Question

A gas mixture consists of $2$ moles of ${O_2}$ and $4$ moles of Ar at temperature T. Neglecting all vibrational modes, the total thermal energy of the system is
A. ${\text{4RT}}$
B. ${\text{15RT}}$
C. ${\text{9RT}}$
D. ${\text{11RT}}$

Answer 1

The thermal energy of the system is the sum total of its kinetic energy, the rotational energy, the kinetic energy and the vibrational energy. Remember the formulas of the internal thermal for mono-atomic and di-atomic molecules.

Complete step by step answer:
A gas is the mixture of two gases oxygen and Argon.
Now, we know that the internal energy of the di-atomic molecule is –
$I.E{._1} = \dfrac{5}{2}nRT$
Given that there are two moles of oxygen.
Therefore, place $n = 2$ in the above equation.
$I.E{._1} = \dfrac{5}{2} \times 2 \times RT$
Simplify –
$I.E{._1} = 5RT{\text{ }}....{\text{ (a)}}$
There are four moles of argon. Argon being mono-atomic molecule
Internal energy, $I.E{._2} = \dfrac{3}{2}nRT$
Therefore, place $n = 4$ in the above equation.
$I.E{._2} = \dfrac{3}{2} \times 4 \times RT \\\ \implies I.E{._2} = 6RT{\text{ }}......{\text{ (b)}} \\\$
Since, the gas is a mixture –
The total internal energy $I.E. = I.E{._1} + I.E{._2}$
By using the values of the equations (a) and (b)
$I.E. = 5RT + 6RT \\\ \implies I.E. = 11RT \\\$
Therefore, the total thermal energy of the system is $11RT.$

So, the correct answer is “Option D”.

Note:
Refer and remember the law of thermodynamics to solve these types of questions. Remember the difference between thermal energy and thermal expansion. Thermal expansion is an increase in the size of the substance when it is heated.