Serway & Jewett — The Kinetic Theory of Gases: Problem 30.

The compressibility \(\kappa\) of a substance is defined as the fractional change in volume of that substance for a given change in pressure: \[ \kappa = -\frac{1}{V} \frac{dV}{dP} \] (a) Explain why the negative sign in this expression ensures \(\kappa\) is always positive. (b) Show that if an ideal gas is compressed isothermally, its compressibility is given by \(\kappa_1 = 1/P\). (c) What If? Show that if an ideal gas is compressed adiabatically, its compressibility is given by \(\kappa_2 = 1/(\gamma P)\). Determine values for (d) \(\kappa_1\) and (e) \(\kappa_2\) for a monatomic ideal gas at a pressure of 2.00 atm.