Question

The specific heat of an ideal gas depends on the temperature as
$A)\text{ }\dfrac{1}{T}$
$B)\text{ }T$
$C)\text{ }\sqrt{T}$
D) Does not depend on temperature

Answer 1

Hint: The specific heat at constant volume is the ratio of the change in internal energy of a gas with change in its temperature and the specific heat at constant pressure can be related to the specific heat at constant volume. By using the formula for the internal energy of a gas and finding its variation with temperature we can solve this problem.

Formula used:
$U=\dfrac{n}{2}RT$
${{C}_{V}}=\dfrac{dU}{dT}$
${{C}_{p}}={{C}_{V}}+R$

Complete step-by-step answer:
We will use the direct formula for the internal energy of a gas to find out its variation with temperature which will give us the value of the specific heat of the gas at constant volume.
The internal energy $U$ of one mole of a gas with $n$ degrees of freedom and at temperature $T$ is given by
$U=\dfrac{n}{2}RT$ --(1)
Where $R=8.314J.mo{{l}^{-1}}{{K}^{-1}}$ is the universal gas constant.
Now, the specific heat constant at constant volume ${{C}_{V}}$ of an ideal gas is the ratio of the change in its internal energy $U$ to the change in temperature $T$
${{C}_{V}}=\dfrac{dU}{dT}$ --(2)
Now, putting (1) in (2), we get
${{C}_{V}}=\dfrac{d\left( \dfrac{n}{2}RT \right)}{dT}=\dfrac{n}{2}R$ --(3)
Also, the specific heat at constant pressure ${{C}_{P}}$ of an ideal gas is related to the specific heat at constant volume ${{C}_{V}}$ by
${{C}_{P}}={{C}_{V}}+R$ --(4)
Now, putting (3) in (4), we get
${{C}_{P}}=\dfrac{n}{2}R+R=\left( \dfrac{n}{2}+1 \right)R$ --(5)
From (3) and (5), we can see that both ${{C}_{V}}$ and ${{C}_{P}}$ have no relation with the temperature $T$ of the gas and are not dependent on it.

So, the correct answer is “Option D”.

Note: The specific heat of an ideal gas at constant pressure is actually the ratio of the change in the enthalpy of the gas with the change in its temperature. Similar to the internal energy, the enthalpy of a gas also depends directly upon its temperature and therefore, similar to the process of finding out ${{C}_{V}}$, the term $T$ vanishes when we differentiate the enthalpy with the temperature. This is the exhaustive method of proving that none of the specific constants depends upon the temperature of the gas.