Carnot Cycle

Reversible heat engine between two reservoirs · η = 1 − T_c/T_h

Input parameters

Operating mode

Working fluid

Gas type

Adiabatic index γ = 1.667 (sets C_v = R/(γ−1)).

Hot-reservoir temperature

T_h of the hot reservoir (T_h > T_c)

3101500

Cold-reservoir temperature

T_c of the cold reservoir (> 0 K)

501000

Amount of substance

Moles of working gas

mol

0.15

Initial volume

Volume at state 1

150

Isothermal expansion ratio

V₂/V₁ on the hot isotherm (> 1)

1.18

Clockwise — the cycle produces work (engine)

Linear axes, drawn to scale: the Carnot cycle is a crescent, not the idealized textbook parallelogram.

Results

Efficiency η

Fraction 0.4 · Carnot upper bound

Heat absorbed Q_h

2882

Heat released |Q_c|

1729

Net work W

1153

Per-process balance

Q, W, ΔU in J · ΔS in J/K

Process	Q	W	ΔU	ΔS
1→2 isothermal	2882	2882	0	5.763
2→3 adiabatic	0	2494	-2494	0
3→4 isothermal	-1729	-1729	0	-5.763
4→1 adiabatic	0	-2494	2494	0
Full cycle	1153	1153	0	0

Fundamentals & Explanation

The Carnot Cycle: the ideal engine between two temperatures

The Carnot cycle is a sequence of four reversible processes that exchanges heat with only two thermal reservoirs: a hot one at temperature T_h and a cold one at T_c. Its importance is theoretical: it sets the maximum possible efficiency of any heat engine working between those two temperatures. No real engine can beat it, because it would have to be reversible.

Process	Type	Heat
1 → 2	Isothermal at T_h (expansion)	Q_h > 0 absorbed
2 → 3	Adiabatic (expansion)	Q = 0
3 → 4	Isothermal at T_c (compression)	Q_c < 0 released
4 → 1	Adiabatic (compression)	Q = 0

Along the isotherms temperature is constant, so all energy enters or leaves as heat; along the adiabats the gas is insulated and only exchanges work, raising or lowering its temperature between T_h and T_c.

Efficiency and Carnot's Theorem

For an ideal gas, the heat of each isotherm is Q = nRT·ln(V_f/V_i). The two adiabats force the volume ratios to match, so the net work and efficiency collapse into a strikingly simple expression:

\eta = \frac{W_{\text{net}}}{Q_h} = 1 - \frac{T_c}{T_h}

Carnot's theorem states that this efficiency depends only on the two temperatures, not on the working substance or the size of the cycle. The practical takeaway: to gain efficiency you must widen the gap T_h − T_c; and since T_c never reaches absolute zero, η is always below 1.

Engine, Refrigerator and Heat Pump

Run clockwise (on the P-V diagram) the cycle is an engine: it absorbs Q_h, delivers work and rejects Q_c. Reversed (counter-clockwise) it becomes a refrigerator or a heat pump: it consumes work to pump heat from the cold reservoir to the hot one. Its performance is measured by the coefficient of performance (COP):

\text{COP}_{\text{ref}} = \frac{T_c}{T_h - T_c},\qquad \text{COP}_{\text{pump}} = \frac{T_h}{T_h - T_c} = \text{COP}_{\text{ref}} + 1

Unlike efficiency, the COP can exceed 1: a refrigerator moves more heat than the work it costs. That is why a heat pump is so efficient for heating.

The P-V and T-S Diagrams

On the P-V diagram the cycle is a closed loop and the enclosed area is the net work. On the T-S diagram (temperature–entropy) the Carnot cycle is always a rectangle: isotherms are horizontal lines and adiabats (isentropic) are vertical. Its area is the net work too:

W_{\text{net}} = (T_h - T_c)\,\Delta S

The entropy exchanged on the hot isotherm, ΔS = nR·ln(V₂/V₁), is returned in full on the cold one, so the entropy of the universe does not change: the cycle is reversible.

Ideal Gas vs. van der Waals

Does anything change for a real gas? With the van der Waals equation, P = nRT/(V−nb) − an²/V², the isotherms are no longer perfect hyperbolas and the P-V loop deforms. And yet —remarkably— the efficiency stays exactly 1 − T_c/T_h. The reversible adiabat of a van der Waals gas is:

T\,(V - nb)^{R/C_v} = \text{const}

and when computing Q_h and Q_c the attraction terms a cancel: the same η remains. Toggle the ideal-cycle overlay on the P-V diagram to see it: same efficiency, different shape. This is precisely the power of Carnot's theorem.

References

Fermi, E. Thermodynamics (Dover, 1956).
Callen, H. B. Thermodynamics and an Introduction to Thermostatistics (Wiley, 1985).
Zemansky & Dittman. Heat and Thermodynamics (McGraw-Hill).

Carnot Cycle

Input parameters

Results

Per-process balance

Fundamentals & Explanation

The Carnot Cycle: the ideal engine between two temperatures

Efficiency and Carnot's Theorem

Engine, Refrigerator and Heat Pump

The P-V and T-S Diagrams

Ideal Gas vs. van der Waals

References

You may also like:

The Carnot Cycle: the ideal engine between two temperatures

Efficiency and Carnot's Theorem

Engine, Refrigerator and Heat Pump

The P-V and T-S Diagrams

Ideal Gas vs. van der Waals

References