Discussion
Consider a closed system is subdivided into two subsystems 1 and 2, which are connected such that internal energy and particles may be exchanged and volume of the two remain constant. Under this condition the minimum value of the quantity \( \left(\frac{dU_{1}}{dN_{1}}\right) \) is given by (here μ's and T's are the respective chemical potential and temperature) :
Answer: A
The minimum value is \( \frac{\mu_{1}T_{2} - \mu_{2}T_{1}}{T_{2} - T_{1}} \).
The whole system is closed, so \( dU_{2} = -dU_{1} \) and \( dN_{2} = -dN_{1} \), with volumes fixed. The total entropy change is
\[ \begin{aligned} dS = &\left(\frac{1}{T_{1}} - \frac{1}{T_{2}}\right) dU_{1} \\ - &\left(\frac{\mu_{1}}{T_{1}} - \frac{\mu_{2}}{T_{2}}\right) dN_{1} \end{aligned} \]
The second law requires \( dS \ge 0 \). Taking \( T_{2} \gt T_{1} \) and particles flowing into subsystem 1 (\( dN_{1} \gt 0 \)), divide through and multiply the fraction by \( T_{1}T_{2} \):
\[ \frac{dU_1}{dN_1}\ge\frac{\mu_1T_2-\mu_2T_1}{T_2-T_1} \]
The other three options change a sign in the numerator or the denominator, which no step of this derivation produces.
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