Question

Two moles of helium gas is mixed with three moles of hydrogen molecules ( taken to be rigid). What is the molar specific heat of mixture at constant volume? (R=$8.3J/mol\,K$)
A.$21.6\,J/mol\,K$
B.$19.7\,J/mol\,K$
C.$17.4J/mol\,K$
D.$15.7\,J/mol\,K$

Answer 1

Hint:. In order to find the answer, calculate the value of ${{f}_{mix}}$ for the mixture of the gases, using the number of moles. Then find the molar specific heat of the mixture using this value of ${{f}_{mix}}$. There is a formula which relates ${{C}_{v}},R\,$ and ${{f}_{mix}}$.

Complete step by step answer:
In order to answer our question, we need to know what degree of freedom ‘f’. Molecular degrees of freedom check with the quantity of the way a molecule within the gas phase may move, rotate, or vibrate in a medium or space. Generally, there are three sorts of degrees of freedom that exist, those being translational, rotational, and vibrational. The quantity of degrees of freedom of every type possessed by a molecule depends on both the quantity of atoms within the molecule and therefore, also the geometry of the molecule, with geometry relating the way within which the atoms are arranged in the three-dimensional space. the quantity of degrees of freedom a molecule possesses by playing a job in estimating the values of varied thermodynamic variables using the equipartition theorem. These molecular degrees of freedom essentially describe how a molecule is in a position to contain and distribute its energy.
There are two types of degrees of freedom-translational and rotational degree of freedom. Translational degrees of freedom define the ability of a gas to move freely in space, in a straight line in all the three coordinates, whereas rotational degrees of freedom represent the number of ways a molecule can rotate about its centre of mass.

Let us find out the ${{f}_{mix}}$ of the mixture. It can be written as:
${{f}_{mix}}=\dfrac{{{n}_{1}}{{f}_{1}}+{{n}_{2}}{{f}_{2}}}{{{n}_{1}}+{{n}_{2}}}$
But ${{f}_{1}}=3$ for helium as it is monoatomic and ${{f}_{2}}=5$ for hydrogen as it is diatomic.
So,
${{f}_{mix}}=\dfrac{2\times 3+3\times 5}{5}=\dfrac{21}{5}$
Now, using ${{f}_{mix}}$, we get the value of${{C}_{v}}$as:
${{C}_{v}}=\dfrac{{{f}_{mix}}R}{2}=\dfrac{21}{5}\times \dfrac{R}{2}$
Which comes out to be $17.4\,J/mol\,K$.
So, the correct answer is “Option C”.

Note: Monoatomic molecules have 3 degrees of freedom as there are 1 translational degree in each i.e x,y,z direction. They have 0 degree of freedom for rotation. Whereas, in diatomic molecules, there are 3 translational degrees of freedom and 2 degrees of freedom for rotation which adds up to 5.