Discussion
Two identical blocks of a metal with heat capacity C are initially at temperatures \( T_{1} \) and \( T_{2} \), \( (T_{1} \lt T_{2}) \). They are brought in thermal contact. When the system reaches equilibrium the change in entropy would be :
Answer: B
The change in entropy is \( C \ln\left(\frac{(T_{1} + T_{2})^{2}}{4T_{1}T_{2}}\right) \).
Identical blocks reach the mean temperature \( T_{f} = \frac{T_{1} + T_{2}}{2} \). Each block's entropy changes by \( C\ln(T_{f}/T_{i}) \):
\[ \begin{aligned} \Delta S &= C \ln\frac{T_{f}}{T_{1}} + C \ln\frac{T_{f}}{T_{2}} \\ &= C \ln\frac{T_{f}^{2}}{T_{1}T_{2}} \\ &= C \ln\frac{(T_{1} + T_{2})^{2}}{4T_{1}T_{2}} \end{aligned} \]
Since the arithmetic mean exceeds the geometric mean, the argument is greater than 1 and \( \Delta S \gt 0 \), as an irreversible process requires. The option with the inverted fraction is negative; the ones with 1 or 2 in place of 4 forget that \( T_{f} \) is a half-sum, squared.
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