An object of mass \(m_1\), specific heat \(c_1\), and temperature \(T_1\) is placed
in contact with a second object of mass \(m_2\), specific heat \(c_2\), and
temperature \(T_2\) > \(T_1\). 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 $$\Delta S = m_1c_1 ln {T
\over T_1} + m_2c_2 ln {T' \over T_2}$$
and show that energy conservation requires that
$$m_1c_1 (T - T_1) = m_2c_2 (T_2 - T')$$
(b) Show that the entropy change \(\Delta S\), 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.