A gas mixture consists of 2 moles of $O_2$ and 4 moles of Ar at temperature T. Neglecting all vibrational mode

Question

A gas mixture consists of 2 moles of $O_2$ and 4 moles of Ar at temperature T. Neglecting all vibrational modes, the total internal energy of the system is:

Accepted Answer

The total internal energy ($U$) of a mixture of ideal gases is the sum of the internal energies of individual gases. The internal energy for $n$ moles of an ideal gas is given by $U = \frac{f}{2}nRT$, where $f$ is the number of degrees of freedom.

1. **For Argon (Ar):** It is a monatomic gas, so it has 3 degrees of freedom ($f=3$) associated with translational motion .
   $n_{Ar} = 4$
   $U_{Ar} = \frac{3}{2} (4) RT = 6RT$

2. **For Oxygen ($O_2$):** It is a diatomic gas. Neglecting vibrational modes, it has 5 degrees of freedom ($f=5$; 3 translational + 2 rotational).
   $n_{O_2} = 2$
   $U_{O_2} = \frac{5}{2} (2) RT = 5RT$

**Total Internal Energy:**
$U_{total} = U_{Ar} + U_{O_2} = 6RT + 5RT = 11RT$.

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