Question

A gas undergoes a process in which its pressure $P$ and volume $V$ are related as $V{P^n} = {\text{constant}}$. The bulk modulus of the gas in the process is
A. $nP$
B. ${P^{1/n}}$
C. $P/n$
D. ${P^n}$

Answer 1

Use the formula for bulk modulus for the material of an object. This formula gives the relation between the change in pressure on the object, original volume of the object and change in volume of the object. Differentiate the given relation between the pressure and volume of the gas in the process and determine the bulk modulus of the gas in the process.

Formula used:
The bulk modulus $\beta$ of material of an object is given by the formula

$\beta = - \dfrac{{VdP}}{{dV}}$ …… (1)
Here, $V$ is the original volume of the object, $dV$ is the change in volume of the object and $dP$ is the change in pressure on the object.

Complete step by step answer:
We have given that the pressure of the gas is $P$ and volume of the gas is $V$.
We have also given that the pressure and volume of the gas in the process are related as
$V{P^n} = {\text{constant}}$
Let us differentiate the above relation with respect to volume.
$\dfrac{d}{{dV}}\left( {V{P^n}} \right) = 0$
$\Rightarrow {P^n}\dfrac{{dV}}{{dV}} + V\dfrac{{d{P^n}}}{{dV}} = 0$
We cannot differentiate the second term in the above expression. Let us use chain rule for the differentiation of the second term in the above equation.
$\Rightarrow {P^n}\left( 1 \right) + V\dfrac{{d{P^n}}}{{dP}}\dfrac{{dP}}{{dV}} = 0$
$\Rightarrow {P^n} + Vn{P^{n - 1}}\dfrac{{dP}}{{dP}}\dfrac{{dP}}{{dV}} = 0$
$\Rightarrow {P^n} + Vn{P^{n - 1}}\left( 1 \right)\dfrac{{dP}}{{dV}} = 0$
$\Rightarrow {P^{n - 1}}\left( {P + nV\dfrac{{dP}}{{dV}}} \right) = 0$
$\Rightarrow nV\dfrac{{dP}}{{dV}} = - P$
$\therefore V\dfrac{{dP}}{{dV}} = - \dfrac{P}{n}$
Substitute $- \beta$ for $V\dfrac{{dP}}{{dV}}$ in the above equation.
$\Rightarrow - \beta = - \dfrac{P}{n}$
$\therefore \beta = \dfrac{P}{n}$
Therefore, the bulk modulus for the gas in the process is $\dfrac{P}{n}$.

Hence, the correct option is C.

Note: In general, we use the formula for the bulk modulus of the material of an object with the positive sign. But actually the formula for the bulk modulus of material of an object includes a negative sign showing that the volume of the object decreases when the pressure is applied on it. Hence, the students should not forget to use this negative sign in the formula for bulk modulus. Otherwise, we will end with the same expression for bulk modulus of the gas in the process but with negative sign which is not in any of the options given.