Question

The gas mixture consists of 3 moles of oxygen and 5 moles of argon at temperature T. Considering only translational and rotational modes , the total internal energy of the system is
(A) 12 RT
(B) 20 RT
(C) 15 RT
(D) 4 RT

Answer 1

We will calculate the internal energies for oxygen and argon separately because one is diatomic the other is monoatomic. The measure of their kinetic energies will give us the internal energies and we will have to calculate degrees of freedom of each gas.

Complete step by step answer:
The internal energy of the system is the measure of its kinetic energy . Kinetic energy is different for different gas molecules. It is different for monoatomic gas while it is different for diatomic gas. The internal energy of a gas is given by :
$U = n\dfrac{f}{2}RT$
where $n =$no. of moles
$f =$ degree of freedom
$T =$ temperature
$R =$universal gas constant
Moles of oxygen= 3
Moles of argon=5
We will now calculate degrees of freedom of each gas.
Oxygen is a diatomic gas and for a diatomic molecule :
Degree of freedom of translational motion of a diatomic molecule =3
Degree of freedom of rotational motion = 2
Total degrees of freedom =5
$\Rightarrow {U_{oxygen}} = 3\dfrac{5}{2}RT \\\ \Rightarrow{U_{oxygen}} = \dfrac{{15}}{2}RT \\\$
Argon is monoatomic gas and for a monatomic gas:
Degree of freedom of translational motion= 3
Degree of freedom of rotational motion=0
Total degrees of freedom=3
$\Rightarrow {U_{argon}} = 5\dfrac{3}{2}RT \\\ \Rightarrow{U_{argon}} = \dfrac{{15}}{2}RT \\\$
The internal energy will be the summation of the two internal energies.
$\Rightarrow {U_{system}} = {U_{oxygen}} + {U_{argon}} \\\ \Rightarrow{U_{system}} = \dfrac{{15}}{2}RT + \dfrac{{15}}{2}RT \\\ \therefore{U_{system}} = 15RT \\\$

Hence , the correct option is C.

Note: The degree of freedom of oxygen is greater than degree of freedom of argon. This implies as the atomicity of an element increases so does its degree of freedom. The internal energies of two isolated systems can be added.The internal energy of a system depends on temperature and degrees of freedom. On increasing the temperature , the internal energy of the system increases .