Question

A gas mixture consists of 2 moles of oxygen and 4 moles of Argon at temperature T. Neglecting all vibrational moles, the total internal energy of the system is:
A. 4 RT
B. 15 RT
C. 9 RT
D. 11 RT

Answer 1

In this question we have been asked to calculate the total energy of the system consisting 2 moles of oxygen and 4 moles of Argon. Now, the ideal gas equation we know is that the internal energy is given as the product of the number of moles, the gas constant R and temperature. We have been asked to neglect the vibrational energy. Therefore, we shall first calculate the individual internal energy for both the given gases. The sum of the internal energies of both gases will be the total internal energy.

Formula used:
$TE=KE+PE+VE$
Where,
TE is total energy
KE is kinetic energy
PE is potential energy
VE is vibrational energy
$U=\dfrac{1}{2}nFRT$
Where,
n is the number of moles
F is the degree of freedom

Complete answer:
In this question we have been given that there are two moles of oxygen and 4 moles of Argon.
Now, to calculate the total internal energy
We know, total energy is given by,
$TE=KE+PE+VE$
Now, it is given that vibrational energy of the gases be neglected. The potential energy of the gases will also be taken as zero, since there is no vertical moment of gases mentioned.
Therefore, total energy will be equal to internal kinetic energy.
We know,
$U=\dfrac{1}{2}nFRT$………………. (1)
Where F is the degree of freedom.
Now, we know that degree of freedom is given by,
$F=2n+1$ ……………………. (2)
We know that oxygen is a diatomic molecule.
Therefore, substituting n=2 in equation (2)
We get,
Degree of freedom for oxygen i.e. ${{F}_{o}}=5$
Substituting the values in equation (1)
We get,
${{U}_{o}}=\dfrac{1}{2}\times 2\times 5RT$
Therefore,
${{U}_{o}}=5RT$………………. (3)
Similarly, for Argon
Argon is a monoatomic molecule. Therefore, n=1
We get,
${{F}_{A}}=3$
Substituting the above values in equation (1)
We get,
${{U}_{A}}=\dfrac{1}{2}\times 4\times 3RT$
Therefore,
${{U}_{A}}=6RT$………….. (4)
Now, total energy is given as
${{U}_{T}}={{U}_{A}}+{{U}_{0}}$
Therefore, from (3) and (4)
We get,
${{U}_{T}}=5RT+6RT$
Therefore,
${{U}_{T}}=11RT$

Therefore, the correct answer is option D.

Note:
The degree of freedom refers to the number of ways a gas molecule can move freely in space. There are three types of degrees of freedom namely, translational rotational and vibrational. The molecule transits around three axis and rotates at two axis perpendicular to the line joining the two atoms in the molecule. The vibrational degree of freedom has only one motion.