GATE 2022 CH – Question 47
A substance at 4 °C has a thermal expansion coefficient $\beta = \frac{1}{v}\left(\frac{\partial v}{\partial T}\right)_P = 0$ K$^{-1}$, an isothermal compressibility, $\kappa_T = -\frac{1}{v}\left(\frac{\partial v}{\partial P}\right)_T = 5 \times 10^{-4}$ Pa$^{-1}$ and a molar volume $v = 18 \times 10^{-6}$ m$^3$ mol$^{-1}$. If $s$ is the molar entropy, then at 4 °C, the quantity $\left[v\left(\frac{\partial s}{\partial v}\right)_T\right]$ evaluated for the substance is __________ J mol$^{-1}$ K$^{-1}$ (rounded off to the nearest integer).
Practise this question in The GATE Grind →
Show answer and explanation
Correct answer: 0
Explanation
By a Maxwell relation $\left(\frac{\partial s}{\partial v}\right)_T = \left(\frac{\partial P}{\partial T}\right)_v = \frac{\beta}{\kappa_T}$. With $\beta = 0$ (water at its density maximum, 4 °C) this is zero, so $v\left(\frac{\partial s}{\partial v}\right)_T = 0$.