Audit entropy on the full isolated thermal boundary

Find the permitted final state

Two bodies exchange energy inside an isolated boundary, with no work or phase change. Constant heat capacities Ch and Cc give Ch(Tf−Th)+Cc(Tf−Tc)=0. Thus Tf is an energy-weighted temperature, not an arbitrary arithmetic mean unless capacities match.

For equal100 J/K capacities initially400 and300 K, Tf=350 K. The hot body loses5000 J and the cold gains5000 J. Conservation closes exactly, but it does not determine whether the entropy change is zero.

Use a reversible comparison path for a state function

Each entropy change is C ln(Tf/Ti). This expression can be evaluated by an imagined reversible path between the same endpoint states even though the actual finite-temperature heat exchange is irreversible. Hot entropy is about−13.3531 J/K; cold entropy is15.4151 J/K; their sum is about2.06193 J/K.

Dividing all finite heat by350 K or one starting temperature would silently replace a changing-temperature process with an inappropriate isothermal approximation. Nor should the hot body alone be called the universe while it exchanges heat with the cold body.

Distinguish an isothermal phase change

At phase coexistence, a reversible isothermal transition does permit ΔS=q_rev/T. For a separate fictional melting transition with latent enthalpy 4.00 kJ/mol at 400 K, the molar entropy change is +10.0 J/(mol K); freezing at that coexistence state reverses its sign. This does not justify dividing a changing-temperature body’s entire heat by one endpoint temperature.

Transfer and check

With Ch=100 J/K at450 K and Cc=50 J/K at300 K, energy gives Tf=400 K. Entropy is100ln(400/450)+50ln(400/300), about2.6058 J/K. Predict the signs before calculating. The graph’s other trial common temperatures are not alternate allowed isolated final states; only the energy-balanced point is feasible.

Source and evidence boundary

Conceptual reference checked 30 October 2026: OpenStax Chemistry 2e. All prose, cases, graphs and questions are original. These supplied thermodynamic, electrical, nuclear or orbital models are not actual measurements, apparatus instructions, synthesis protocols or practical certification. Retain conventions, independent constraints, changed-case reasoning and limits. No heating, wiring, electrolysis, source handling or chemical manipulation is requested.