# Relating the maximum entropy with the pressure given in terms of volume

For a monoatomic ideal gas let the pressure of gas in a container be given as $p=p_0- \alpha V$ Where $p_0$ and $\alpha$ are constants and $V$ is the volume of gas. Then find the volume at which entropy of gas is maximum.

The major problem I have here is how can one just relate the entropy with only the value of pressure of gas in terms of volume. I have tried differentiating both sides to get $\frac {dP}{dV}$. Also used some formulas but they were of no use. Somebody please help.

For an ideal gas, the temperature is pV/nR. So, $$T=\frac{(p_0-\alpha V)V}{nR}$$So the entropy of the ideal gas is $$S=const+nR(1.5\ln{T}+\ln{V})=const'+nR(1.5\ln[(p_0-\alpha V)V]+\ln{V})$$
• The answer to you first question is Yes. For your second comment, I was hoping you would recognize this from the equation for the entropy change of an ideal gas: $\Delta S=nC_v\ln{(T_2/T_1)}+nR\ln{(V_2/V_1)}$ – Chet Miller Jan 28 '18 at 13:02