Question

A gas mixture consists of 2 moles of oxygen and 4 moles of argon at temperature $T$. Neglecting all vibrational modes, the total internal energy of the system is
A. $4RT$
B. $5RT$
C. $15RT$
D. $11RT$

Answer 1

Use the formula for internal energy of the monatomic gas and internal energy of the diatomic gas in terms of number of moles of the gas, gas constant and temperature of the gas. Substitute the values of the number of moles of the oxygen and argon gas in the formula for internal energy of the diatomic and monatomic gas respectively. The total internal energy of the system is the sum of the internal energies of these two gases.

Formulae used:
The internal energy ${U_{{\text{monoatomic}}}}$ of a monatomic gas is given by
${U_{{\text{monoatomic}}}} = \dfrac{3}{2}nRT$ …… (1)
Here, $n$ is the number of moles of the monatomic gas, $R$ is the gas constant and $T$ is the temperature of the gas.
The internal energy ${U_{{\text{diatomic}}}}$ of a diatomic gas is given by

${U_{{\text{diatomic}}}} = \dfrac{5}{2}nRT$ …… (2)
Here, $n$ is the number of moles of the diatomic gas, $R$ is the gas constant and $T$ is the temperature of the gas.

Complete step by step answer:
We have given that the number of moles of the oxygen gas are 2 and the number of moles of the argon gas are 4.
${n_{{\text{oxygen}}}} = 2\,{\text{mol}}$
${n_{{\text{argon}}}} = 4\,{\text{mol}}$
We have asked to determine the total internal energy of the gas mixture of oxygen and argon. The oxygen gas exists as a diatomic gas. Hence, the internal energy of the oxygen is
${U_{{\text{oxygen}}}} = \dfrac{5}{2}{n_{{\text{oxygen}}}}RT$
Substitute $2\,{\text{mol}}$ for ${n_{{\text{oxygen}}}}$ in the above equation.
${U_{{\text{oxygen}}}} = \dfrac{5}{2}\left( {2\,{\text{mol}}} \right)RT$
$\Rightarrow {U_{{\text{oxygen}}}} = 5RT$
The argon gas exists as a diatomic gas. Hence, the internal energy of the argon is
${U_{{\text{argon}}}} = \dfrac{3}{2}{n_{{\text{argon}}}}RT$
Substitute $4\,{\text{mol}}$ for ${n_{{\text{argon}}}}$ in the above equation.
${U_{{\text{argon}}}} = \dfrac{3}{2}\left( {4\,{\text{mol}}} \right)RT$
${U_{{\text{argon}}}} = 6RT$

The total internal energy $U$ of the system is the sum of the internal energy of the oxygen gas in the mixture and internal energy of the argon gas in the mixture.
$U = {U_{{\text{oxygen}}}} + {U_{{\text{argon}}}}$
Substitute $5RT$ for ${U_{{\text{oxygen}}}}$ and $6RT$ for ${U_{{\text{argon}}}}$ in the
above equation.
$U = 5RT + 6RT$
$\Rightarrow U = 11RT$
Therefore, the total internal energy of the system of the gas is $11RT$.
Hence, the correct option is D.

Note: The students should keep in mind that the formula for internal energy of the oxygen and argon gas is different and not the same because oxygen is diatomic gas and argon is monoatomic gas. The number of degrees of freedom for monoatomic and diatomic gases are different. Hence, the internal energy of monoatomic and diatomic gas is also different.