Clausius relation for reversible heat transfer: Select the correct differential relation between heat interaction and entropy at absolute temperature T.

Introduction / Context:
The Clausius definition of entropy connects infinitesimal reversible heat transfer to temperature. Recognizing the correct differential relation is crucial for deriving property changes, performing cycle analyses, and understanding the second law's mathematical form.

Given Data / Assumptions:

• Reversible heat transfer at system boundary.
• Absolute temperature T is uniform at the boundary where δQ is exchanged.
• Infinitesimal process step permitting differential notation.

Concept / Approach:
For a reversible process step, the entropy change is defined by ds = δQ_rev / T. Rearranging yields the differential heat interaction δQ_rev = T * ds. This is a definition-based identity for reversible transfer; for irreversible processes, δQ ≠ T ds, but the inequality ∮(δQ/T) ≤ 0 still holds when integrating around a cycle (Clausius inequality).

Step-by-Step Solution:

Verification / Alternative check:
Units check: [δQ] = energy, [T * ds] = temperature * entropy change = energy, confirming consistency.

Why Other Options Are Wrong:

• T/ds and ds/T: Invert the definition and are dimensionally incorrect for heat.
• None of these: Incorrect because the correct relation is provided in option A.

Common Pitfalls:
Applying δQ = T ds to irreversible steps; in that case only ds ≥ δQ/T applies when considering system plus surroundings.

Final Answer:
δQ = T * ds