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Step-by-Step Solution
Step 1: Determine the degrees of freedom for each gas
• Oxygen (O₂) is a diatomic molecule. If we consider only translational and rotational modes (no vibrational modes at the given conditions), it has 3 translational and 2 rotational degrees of freedom. Thus, total degrees of freedom, $f_1 = 5$.
• Argon (Ar) is a monoatomic gas. It has 3 translational degrees of freedom and no rotational degrees of freedom. Thus, total degrees of freedom, $f_2 = 3$.
Step 2: Write the formula for the internal energy
The internal energy $U$ of a gas with $f$ degrees of freedom for $n$ moles at absolute temperature $T$ is given by:
$$ U = \frac{f}{2} \, n \, R \, T $$
Step 3: Calculate the internal energy contributed by oxygen
The number of moles of oxygen is $n_1 = 3$. Substituting $f_1 = 5$ and $n_1 = 3$ into the formula:
$$ U_{\text{O}_2} = \frac{5}{2} \times 3 \times R \times T
= \frac{15}{2} \, R \, T
= 7.5 \, R \, T $$
Step 4: Calculate the internal energy contributed by argon
The number of moles of argon is $n_2 = 5$. Substituting $f_2 = 3$ and $n_2 = 5$:
$$ U_{\text{Ar}} = \frac{3}{2} \times 5 \times R \times T
= \frac{15}{2} \, R \, T
= 7.5 \, R \, T $$
Step 5: Find the total internal energy of the mixture
The total internal energy is the sum of the internal energies of oxygen and argon:
$$ U_{\text{total}} = U_{\text{O}_2} + U_{\text{Ar}}
= 7.5 \, R \, T + 7.5 \, R \, T
= 15 \, R \, T $$
Conclusion
Therefore, the total internal energy of the given gas mixture is $15 \, R \, T$.
