directly to the reservoir temperatures:
$$
\frac{Q_C}{Q_H} = \frac{T_C}{T_H}
$$
So the Carnot efficiency is
$$
e_{\text{Carnot}} = 1 - \frac{T_C}{T_H}
$$
Here $T_H$ and $T_C$ must be measured in kelvin.
The maximum possible efficiency for any engine operating between two temperatures is
$$
e_{\text{Carnot}} = 1 - \frac{T_C}{T_H}
$$
Temperatures must be absolute temperatures, in kelvin.
What the Formula Means
The Carnot efficiency depends only on the temperatures of the two reservoirs. It does not depend on the working substance, as long as the cycle is reversible.
This result has two very important consequences. First, no engine can be more efficient than a Carnot engine operating between the same two temperatures. Second, to increase efficiency, you want $T_H$ as large as possible and $T_C$ as small as possible.
However, even in principle, the efficiency can never reach 100 percent unless $T_C = 0 \text{ K}$, which is not physically attainable.
Why the Carnot Cycle Is Reversible
A process is reversible if it can be undone without leaving any net change in the system or surroundings. In the Carnot cycle, this requires each step to happen infinitely slowly, with only tiny temperature and pressure differences.
That makes the Carnot cycle an ideal model. Real engines involve friction, turbulence, rapid combustion, and finite temperature differences, so they are irreversible and less efficient.
Comparison With Real Engines
Real heat engines, such as car engines and steam turbines, are designed for practical power output, not perfect reversibility. The Carnot cycle gives a benchmark for comparison.
| Engine type | Reversible? | Can reach Carnot efficiency? |
|---|---|---|
| Carnot engine | Yes, ideally | Yes, by definition |
| Real engine | No | No |
| Engine with friction or heat loss | No | No |
The Carnot Principle
The Carnot principle states that no engine operating between two heat reservoirs can be more efficient than a reversible engine operating between the same reservoirs. All reversible engines between the same two temperatures have the same efficiency.
This is a profound result because it shows that the maximum efficiency is universal.
Carnot principle:
No heat engine working between $T_H$ and $T_C$ can be more efficient than a reversible engine between the same temperatures.
A Simple Numerical Example
Suppose a heat engine operates between
$$
T_H = 500 \text{ K}, \qquad T_C = 300 \text{ K}
$$
Then its maximum possible efficiency is
$$
e_{\text{Carnot}} = 1 - \frac{300}{500} = 1 - 0.6 = 0.4
$$
So
$$
e_{\text{Carnot}} = 40\%
$$
This means that even an ideal engine could convert at most 40 percent of the absorbed heat into useful work. The remaining 60 percent must be rejected to the cold reservoir.
If this engine absorbs $Q_H = 1000 \text{ J}$, then
$$
W_{\text{net}} = e Q_H = 0.4 \times 1000 = 400 \text{ J}
$$
and
$$
Q_C = Q_H - W_{\text{net}} = 1000 - 400 = 600 \text{ J}
$$
The Cycle as a Standard of Perfection
The Carnot cycle is important because it tells us what nature allows, not because it is easy to build. It provides the ideal standard for heat engines and plays a central role in thermodynamics.
Whenever we ask how efficient an engine can be, the Carnot cycle gives the answer. It reminds us that some heat must always be expelled, and that temperature differences set the ultimate limits of energy conversion.