Discussion
One block of certain metal is at temperature \(T_{1}\) and a second identical block is at a temperature \(T_{2}\). These blocks are brought in contact and the system is thermally isolated from the surroundings. Assume that the heat capacity at constant volume C of each block is independent of temperature T. The increase in the entropy of the universe, when the system comes to an equilibrium is :
Answer: C
The entropy increase is C ln\(\left[(T_{1} + T_{2})^{2}/4T_{1}T_{2}\right]\).
Identical blocks with equal heat capacity reach the final temperature \(T_{f} = (T_{1} + T_{2})/2\). Each block's entropy change is \(C\ln(T_{f}/T_{i})\), so:
\[ \begin{aligned} \Delta S &= C\ln\frac{T_{f}}{T_{1}} + C\ln\frac{T_{f}}{T_{2}} = C\ln\frac{T_{f}^{2}}{T_{1}T_{2}} \\ &= C\ln\frac{(T_{1} + T_{2})^{2}}{4T_{1}T_{2}} \end{aligned} \]
Because the arithmetic mean exceeds the geometric mean, the argument is greater than 1 and \(\Delta S \gt 0\), as required for an irreversible process. Option D is the inverse ratio, which would give a decrease.
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