Question

One mole of ideal monatomic gas ($\gamma =$5/3) is mixed with one mole of diatomic gas ($\gamma=$7/5). $\gamma$ denotes the ratio of specific heat at constant pressure, to that at constant volume. Find $\gamma$ for the mixture.
A. 3/2
B. 23/15
C. 35/23
D. 4/3

Answer 1

The degrees of freedom for a monatomic gas is $3$ and that for a diatomic gas is $5$.
The ratio of specific heats $\gamma$ for an ideal gas is
$\gamma =1+\dfrac{2}{f}$

Let ${{N}_{1}}$ moles of an ideal gas with ${{f}_{1}}$ degrees of freedom per molecule be mixed with ${{N}_{2}}$ moles of another ideal gas with ${{f}_{2}}$ degrees of freedom per molecule at a particular temperature. The ratio of specific heats $\gamma$ for the mixture is

$\gamma =\dfrac{{{N}_{1}}(2+{{f}_{1}})+{{N}_{2}}(2+{{f}_{2}})}{{{N}_{1}}{{f}_{1}}+{{N}_{2}}{{f}_{2}}}$

Complete step by step answer:
For the monatomic gas,
Number of moles, ${{N}_{1}}=1$
$\gamma =\dfrac{5}{3}$
Calculate the degrees of freedom ${{f}_{1}}$ of the monatomic gas by substituting the value of $\gamma$ in the $\gamma$-$f$ relation:
\dfrac{5}{3}=1+\dfrac{2}{{{f}_{1}}} \\\
$\implies \dfrac{5}{3}-1=\dfrac{2}{{{f}_{1}}} \\\$
$\implies \dfrac{2}{3}=\dfrac{2}{{{f}_{1}}} \\\$
$\implies {{f}_{1}}=3 \\\$

For the diatomic gas,
Number of moles, ${{N}_{2}}=1$
$\gamma =\dfrac{7}{5}$
Calculate the degrees of freedom ${{f}_{2}}$ of the monatomic gas by substituting the value of $\gamma$ in the $\gamma$-$f$ relation:
\dfrac{7}{5}=1+\dfrac{2}{{{f}_{2}}} \\\
$\implies \dfrac{7}{5}-1=\dfrac{2}{{{f}_{2}}} \\\$
$\implies \dfrac{2}{5}=\dfrac{2}{{{f}_{2}}} \\\$
$\implies {{f}_{2}}=5 \\\$

Now, substituting the values of ${{N}_{1}}$ ,${{N}_{2}}$and ${{f}_{1}}$ ,${{f}_{2}}$ in the formula for the ratio of specific heats $\gamma$ for the mixture:

$\gamma =\dfrac{1(2+{{3}_{1}})+1(2+5)}{(1)(3)+(1)(5)}\\\$
$\implies \gamma =\dfrac{12}{8} \\\$
$\implies \gamma =\dfrac{3}{2} \\\$

So, the correct answer is “Option A”.

Additional Information:
In monatomic gases, the atoms are not bound to each other and free to move in the three dimensional space. So, they have 3 degrees of freedom (translational). Only noble gases are monatomic at standard temperature and pressure.
In diatomic gases, two atoms are bonded to each other by a rigid bond. They have 5 degrees of freedom (3 translational and 2 rotational). Hydrogen, oxygen, nitrogen, etc exist as diatomic molecules.

Note:
The ratio of specific heats is a constant independent of the temperature. In addition, the expression holds true only for ideal gas mixture.