Carnot Cycle- Khan Academy Thermodynamics Tutorial

What Is the Carnot Cycle?

The Carnot cycle is the most efficient heat engine cycle possible. It sets the theoretical limit for how well any heat engine can convert heat energy into work. No real engine can beat it, but understanding it helps you see why your car's engine or refrigerator has inherent efficiency limits.

Nicolas Léonard Sadi Carnot developed this concept in 1824. He asked a simple question: what makes some heat engines better than others? His answer defined the foundation of classical thermodynamics.

The Four Processes of the Carnot Cycle

Every Carnot engine goes through four reversible processes. "Reversible" means the system can return to its starting point without leaving any net change in the surroundings. This is an idealization—real processes always have some irreversibility—but it gives us a benchmark.

1. Isothermal Expansion (Heat Absorption)

The working substance (usually a gas) expands while absorbing heat from a hot reservoir. The temperature stays constant during this process, which means the entropy increases. The gas does work on its surroundings.

This is your engine taking in thermal energy. The gas pushes against a piston, converting heat into mechanical motion.

2. Adiabatic Expansion (Temperature Drop)

The gas continues expanding, but now with no heat exchange. Because it does work without taking in energy, its temperature drops. The system reaches the cold reservoir temperature by the end of this step.

Think of it as the gas "spending" its thermal energy to keep pushing the piston.

3. Isothermal Compression (Heat Rejection)

The gas is compressed while in contact with the cold reservoir. Heat flows out of the system into the cold reservoir. The temperature stays constant, but entropy decreases. Work is done on the gas.

This step is necessary—you cannot convert all heat to work. Some thermal energy must be discarded.

4. Adiabatic Compression (Temperature Rise)

The gas is compressed further, with no heat exchange. The temperature rises back to the hot reservoir temperature. The system returns to its original state, ready to repeat the cycle.

This completes one full cycle. The net work output equals the difference between heat absorbed and heat rejected.

Carnot Efficiency Formula

The efficiency of a Carnot engine depends only on the temperatures of the hot and cold reservoirs:

η = 1 - (Tc / Th)

Where:

• η = efficiency (as a decimal, multiply by 100 for percentage)
• Tc = absolute temperature of cold reservoir
• Th = absolute temperature of hot reservoir

Always use Kelvin for these calculations. Celsius or Fahrenheit will give you wrong answers.

Example Calculation

A heat engine operates between a hot reservoir at 500 K and a cold reservoir at 300 K.

η = 1 - (300/500) = 1 - 0.6 = 0.4 = 40%

This means 40% of the heat energy becomes work. The remaining 60% must be rejected to the cold reservoir.

Why Higher Temperature Differences Matter

Look at the efficiency formula. The larger the ratio Tc/Th, the lower your efficiency. To maximize efficiency, you want:

• The hottest possible hot reservoir (higher Th)
• The coldest possible cold reservoir (lower Tc)

Car engines run hotter today than 50 years ago for this exact reason. Higher operating temperatures mean better thermal efficiency. Jet engines operate at extremely high temperatures, which is why they're more efficient than piston engines.

Real Engines vs. Carnot Efficiency

No real engine achieves Carnot efficiency. Here's why:

• Friction — Always present, converts mechanical energy to heat
• Heat losses — Some heat escapes to surroundings instead of doing work
• Irreversibilities — Real processes are never perfectly reversible
• Material limits — Components melt or fail if temperatures get too extreme

Typical real-world efficiencies compared to Carnot:

Real engines typically achieve 50-70% of Carnot efficiency. Closing this gap requires better materials, lower friction, and improved combustion.

The Carnot Refrigerator and Heat Pump

The Carnot cycle works in reverse. Instead of converting heat to work, you use work to move heat from a cold place to a hot place.

A Carnot refrigerator has a coefficient of performance (COP):

COP = Tc / (Th - Tc)

A Carnot heat pump has a different COP:

COP = Th / (Th - Tc)

The heat pump COP can exceed 1 because it's moving heat rather than converting it. This is why heat pumps can be 300-500% efficient at heating—they move three to five units of heat for every unit of electrical work input.

Key Takeaways

• The Carnot cycle defines the maximum possible efficiency for a heat engine
• Efficiency depends only on reservoir temperatures, not on the working substance
• Higher temperature differences always mean better efficiency
• No real engine can match Carnot efficiency because irreversibilities are unavoidable
• The same principles apply to refrigerators and heat pumps, just reversed

Getting Started: Solving Carnot Problems

Step 1: Convert all temperatures to Kelvin. Add 273 to Celsius values.

Step 2: Identify Th (hot reservoir) and Tc (cold reservoir) from the problem.

Step 3: Apply the efficiency formula: η = 1 - (Tc/Th)

Step 4: For work output: W = η × Qh (where Qh is heat absorbed from hot reservoir)

Step 5: For heat rejected: Qc = Qh - W

Practice with different temperature pairs until the calculations become automatic. The math is straightforward—most mistakes come from using wrong temperature units or misidentifying which reservoir is which.