Change in Entropy — Question 10

Question 10

Equal-entropy endpoints do not determine the heat-transfer history

For a fixed amount of ideal gas with constant molar capacities, use $\Delta S=nC_V\ln(T_f/T_i)+nR\ln(V_f/V_i)$ and $C_p=C_V+R$, with $R=8.314$ J/(mol K). A .500 mol monatomic gas starts at 300 K. Its final volume is half its initial volume and its final entropy equals its initial entropy.

1. Find the required final temperature.

2. Construct a reversible two-leg path: first isothermal compression at 300 K, then fixed-volume heating. Calculate entropy changes on the legs and determine whether zero net entropy change forces zero heat on every leg.

Question 10 - Solution

Equal-entropy endpoints do not determine the heat-transfer history

(a) Method and calculation

$$0=\tfrac32nR\ln(T_f/300)+nR\ln(1/2).$$$$T_f=300\,2^{2/3}\approx476.220\,\mathrm K.$$

(b) Interpretation and further calculation

$$\Delta S_1=-nR\ln2,\quad\Delta S_2=\tfrac32nR\ln(2^{2/3})=nR\ln2.$$$$Q_1=-nR(300)\ln2,\quad Q_2=\tfrac32nR(T_f-300)>0.$$Each leg transfers heat although the endpoint entropy difference is zero. A reversible adiabatic path is another route to these endpoints, not the only possible route.

Physical check

Isentropic endpoints do not establish that the actual process was adiabatic.