Carnot cycle

The tool compares the four reversible processes of an ideal-gas Carnot cycle: 1→2 isothermal expansion at the hot temperature Tₕ, 2→3 adiabatic expansion to T꜀, 3→4 isothermal compression at T꜀, and 4→1 adiabatic compression back to Tₕ. Isothermal stages exchange heat with the named reservoir; adiabatic stages are thermally isolated. Temperatures are absolute kelvin. The inputs require 100 ≤ T꜀ ≤ Tₕ ≤ 1200 K and 0.2 ≤ n ≤ 5 mol.

The model fixes R = 8.31446261815324 J/(mol·K), γ = 1.4, Cᵥ = R/(γ−1), V₁ = 0.02 m³ and V₂ = 0.05 m³. The expansion ratio is r = V₂/V₁ = 2.5. With a = (Tₕ/T꜀)^1/(γ−1), the other volumes are V₃ = aV₂ and V₄ = aV₁; every pressure follows p = nRT/V. The adiabatic invariants are pV^γ and TV^γ−1. Constant heat capacities and γ = 1.4 are idealizations, not a real-gas equation valid for every temperature.

Heat Q is positive into the gas; work W is positive when done by the gas. Thus ΔU = Q−W. On 1→2, Q = W = nRTₕ ln r, ΔU = 0 and ΔS = nR ln r. On 2→3, Q = 0, W = nCᵥ(Tₕ−T꜀), ΔU = −W and ΔS = 0. On 3→4, Q = W = −nRT꜀ ln r, ΔU = 0 and ΔS = −nR ln r. On 4→1, Q = 0, W = nCᵥ(T꜀−Tₕ), ΔU = −W and ΔS = 0. Over the whole cycle, ΔU and ΔS sum to zero.

The main heat values are magnitudes: Qₕ = nRTₕ ln r enters the gas and Q꜀ = nRT꜀ ln r leaves it. The cold process therefore has negative Q in the signed table. Net work is W = nR(Tₕ−T꜀) ln r and efficiency is η = (Tₕ−T꜀)/Tₕ. These values come from the model equations, not an independent experiment. At Tₕ = T꜀ the isotherms overlap, both adiabats reduce to points, and net work and efficiency are zero. Corners 1/4 and 2/3 coincide; this is a degenerate limit, not a useful engine with finite power.

Calculate constructs the closed cycle at corner 1. Playback moves a cursor through it in eight display seconds, two per process, with logarithmic interpolation of volume. It supplies no physical cycle time or power. A frame advances at most 0.25 seconds; pause, hiding or leaving the tool discards elapsed time rather than catching up. Next corner pauses and advances to the next exact corner. At an inner corner, the displayed process is the following process; the final point belongs to 4→1. Play at the end restarts from corner 1. The slider selects an existing state without changing the cycle.

The linear diagram contains the entire cycle; its enclosed p–V area equals net work. The optional log/log view helps separate values at large temperature ratios, but area in that view is not work. Diagram and tables use litres and kilopascals; the model uses SI units. Editing inputs clears old results; Reset keeps the inputs. Returning within the app preserves the calculated cycle and view, paused. Full reloads restore valid inputs only.

Sources: OpenStax, University Physics Volume 2, §4.5: The Carnot Cycle and §3.6: Adiabatic Processes for an Ideal Gas. The fixed geometry, parameter ranges and playback timing are choices of this tool.