T-S Diagrams: Temperature-Entropy Diagrams

Core Idea

A T-S diagram (temperature vs. entropy) plots thermodynamic processes and cycles with temperature on the vertical axis and entropy on the horizontal; the area under a curve equals the heat transferred. For a reversible process, đQ_rev = T dS, so the area under a T-S curve directly gives heat transfer, making these diagrams particularly useful for steam cycles and heat engine analysis. T-S diagrams complement P-V diagrams in understanding thermodynamic cycles from different perspectives.

How It's Best Learned

Plot ideal cycles on T-S diagrams. Calculate heat transfer from areas. Compare reversible and irreversible process paths.

Common Misconceptions

• Thinking T-S and P-V diagrams show the same cycles (they look different but represent the same physics).
• Forgetting that only the area between the curve and the S-axis equals Q (not area between any reference line).
• Confusing entropy increase with heat transfer direction.

Explainer

The T-S diagram builds directly on your understanding of entropy as a state function and the second law. You know that entropy measures the dispersal of energy in a system, and that entropy cannot spontaneously decrease in an isolated system. The T-S diagram gives you a graphical language to reason about heat and thermodynamic cycles using these ideas directly — it is to heat what the P-V diagram is to work.

The key relationship is δQ_rev = T dS: for a reversible process, the infinitesimal heat transferred equals the absolute temperature multiplied by the change in entropy. On a T-S diagram, with temperature on the vertical axis and entropy on the horizontal, this expression becomes an area. Specifically, the heat transferred during any reversible process is the area under the curve traced on the T-S diagram. A reversible isothermal process (constant temperature) appears as a horizontal line — temperature is fixed, entropy increases as heat flows in. The heat absorbed is simply T × ΔS, readable directly as the rectangle under that line. A reversible adiabatic (isentropic) process appears as a vertical line: no heat flows, so entropy is constant, but temperature rises or falls.

When a complete thermodynamic cycle is plotted, it forms a closed loop. The area enclosed by the loop equals the net work output of the cycle — because net work equals heat in minus heat rejected, and each quantity is the area under its respective portion of the curve. The Carnot cycle is particularly elegant in T-S space: two horizontal isotherms (at T_H and T_C) connected by two vertical isentropes form a rectangle. The efficiency η = (T_H − T_C)/T_H is immediately visible as the ratio of the rectangle's height span to its top edge. No algebra required — the diagram makes the physics legible at a glance.

The T-S diagram does not replace the P-V diagram; the two represent the same physical processes from different perspectives. A P-V diagram emphasizes mechanical work; a T-S diagram emphasizes heat transfer and thermal efficiency. Using both together — plotting a cycle on each — gives you a complete picture of how a heat engine converts thermal energy into mechanical work and where the unavoidable losses to the cold reservoir appear. Real cycles (Rankine, Brayton) deviate from ideal rectangles in T-S space, and the shape of those deviations tells you precisely where irreversibility is eating into efficiency.