Change in Entropy — Question 1

Question 1

Temperature and volume contributions to gas entropy add independently

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 gas has $C_V=5R/2$. It changes from $(300\,\mathrm K,V_i)$ to $(450\,\mathrm K,2V_i)$.

1. Calculate the temperature and volume contributions and their sum.

2. Construct two reversible reference legs that give the same sum, first heating at fixed volume and then expanding isothermally. Find the pressure ratio between the actual endpoints.

Question 1 - Solution

Temperature and volume contributions to gas entropy add independently

(a) Method and calculation

$$\Delta S_T=.500(\tfrac52)R\ln(450/300),\quad\Delta S_V=.500R\ln2.$$$$\Delta S=\tfrac12R[\tfrac52\ln(3/2)+\ln2]\approx7.09521\,\mathrm{J/K}.$$

(b) Interpretation and further calculation

$$\Delta S_1=\int_{300}^{450}\frac{nC_V}{T}dT,\quad\Delta S_2=\int_{V_i}^{2V_i}\frac{nR}{V}dV.$$$$\Delta S_1+\Delta S_2=\Delta S,\quad p_f/p_i=(450/300)/2=.750.$$The endpoint formula applies even if the actual process is irreversible; the reference legs only evaluate the state-function difference.

Physical check

Temperature rises while pressure falls here because volume grows by a larger factor.