Question

A gas mixture consists of $2$ moles of oxygen and $4$ moles of argon at temperature $T$ . Neglecting all vibrational modes we calculated the total internal energy of the system to be $xRT$ . What is the value of $x$ ?

Answer 1

The internal energy of n moles of a gas is $U=\frac{1}{2}\textit{nFRT}$
where F = number of degrees of freedom.
The internal energy of 2 moles of oxygen at temperature T is
$\textit{U}_{1}=\frac{1}{2}\times 2\times 5\textit{RT}=5\textit{RT}$ (F = 5 for oxygen molecule)
The total internal energy of 4 moles of argon at temperature T is
$U_{2}=\frac{1}{2}\times 4\times 3\textit{RT}=6\textit{RT}$ (F = 3 for argon molecule)
Total internal energy $=U_{1}+U_{2}=11\textit{RT}$
$\Rightarrow x=11$