Thermodynamic diagrams – area under the T–s curve On a temperature–entropy (T–s) diagram, the area under the process curve between two entropy states represents which thermodynamic quantity for that process?

Introduction / Context:
Graphical thermodynamics uses property diagrams to visualize processes. The temperature–entropy (T–s) plane is especially useful because heat transfer appears directly as geometric area.

Given Data / Assumptions:

• Closed system or control mass undergoing a quasi-equilibrium process.
• Entropy is a state property; temperature is absolute temperature.
• Sign convention: heat added to the system is positive.

Concept / Approach:
The fundamental relation for a reversible differential heat transfer is δQ_rev = T * dS. Integrating along a reversible path gives Q_rev = ∫ T dS, which corresponds geometrically to the area under the T–s curve between s1 and s2. For an irreversible process, the integral of T dS along a suitably defined reversible path connecting the same endpoints represents the heat transferred; sign indicates absorbed (positive area) or rejected (negative area).

Step-by-Step Solution:
1) Write δQ_rev = T * dS for reversible segments.2) Integrate from s1 to s2: Q = ∫(s1→s2) T ds.3) Interpret on the diagram: the area under the curve equals Q, positive if the curve lies above the axis as s increases (heat in), negative if heat out.4) Conclude that the area represents heat transfer, with sign determined by process direction.

Verification / Alternative check:
For isothermal process at temperature T0, area = T0 * (s2 − s1), exactly matching the textbook formula for reversible isothermal heat exchange.

Why Other Options Are Wrong:

• “Heat absorbed” or “heat rejected” alone miss the directionality; the area can represent either depending on process direction.
• “Shaft work only” and “change in internal energy only” are not generally represented by the T–s area.

Common Pitfalls:
Assuming the area applies only to reversible processes; with care, the integral concept still links heat to a reversible path between the same states.

Final Answer:
either heat absorbed or heat rejected (signed by direction)