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Premium experiment • reversible heat-engine limit

PHYSICS • THERMODYNAMICS

Carnot Heat Engine & PV Diagrams

Trace the two isothermal and two adiabatic legs of an ideal reversible engine while comparing heat input, work and the Carnot efficiency limit.

● REVERSIBLE CYCLEPressure–volume diagram
Efficiency
Net work
Heat in
Heat out

Reasoning inspector

live model state
Hot reservoir—
Cold reservoir—
Tc / Th—
Carnot efficiency—
Heat absorbed Qin—
Heat rejected Qout—
Net work—
Entropy in—
Entropy out—
Isothermal volume ratio—
γ—

Maximum reversible efficiency

The Carnot cycle sets an upper bound for any heat engine operating between the same hot and cold reservoir temperatures.

LEARN WITH THE SIMULATION

Understand Carnot Heat Engine & PV Diagrams by experimenting.

START HERE

The Carnot cycle is the reversible benchmark for every heat engine operating between two thermal reservoirs. This laboratory draws the four ideal processes on a PV plane while calculating heat absorbed, heat rejected, net work, efficiency and reversible entropy transfer. The premium challenge separates two ideas that are often conflated: the reservoir temperature ratio sets the Carnot efficiency, while the chosen volume ratio changes the size of the cycle and its heat and work amounts without changing that efficiency.

BUILD THE IDEA

What you should understand first

  • Compute Carnot efficiency η = 1 − Tc/Th using absolute temperatures.
  • Identify the two isothermal and two adiabatic legs of the reversible cycle.
  • Relate net work to Qin − Qout and to the enclosed PV-cycle area.
  • Verify reversible entropy balance through Qin/Th = Qout/Tc.

TRY IT YOURSELF

Can you change the work output substantially while leaving Carnot efficiency unchanged?

  1. Choose hot and cold reservoir temperatures and record efficiency, Qin, Qout and work.
  2. Change only the isothermal volume ratio.
  3. Observe how the PV loop area, heat transfer and work change.
  4. Verify that efficiency stays fixed until the reservoir temperature ratio changes.
Read the visual:

For the ideal gas cycle used here, changing the isothermal expansion ratio scales heat input, heat rejection and net work through ln(V₂/V₁), but their ratio remains η = 1 − Tc/Th. Efficiency is therefore set by reservoir temperatures rather than by the geometric size of the cycle alone.

WHY IT WORKS 01

The temperature ratio determines the reversible efficiency ceiling

A reversible Carnot engine exchanges heat isothermally with reservoirs at Th and Tc and moves between those isotherms adiabatically. Reversibility requires Qin/Th = Qout/Tc, so Qout/Qin = Tc/Th. Substituting that into η = 1 − Qout/Qin gives η = 1 − Tc/Th, showing directly why no cycle-detail parameter appears in the ideal efficiency formula.

WHY IT WORKS 02

The PV loop area is the net boundary work

On a pressure-volume diagram, expansion contributes positive work and compression contributes negative work. The closed-loop integral ∮P dV equals the net work output. In the ideal Carnot model, the isothermal heat transfers scale with nRT ln(V₂/V₁), so changing the volume ratio expands or contracts the loop area while leaving the fraction W/Qin controlled only by the reservoir temperatures.

WHY IT WORKS 03

Entropy transfer closes exactly for the reversible cycle

During the hot isotherm the working substance receives Qin at temperature Th, corresponding to entropy transfer Qin/Th. During the cold isotherm it rejects Qout at Tc. For a reversible complete cycle those magnitudes are equal, so the working substance returns to its initial entropy and no entropy is generated internally. Real engines generate entropy and therefore fall below the Carnot efficiency.

DO NOT MIX THESE UP

Common misconception

Carnot efficiency is a theoretical upper bound for reversible engines, not the efficiency normally achieved by practical machines. Increasing the cycle volume ratio can increase work per modeled cycle without violating the Carnot limit because both heat input and work scale together. Temperatures must be absolute; substituting Celsius values into 1 − Tc/Th is physically incorrect.

PRACTICE CHALLENGE

Can you reach a target efficiency and then change work output without moving away from that efficiency?

  • Set a target efficiency in the premium controls.
  • Adjust Th and Tc until the target is reached.
  • Change only the isothermal volume ratio and observe Qin, Qout and work.
  • Explain why efficiency remains fixed while the PV-loop area and energy transfers change.
Reveal the reasoning

The target efficiency is reached by setting the reservoir ratio so that 1 − Tc/Th matches the target. Changing the volume ratio modifies ln(V₂/V₁), which scales heat and work amounts, but it does not change Tc/Th. Therefore the reversible efficiency remains unchanged even though the cycle area and work per cycle move.

MODEL BOUNDARIES

What this simulation assumes

Every simulation simplifies reality or a mathematical system. These are the assumptions and limits to keep in mind while interpreting the results.

Model assumptions

  • The working fluid is an ideal gas in a fully reversible Carnot cycle.
  • The cycle contains two isothermal and two adiabatic legs with constant heat-capacity ratio gamma.
  • Reservoir temperatures remain fixed and the isothermal volume ratio is identical on the hot and cold legs.

Model limitations

  • Friction, finite heat-transfer rates, pressure losses, dead volume and real-gas behavior are omitted.
  • The PV plot represents an ideal quasistatic cycle rather than a specific engine mechanism.

CONNECT IT

Where this matters

  • Heat-engine performance limits
  • Thermodynamic cycle analysis
  • Power-system benchmarking
  • Understanding reversibility and entropy transfer

WORDS THAT MATTER

Key vocabulary

Carnot efficiency
Maximum reversible heat-engine efficiency η = 1 − Tc/Th between two reservoir temperatures.
Isothermal process
Process carried out at constant temperature while heat and work may be exchanged.
Adiabatic process
Process with no heat transfer between system and surroundings.
Reversible process
Ideal limiting process that can be reversed without net entropy generation.
Entropy transfer
Heat transfer divided by absolute boundary temperature for a reversible transfer.