Problem 1.46 (Schroeder’s Intro to Thermal Physics)

Problem 1.46

Measured heat capacities of solids and liquids are almost always at constant pressure, not constant volume. To see why, estimate the pressure needed to keep $V$ fixed as $T$ increases, as follows.

(a) First imagine slightly increasing the temperature of a material at constant pressure. Write the change in volume, $\mathrm{d}V_1$ , in terms of $\mathrm{d}T$ and the thermal expansion coecient $\beta$ introduced in Problem 1.7.
(b) Now imagine slightly compressing the material, holding its temperature fixed. Write the change in volume for this process, $\mathrm{d}V_2$ , in terms of $\mathrm{d}P$ and the isothermal compressibility $\kappa_T$ , defined as
$\kappa_T \equiv -\dfrac{1}{V}(\dfrac{\partial V}{\partial P})_T$ .
(This is the reciprocal of the isothermal bulk modulus defined in Problem 1.39.)

(c) Finally, imagine that you compress the material just enough in part (b) to offset the expansion in part (a). Then the ratio of $\mathrm{d}P$ to $\mathrm{d}T$ is equal to $(\partial P/ \partial T)_V$ , since there is no net change in volume. Express this partial derivative in terms of $\beta$ and $\kappa_T$ . Then express it more abstractly in terms of the partial derivatives used to define $\beta$ and $\kappa_T$ . For the second expression you should obtain
$(\dfrac{\partial P}{\partial T})_V = - \dfrac{(\partial V / \partial T)_P}{(\partial V /\partial P)_T}$
This result is actually a purely mathematical relation, true for any three quantities that are related in such a way that any two determine the third.

(d) Compute $\beta$ , $\kappa T$ , and $(\partial P/ \partial T)_V$ for an ideal gas, and check that the three expressions satisfy the identity you found in part (c).
(e) For water at $25^oC$ , $\beta = 2.57 \cdot 10^{-4} K^{-1}$ and $\kappa_T = 4.52 \cdot 10^{-10} Pa^{-1}$ . Suppose you increase the temperature of some water from $20^oC$ to $30^oC$ . How much pressure must you apply to prevent it from expanding? Repeat the calculation for mercury, for which (at $25^oC$ ) $\beta=1.81 \cdot 10^{-4} K^{-1}$ and $\kappa_T = 4.04 \cdot 10^{-11} Pa^{-1}$ . Given the choice, would you rather measure the heat capacities of these substances at constant $V$ or at constant $P$ ?