Question

One mole of an ideal diatomic gas whose pressure changes with volume as $P=\alpha V$, where $\alpha$ is constant. The molar heat capacity of the gas is
A. R
B. 2R
C. 3R
D. 4R

Answer 1

Hint : Firstly find the power at which the volume of the given ideal gas is raised to. Now, you could recall the expression for molar heat capacity in terms of molar specific heat capacity at constant volume, universal gas constant and the power mentioned above. You could substitute for $C_V$ of an ideal diatomic gas and thus find the answer.
Formula used:
Molar specific heat capacity,
$C=C_V-{\displaystyle \frac{R}{x-1}}$
Molar specific heat capacity at constant volume,
$C_V={\displaystyle \frac{d}{2}}R$
Complete answer:
In the question, we are given an ideal diatomic gas whose pressure variation is given by,
$P=\alpha V$
Where, $\alpha$ is some constant. We are supposed to find the molar heat capacity of the gas using this information.
So,
$P{V}^{-1}=\alpha$ …………………………. (1)
But we know that,
$P{V}^x=k$ ……………………………….. (2)
Where, k is a constant.
Then, by comparing (1) and (2), we get,
$x=-1$ …………………………….. (3)
We have the expression for molar specific heat capacity of a gas is given by,
$C=C_V-{\displaystyle \frac{R}{x-1}}$ …………………………………. (4)
Where, $C_V$ is the specific heat at constant volume, R is the universal gas constant and x is the power at which volume of the gas is raised to in equation (2).
But we know that molar specific heat capacity at constant volume for an ideal gas will be given by,
$C_V={\displaystyle \frac{d}{2}}R$
Where, d is known to be the number of degrees of freedom. As the given gas is ideal diatomic gas, the specific heat capacity at constant volume will be given by,
$C_V={\displaystyle \frac{5}{2}}R$…………………………………….. (5)
So, we could now substitute (3) and (5) in equation (4) to get,
$C={\displaystyle \frac{5}{2}}R-{\displaystyle \frac{R}{-1-1}}$
$\Rightarrow C={\displaystyle \frac{5}{2}}R+{\displaystyle \frac{R}{2}}$
$\therefore C=3R$
Therefore, we found the molar heat capacity of the gas to be 3R.

Hence, option C is found to be the right answer.
Note:
Heat capacity could be simply defined as the ratio of the amount of heat energy that is supplied to the substance to the resultant temperature change. If we specified the amount of substance that is, if the number of moles were specified, we get a molar specific heat capacity of the substance. SI unit of molar heat capacity is $J/{}^{\circ }C.mol$.