For a sample of perfect gas when its pressure is changed isothermally from $p_i$ to $p_f$, the entropy change

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:

Accepted Answer

For an ideal (perfect) gas, the change in entropy as a function of temperature and pressure is given by the general equation:
$\Delta S = nC_p \ln\left(\frac{T_f}{T_i}\right) + nR \ln\left(\frac{p_i}{p_f}\right)$
For an isothermal process, the temperature remains constant ($T_i = T_f$), so $\ln\left(\frac{T_f}{T_i}\right) = \ln(1) = 0$.
Substituting this into the equation, the first term becomes zero, and we are left with:
$\Delta S = nR \ln\left(\frac{p_i}{p_f}\right)$.

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