QJEEM23Ph4d1a: JEE Main 2023 Physics - Class 11 JEEM23Ph4d1a Solution: Under isothermal condition, the pressure of a gas is given by , …

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:

$B = -V \frac{\Delta P}{\Delta V}$

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:

$P = aV^{-3}$

Differentiate $P$ with respect to $V$ :

$\frac{dP}{dV} = -3aV^{-4}$

Now we can use the definition of the bulk modulus:

$B = -V \frac{\Delta P}{\Delta V} = -V \frac{dP}{dV}$

Plug in the value for $\frac{dP}{dV}$ :

$B = -V(-3aV^{-4})$

Simplify the expression:

$B = 3aV^{-3}$

Notice that $3aV^{-3}$ is equal to $3P$ , since $P = aV^{-3}$ :

$B = 3P$

Therefore, the bulk modulus at constant temperature is equal to 3P.