A gas mixture consists of $2$ moles of oxygen and $4$ moles of argon at temperature $T$. Neglecting all vibrational modes, what is the total internal energy of the system?

Solution

1. Oxygen is diatomic, so its degrees of freedom $f_1=5$. Internal energy $$\begin{array}{rl}{U}_1& =n_1(f_1/2)RT\\ & =2(5/2)RT\\ & =5RT\end{array}$$.
2. Argon is monatomic, so its degrees of freedom $f_2=3$. Internal energy $$\begin{array}{rl}{U}_2& =n_2(f_2/2)RT\\ & =4(3/2)RT\\ & =6RT\end{array}$$.
3. Total internal energy $$\begin{array}{rl}U& =U_1+U_2\\ & =5RT+6RT\\ & =11RT\end{array}$$.
4. Hence the answer is (A).