Chapter 20, Problem 20.53P

CALC An object of mass m₁, specific heat c₁, and temperature T₁ is placed in contact with a second object of mass m₂ specific heat c₂, and temperature T₂ > T₁. As a result, the temperature of the first object increases to T and the temperature of the second object decreases to T^′. (a) Show that the entropy increase of the system is

$\Delta S=m_1{c}_1\ln \frac{T}{T_1}+m_2{c}_2\ln \frac{T^{\prime }}{T_2}$

and show that energy conservation requires that

$m_1{c}_1\left(T-T_1\right)=m_2{c}_2\left(T_2-T^{\prime }\right)$

(b) Show that the entropy change ΔS. considered as a function of T, is a maximum if T = T^′, which is just the condition of thermodynamic equilibrium. (c) Discuss the result of part (b) in terms of the idea of entropy as a measure of randomness