Question

The bulk modulus of a gas is defined as $B=-V{\displaystyle \frac{dP}{dV}}$ . For an adiabatic process the variation of B is proportional to $P^n$ for an ideal gas, n is :
$\begin{array}{rl}& \text{A}\text{. zero}\\ & \text{B}\text{. 1}\\ & \text{C}\text{. }{\displaystyle \frac{\text{5}}{\text{3}}}\\ & \text{D}\text{. 2}\end{array}$

Answer 1

Hint:Use ideal gas law for the adiabatic process. Express P in terms of volume and take the derivative of the pressure respect to the volume and apply the derivative in the given equation to get the value of B in terms of pressure to get the value of n


Complete step by step solution:
We know an ideal gas in the adiabatic process follows the Law $P{V}^{\gamma }=k$ where k is the constant.
here P is the pressure exerted by the gas and V is the volume occupied by the gas.
So now we can express the pressure in terms of volume as:
$P={\displaystyle \frac{k}{V^{\gamma }}}$
The derivative of the $x^n$ is $n\times x^{n-1}$.
Applying the same result we can calculate the derivative of the P. Now no, we can take the derivative of pressure with respect to volume as:
${\displaystyle \frac{dP}{dV}}=-k\gamma V^{-\gamma -1}$
Now substitute the value of the derivative of P in the given equation we get
$B=-V{\displaystyle \frac{dP}{dV}}=k\gamma {\displaystyle \frac{P}{k}}=\gamma P$
Thus we found The B in terms of pressure and we have seen that B is directly proportional to the pressure
Hence we can write $B=K\times P$.
As the power of the P term is 1 we get the value of $n=1$.

Thus we have used ideal gas law for the adiabatic process and found the value of ‘n’ as ‘1’.


Note: We need to use the calculus here to calculate the value of the derivative and here we have to be careful about the formulae of the derivatives to get the correct result. Take or make notes of the derivatives as a list to remember and apply the derivatives in the problems.