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Step-by-Step Solution
Step 1: Identify the gases and their respective degrees of freedom
We have a mixture of two types of gases:
• Oxygen (O2) is a diatomic gas, which (when vibrational modes are neglected) has 5 degrees of freedom.
• Argon (Ar) is a monoatomic gas, which has 3 degrees of freedom.
Step 2: Note the number of moles of each gas
Given:
• Number of moles of O2, $n_{O_2} = 2$
• Number of moles of Ar, $n_{Ar} = 4$
Step 3: Recall the formula for total internal energy
The internal energy $U$ of an ideal gas is given by:
$U = \dfrac{f}{2} \, n R T$
where:
• $f$ = degrees of freedom of the gas,
• $n$ = number of moles of the gas,
• $R$ = gas constant,
• $T$ = absolute temperature.
Step 4: Calculate the internal energy contribution from each gas
For O2 (diatomic gas with $f = 5$):
$U_{O_2} = \dfrac{5}{2} \times \left(n_{O_2}\right) R T
= \dfrac{5}{2} \times 2 \, R T
= 5RT.$
For Ar (monoatomic gas with $f = 3$):
$U_{Ar} = \dfrac{3}{2} \times \left(n_{Ar}\right) R T
= \dfrac{3}{2} \times 4 \, R T
= 6RT.$
Step 5: Find the total internal energy
Total internal energy of the mixture:
$U_{\text{total}} = U_{O_2} + U_{Ar}
= 5RT + 6RT
= 11RT.$
Final Answer
The total internal energy of the mixture is $11RT$.
