Blundell Thermal Physics · Problem 12.2

Isothermal and adiabatic processes

Physics — Statistical Mechanics · adiabatic-expansion-general-gas-law

Problem

Assume that gases behave according to a law given by

pV = f(T)

, where

f(T)

is a function of temperature. Show that this implies

\begin{aligned} \left(\frac{\partial p}{\partial T}\right)_V &= \frac{1}{V}\frac{\mathrm{d}f}{\mathrm{d}T}, \tag{12.25}\\ \left(\frac{\partial V}{\partial T}\right)_p &= \frac{1}{p}\frac{\mathrm{d}f}{\mathrm{d}T}. \end{aligned}

Show also that

\begin{aligned} \left(\frac{\partial Q}{\partial V}\right)_p &= C_p\left(\frac{\partial T}{\partial V}\right)_p, \tag{12.27}\\ \left(\frac{\partial Q}{\partial p}\right)_V &= C_V\left(\frac{\partial T}{\partial p}\right)_V. \end{aligned}

In an adiabatic change, we have that

\mathrm{d}Q = \left(\frac{\partial Q}{\partial p}\right)_V \mathrm{d}p + \left(\frac{\partial Q}{\partial V}\right)_p \mathrm{d}V = 0. \tag{12.29}

Hence show that

pV^\gamma

is a constant.