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Step-by-Step Detailed Solution
Step 1: Identify the gases and their mole quantities
The mixture consists of 3 moles of oxygen (O2) and 5 moles of argon (Ar). Both gases are assumed to be ideal.
Step 2: Recall the formula for total internal energy of ideal gases
The total internal energy $U$ of a mixture of ideal gases can be written as:
$ U = \sum \left( \frac{f}{2} n \, R T \right), $
where $f$ is the number of degrees of freedom of each gas, $n$ is the number of moles of that gas, $R$ is the universal gas constant, and $T$ is the temperature.
Step 3: Determine the degrees of freedom for each gas
For a diatomic gas (like O2) with a “rigid” bond (neglecting vibrational degrees of freedom), $f = 5$.
For a monoatomic gas (like Ar), $f = 3$.
Step 4: Substitute the values into the formula
For O2 (diatomic, $f = 5$, 3 moles):
$ U_{\text{O}_2} = \frac{5}{2} \times 3 \, R T = \frac{15}{2} \, R T. $
For Ar (monoatomic, $f = 3$, 5 moles):
$ U_{\text{Ar}} = \frac{3}{2} \times 5 \, R T = \frac{15}{2} \, R T. $
Step 5: Calculate the total internal energy
$ U_{\text{total}} = U_{\text{O}_2} + U_{\text{Ar}}
= \frac{15}{2} R T + \frac{15}{2} R T
= 15 \, R T. $
Step 6: Express the answer in units of RT
The total internal energy in units of $RT$ is $15 \, RT$, which matches the correct choice.
