Question

The molar specific heat of a gas as given from the kinetic theory is ${\displaystyle \frac{5}{2}}R$. If it is not specified whether it is $C_p$ or $C_v$, one could conclude that the molecules of the gas-
(A) Are definitely monoatomic
(B) Are definitely rigid diatomic
(C) Are definitely non-rigid diatomic
(D) Can be monatomic or rigid diatomic

Answer 1

Hint: The relation between the molar specific heat of a gas at constant volume($C_v$) and at constant pressure($C_p$) is given by- $C_p=C_v+R$, Where R is the gas constant. The values corresponding to both monoatomic and diatomic gases can be calculated and compared to give the answer.

Step by step answer
According to the equipartition theorem, for a gas, each degree of freedom contributes internal energy equal to ${\displaystyle \frac{1}{2}}RT$ per mole. Thus,
For a monatomic gas, degree of freedom, $f=3$
The internal energy can be given by, $U=3\times {\displaystyle \frac{1}{2}}RT={\displaystyle \frac{3}{2}}RT$
The molar specific heat at constant volume, $C_v$ is the defined as the change in internal energy per unit temperature so can be written as, $C_v={\displaystyle \frac{3}{2}}R$
Now, the relation between $C_p$ and $C_v$ is given by-
$C_p-C_v=R$
$C_p=C_v+R$
$C_p$for a monatomic gas is given by,
$C_p={\displaystyle \frac{3}{2}}R+R$
$C_p={\displaystyle \frac{5}{2}}R$
Now a diatomic gas can have two extra degrees of freedom due to rotation along two independent axes. This makes the total degrees of freedom for a linear diatomic gas as, $f=5$
From this we get $C_v={\displaystyle \frac{5}{2}}R$
And the $C_p$ can be calculated as-
$C_p={\displaystyle \frac{5}{2}}R+R$
$C_p={\displaystyle \frac{7}{2}}R$
For a non-rigid diatomic molecule the degree of freedom is 6.
So $C_v=3R$
And $C_p=4R$
In monoatomic, $C_p={\displaystyle \frac{5}{2}}R$, and in diatomic, $C_p={\displaystyle \frac{5}{2}}R$ but in non-rigid diatomic, $C_P\ne C_V\ne {\displaystyle \frac{5}{2}}R$.
Since the specific heat can be both $C_p$ and $C_v$ therefore it can be a monatomic or rigid diatomic gas.

Option (D) is correct.

Note: A rigid diatomic molecule is defined as a molecule which does not possess any vibrational energy. A non-rigid diatomic molecule on the other hand has vibrational energy and thus has an extra degree of freedom, making the total number of degrees of freedom as 6.