Under isothermal condition, the pressure of a gas is given by $\mathrm{P}=a \mathrm{~V}^{-3}$, where $a$ is a constant and $\mathrm{V}$ is the volume of the gas. The bulk modulus at constant temperature is equal to

The bulk modulus ($B$) of a substance is defined as the ratio of the infinitesimal pressure increase ($\Delta P$) to the relative decrease in volume ($\frac{-\Delta V}{V}$) at constant temperature: <br/><br/> $B = -V \frac{\Delta P}{\Delta V}$ <br/><br/> To find the bulk modulus for the given pressure-volume relationship, we first need to find the differential change in pressure with respect to volume: <br/><br/> $P = aV^{-3}$ <br/><br/> Differentiate $P$ with respect to $V$: <br/><br/> $\frac{dP}{dV} = -3aV^{-4}$ <br/><br/> Now we can use the definition of the bulk modulus: <br/><br/> $B = -V \frac{\Delta P}{\Delta V} = -V \frac{dP}{dV}$ <br/><br/> Plug in the value for $\frac{dP}{dV}$: <br/><br/> $B = -V(-3aV^{-4})$ <br/><br/> Simplify the expression: <br/><br/> $B = 3aV^{-3}$ <br/><br/> Notice that $3aV^{-3}$ is equal to $3P$, since $P = aV^{-3}$: <br/><br/> $B = 3P$ <br/><br/> Therefore, the bulk modulus at constant temperature is equal to 3P.