Question

A gas undergoes a change according to the law \[P = {P_0}{e^{\alpha V}}\]. Calculate the bulk modulus of the gas.
A. \[P\]
B. \[\alpha PV\]
C. \[\alpha V\]
D. \[\dfrac{{PV}}{\alpha }\]

Answer 1

Hint:In order to proceed with the question let us see the data that they have given and using the data we will solve it. They are saying that, if a gas undergoes a change in volume or pressure, then using the above equation we need to find the bulk modulus of a gas. Bulk modulus defines the ability of a material to resist deformation whenever there is a volume change.

Formula Used:
The formula to find the bulk modulus is given by,
\[B = V\dfrac{{dP}}{{dV}}\]……… (1)
Where,
\[V\] is the actual volume of a substance.
\[dP\] is a change in pressure.
\[dV\] is a change in volume.

Complete step by step solution:
To find the bulk modulus of gas, consider the equation
\[P = {P_0}{e^{\alpha V}}\]……. (2)
Differentiate the above equation with respect to volume V, then we get,
\[\dfrac{{dP}}{{dV}} = {P_0}{e^{\alpha V}}\alpha \]
From equation (2) we can write as,
\[\dfrac{{dP}}{{dV}} = P\alpha \]
Now multiply the above equation both sides by volume V we get,
\[V\dfrac{{dP}}{{dV}} = P\alpha V\]
From equation (1) we can write as,
\[ \therefore B = \alpha PV\]
Therefore, the bulk modulus of gas is \[\alpha PV\].

Hence, option B is the correct answer.

Note:There are three types of moduli of elasticity : Bulk modulus, rigidity modulus, and Young’s modulus. The bulk modulus of air at normal pressure is 1.4 for air. So, the bulk modulus of gas is based on its pressure. The atmospheric pressure of air at Standard temperature and pressure is \[1.0132 \times {10^5}N{m^{ - 2}}\] and the bulk modulus in the order of \[2.15 \times {10^9}N{m^{ - 2}}\].