Bulk Modulus of Gases

Problems from IIT JEE

Problem (IIT JEE 1998): A given quantity of an ideal gas is at pressure $p$ and absolute temperature $T$. The isothermal bulk modulus of the gas is,

  1. $\frac{2}{3}p$
  2. $p$
  3. $\frac{3}{2}p$
  4. $2p$

Solution: Isothermal bulk modulus is ratio of volumetric stress to volumetric strain at constant temperature i.e., \begin{align} B=-\frac{\mathrm{d}p}{\mathrm{d}V/V}=\left. -V\frac{\mathrm{d}p}{\mathrm{d}V}\right|_{T}. \end{align} Differentiate the ideal gas equation, $pV=nRT$, at constant $T$ to get, \begin{align} \left.\frac{\mathrm{d}p}{\mathrm{d}V}\right|_{T}=-\frac{p}{V}. \end{align} Substitute ${\mathrm{d}p}/{\mathrm{d}V}$ from second equation into first equation to get $B=p$.