Question
For a sample of perfect gas when its pressure is changed isothermally from $${p_i}$$ to $${p_f},$$ the entropy change is given by
A.
$$\Delta S = nR\,\ln \left( {\frac{{{p_f}}}{{{p_i}}}} \right)$$
B.
$$\Delta S = nR\,\ln \left( {\frac{{{p_i}}}{{{p_f}}}} \right)$$
C.
$$\Delta S = nRT\,\ln \left( {\frac{{{p_f}}}{{{p_i}}}} \right)$$
D.
$$\Delta S = RT\,\ln \left( {\frac{{{p_i}}}{{{p_f}}}} \right)$$
Answer :
$$\Delta S = nR\,\ln \left( {\frac{{{p_i}}}{{{p_f}}}} \right)$$
Solution :
$$\eqalign{
& {\text{Entropy change is given as,}} \cr
& \Delta S = n{C_p}\,\ln \frac{{{T_f}}}{{{T_i}}} + nR\,\ln \frac{{{p_i}}}{{{p_f}}}\,\,\,...{\text{(i)}} \cr
& {\text{For isothermal process, }}{T_i} = {T_f} \cr
& \therefore \,\,n{C_p}\,\ln \frac{{{T_f}}}{{{T_i}}} = n{C_p}\,\ln \frac{{{T_i}}}{{{T_i}}} = 0\,\,\left[ {\ln \,1 = 0} \right] \cr
& {\text{From Eq}}{\text{. (i)}}\,\,\Delta S = nR\,\ln \,\frac{{{p_i}}}{{{p_f}}} \cr} $$