The Carnot Cycle

The Carnot cycle is the most important thermodynamic cycle in physics. It's the theoretical maximum for the efficiency of any heat engine operating between two temperatures. No real engine can beat a Carnot engine operating between the same two reservoirs — not a gasoline engine, not a steam turbine, not a fuel cell, not anything. The Carnot cycle is the absolute ceiling.

It was conceived by Sadi Carnot in 1824, before anyone even knew what the first or second law of thermodynamics was. Carnot was trying to figure out how efficient a steam engine could be, and in the process invented an idealized engine that's still the benchmark against which all real engines are compared.

The four steps

A Carnot engine repeatedly runs through four reversible steps on a PV diagram. Each step is either isothermal (constant T) or adiabatic (Q = 0). The cycle is:

1. Isothermal expansion at $T_{h}$ — the gas is in contact with the hot reservoir, absorbs heat $Q_{h}$, and expands, doing work on the surroundings. Temperature stays at $T_{h}$. On the PV diagram: a hyperbolic curve falling along an isotherm from point A to point B.

2. Adiabatic expansion — the gas is disconnected from both reservoirs and continues to expand. Since no heat can flow, the expansion cools the gas from $T_{h}$ down to $T_{c}$ (the cold reservoir temperature). From point B to point C, following an adiabatic curve that's steeper than the isotherm.

3. Isothermal compression at $T_{c}$ — the gas is put in contact with the cold reservoir, dumps heat $Q_{c}$ into it, and is compressed. Temperature stays at $T_{c}$. From point C to point D, along the lower isotherm.

4. Adiabatic compression — the gas is disconnected from both reservoirs and compressed further. The work done on it raises its temperature from $T_{c}$ back up to $T_{h}$. From point D back to point A, completing the cycle.

The enclosed area on the PV diagram is the net work done by the engine per cycle.

Why each step matters

You might wonder why Carnot chose this specific sequence. The key insight is that the engine must return to its starting state after each cycle — otherwise it's not an engine, it's a one-shot expansion. And for the engine to be as efficient as possible, each step must be reversible, meaning infinitely slow and with infinitesimal temperature differences at the heat-exchange interfaces.

Isothermal + adiabatic + isothermal + adiabatic is the only combination of reversible steps that can take a gas from hot isotherm to cold isotherm and back. That's the mathematical reason the Carnot cycle has the shape it does.

The Carnot efficiency formula

The efficiency of any heat engine is:

$\eta = \dfrac{W_{\text{net}}}{Q_{h}}$

where $W_{\text{net}}$ is the net work output per cycle and $Q_{h}$ is the heat absorbed from the hot reservoir. By energy conservation, $W_{\text{net}} = Q_{h} - Q_{c}$, so:

$\eta = 1 - \dfrac{Q_{c}}{Q_{h}}$

For a Carnot engine specifically, because the heat transfers happen at constant temperatures, you can show that:

$\dfrac{Q_{c}}{Q_{h}} = \dfrac{T_{c}}{T_{h}}$

This is a famous result. It lets us write the Carnot efficiency as a function of temperatures alone:

$\boxed{\;\eta_{\text{Carnot}} = 1 - \dfrac{T_{c}}{T_{h}}\;}$

Temperatures must be in kelvin (absolute temperature). And this efficiency is the absolute maximum — no engine operating between $T_{h}$ and $T_{c}$ can do better.

Worked example: $T_{h} = 600\;\text{K}$, $T_{c} = 300\;\text{K}$

Step 1: Compute the Carnot efficiency.

$\eta = 1 - \dfrac{T_{c}}{T_{h}} = 1 - \dfrac{300}{600} = 1 - 0.5 = 0.5$

The maximum theoretical efficiency is 50%.

That means if the engine absorbs $1000\;\text{J}$ of heat from the hot reservoir, it can convert at most $500\;\text{J}$ of that into useful work. The other $500\;\text{J}$ must be dumped into the cold reservoir as waste heat. There's no way around it — the second law of thermodynamics forbids any higher efficiency.

Step 2: Translate into heat flows.

If $Q_{h} = 1000\;\text{J}$, then:

$Q_{c} = Q_{h} \cdot \dfrac{T_{c}}{T_{h}} = 1000 \cdot \dfrac{300}{600} = 500\;\text{J}$

$W_{\text{net}} = Q_{h} - Q_{c} = 1000 - 500 = 500\;\text{J}$

Step 3: Check the sign conventions.

$Q_{h}$ is absorbed (positive, into the engine). $Q_{c}$ is expelled (negative, out of the engine — but traditionally we write its magnitude). $W$ is work done by the engine (positive output).

Balance: heat in minus heat out equals work done. $1000 - 500 = 500\;\text{J}$. Energy is conserved. ✓

Why can't we do better?

The Carnot efficiency is the ceiling because of the second law of thermodynamics. Here's the intuition:

The second law says that heat naturally flows from hot to cold and never spontaneously from cold to hot. To get the cold reservoir to give up heat (negative-$Q_{c}$ flow), you'd have to do work on it — and that work would come out of the engine's output, reducing efficiency, not increasing it.

Carnot's engine achieves the ceiling by making every heat transfer reversible — infinitely slow, infinitesimal temperature differences. Real engines fall short because:

• Friction. Moving parts waste energy as heat.
• Rapid heat transfer. Real combustion happens in milliseconds, not quasi-statically. Fast = irreversible = inefficient.
• Finite temperature differences. Real engines need a substantial $\Delta T$ between the gas and the reservoirs to move heat at a reasonable rate, but that same $\Delta T$ represents wasted potential.
• Incomplete cycles. Real gases don't follow the idealized isotherm/adiabat curves perfectly.

A typical gasoline engine achieves maybe 25-35% efficiency. A modern combined-cycle power plant (gas turbine + steam turbine) achieves about 60%. A nuclear reactor's steam cycle manages around 33%. The Carnot limit for each is higher than these, but they fall well short of the theoretical maximum due to real-world irreversibilities.

The reversed Carnot cycle — refrigerators and heat pumps

Run the cycle backwards — compress at $T_{c}$, adiabatic compression to $T_{h}$, expand at $T_{h}$, adiabatic expansion to $T_{c}$ — and you get a refrigerator. Now you're using work (external input) to pump heat from the cold side to the hot side, against the natural flow.

The coefficient of performance (COP) of a Carnot refrigerator is:

$\text{COP}_{\text{ref}} = \dfrac{Q_{c}}{W} = \dfrac{T_{c}}{T_{h} - T_{c}}$

For a fridge keeping food at $T_{c} = 275\;\text{K}$ in a $T_{h} = 300\;\text{K}$ kitchen, the Carnot COP is $275 / 25 = 11$. That means every joule of electrical work pumped into the fridge can move 11 joules of heat out of the cold interior. Real fridges achieve COPs of about 2-4, much lower than the Carnot limit but still quite good.

Heat pumps are the same thing used for heating — they pump heat into a warm room from the cold outdoors, and their Carnot COP (for heating) is $T_{h}/(T_{h} - T_{c})$. In a mild climate a heat pump can deliver 4-5 units of heat to your living room for every unit of electrical energy you put in — far more efficient than a resistance heater.

Real-world significance

The Carnot efficiency appears everywhere energy is extracted as work from a temperature difference:

• Thermal power plants. A coal, gas, oil, or nuclear plant heats water into high-pressure steam. The steam drives turbines (work output), and then the exhaust steam is condensed in a cooling tower. The Carnot efficiency limit based on steam temperature (~600°C at the boiler) and cooling water temperature (~30°C) is about 65%, and real plants achieve perhaps 40-45%.
• Car engines. Combustion temperature in a gasoline engine is ~2500 K; exhaust is ~800 K. Carnot limit: 68%. Real gasoline engines: 25-35%.
• Solar thermal. Concentrated solar collectors can heat water or another fluid to ~400-500°C. The available Carnot efficiency is high, but practical engineering losses reduce it to perhaps 20-25%.
• Geothermal. Low temperature differences (~150°C source, ~30°C ambient) mean low Carnot efficiency (~28%), which is why geothermal electric plants are less efficient than fossil fuel plants.
• Body heat. Your body's metabolism runs at ~310 K (37°C) and you expel waste heat to the environment at ~295 K (22°C). The Carnot efficiency for a "body-heat engine" would be about 4.8%, which is why your body doesn't work by extracting useful motion from heat differences — it uses chemical energy directly.

Try it

• Drag $T_{h}$ higher and higher. The efficiency approaches 100% as $T_{h} \to \infty$. But in practice, materials can't withstand infinitely high temperatures. The upper limit is set by the melting or degradation point of the engine components.
• Drag $T_{c}$ toward $T_{h}$. As the two temperatures get close, efficiency drops to zero. No temperature difference = no free work available.
• Animate the cycle by enabling auto-play. Watch the state point trace through the four steps in order: isothermal expand → adiabatic expand → isothermal compress → adiabatic compress.
• Turn on "Show work area". The enclosed area on the PV diagram is the net work per cycle. When the cycle is thin (temperatures close together), the area is small and work output is small. When it's fat, the engine is powerful.
• Try the extreme case $T_{c} = 0$. The Carnot formula gives efficiency = 100%, but reaching absolute zero is physically impossible (it would take infinite work to extract the last bit of heat from the cold reservoir). This is the third law of thermodynamics in a nutshell.
• Compare to real engines: a modern gas turbine operates between roughly $T_{h} \approx 1600$ K and $T_{c} \approx 300$ K. Carnot says 81% efficiency; real performance is about 60%. Most of the gap comes from irreversibilities and material limits.