Question

A gas mixture consists $3moles$ of oxygen and $5moles$ argon at temperature $T$. Assuming the gases to be ideal and the oxygen bond to be rigid, the total internal energy (in units of $RT$) of the mixture is:

• A. $11$
• B. $13$
• C. $15$
• D. $20$

Answer 1

The correct option is C

$15$

Internal energy:

1. The internal energy of a system is the measure of its kinetic energy.
2. Kinetic energy is different for different gases, it is based on the number of atoms present in them.
3. The number of moles of oxygen, $n=3moles$, the number of moles of argon, $n=5moles$, and the temperature of both the gases is $T$.
4. The internal energy of a gas is given by the formula:$\mathrm{U}=\mathrm{n}\frac{\mathrm{f}}{2}\mathrm{RT}$ Where, $\mathrm{U}$ is the internal energy, $\mathrm{n}$ is the number of moles, $\mathrm{f}$ is degree of freedom, $\mathrm{R}$ is gas constant and $\mathrm{T}$is temperature.
5. First, let's determine the degree of freedom for the gases.
6. For oxygen: Oxygen is a diatomic gas, $Degreeoffreedomfortranslationalmotionofadiatomicmolecule=3$ $Degreeoffreedomforrotationalmotionofadiatomicmolecule=2$.
7. Hence,$Totaldegreesoffreedomforoxygen=5$
8. For argon: Argon is a monoatomic gas, $Degreeoffreedomfortranslationalmotionofamonoatomicmolecule=3$$Degreeoffreedomforrotationalmotionofamonoatomicmolecule=0$
9. Hence, $Totaldegreeoffreedomforargon=3$
10. Let us determine the internal energy of the gases: ${\mathrm{U}}_{\mathrm{oxygen}}=3\frac{5}{2}\mathrm{RT}=\frac{15}{2}\mathrm{RT}$and ${\mathrm{U}}_{\mathrm{argon}}=5\frac{3}{2}\mathrm{RT}=\frac{15}{2}\mathrm{RT}$
11. Now, the total internal energy is: ${\mathrm{U}}_{\mathrm{total}}={\mathrm{U}}_{\mathrm{oxygen}}+{\mathrm{U}}_{\mathrm{argon}}$
12. Let's put the values in above equation ${\mathrm{U}}_{\mathrm{total}}=\frac{15}{2}\mathrm{RT}+\frac{15}{2}\mathrm{RT}=15\mathrm{RT}$.
13. Hence, the total internal energy of the mixture of gases is $15RT$.
14. Option C is correct.