Two bodies in contact: ΔS of the universe

Problem

Two identical bodies of mass $m=1\ \text{kg}$ and specific heat $c=1000\ \text{J/(kg·K)}$, one at $T_1=400\ \text{K}$ and the other at $T_2=300\ \text{K}$, are put in thermal contact until they reach a common temperature. Calculate the total entropy change of the universe. Is it consistent with the second law?

Solution

Equal masses and specific heats $\Rightarrow$ the equilibrium temperature is the arithmetic mean: $T_e=\frac{T_1+T_2}{2}=\frac{400+300}{2}=350\ \text{K}$ Each body exchanges heat reversibly with itself, so $\Delta S=mc\ln\dfrac{T_e}{T_i}$. Hot body (cools down): $\Delta S_1=mc\ln\dfrac{350}{400}=1000\cdot\ln 0{,}875\approx -133{,}5\;\text{J/K}$. Cold body (warms up): $\Delta S_2=mc\ln\dfrac{350}{300}=1000\cdot\ln 1{,}1\overline{6}\approx +154{,}2\;\text{J/K}$. Sum: $\Delta S_\text{univ}=\Delta S_1+\Delta S_2\approx -133{,}5+154{,}2=+20{,}7\;\text{J/K}$ The result is positive: the process is irreversible and consistent with the second law. $\ev{\Delta S_\text{univ}\approx +21\;\text{J/K}>0}$

Collegamenti

Argomenti: Entropy and the second law Concetti: Entropy · Second law of thermodynamics Competenze: Entropy balance