Animated Solution for Physics - Thermodynamics: Consider a mixture of $n$ moles of helium gas and $2n$ moles of oxygen gas (molecules taken to be rigid) as an ideal gas. Its $C_p/C_V$ value will be

Imagine you are in a laboratory, and you have two separate containers. One holds $n$ moles of Helium (He), a light, monoatomic gas. The other holds $2n$ moles of Oxygen (O${}_2$), a rigid diatomic gas.

When we mix these two gases together, they form a new thermodynamic system. Our goal is to find the ratio of specific heats, ${\gamma }_{mix}$, for this newly formed mixture. The ratio of specific heats is a fundamental property that dictates how the gas behaves during adiabatic processes and determines the speed of sound within it.

To find the $\gamma$ of a gas mixture, we cannot simply take the average of their individual $\gamma$ values. That is a classic trap! Instead, we must go back to the foundational definitions. The ratio ${\gamma }_{mix}$ is defined as the total molar heat capacity at constant pressure divided by the total molar heat capacity at constant volume:

Since heat capacity is an extensive property, the total heat capacity of the mixture is just the sum of the heat capacities of its components. Therefore, we can write:

Let's analyze our gases one by one. First, Helium is a monoatomic gas. It only has 3 translational degrees of freedom ($f=3$). Therefore, its specific heats are:

Next, we have Oxygen. The problem explicitly states that the molecules are rigid, meaning we do not consider vibrational degrees of freedom. A rigid diatomic gas has 3 translational and 2 rotational degrees of freedom, giving $f=5$. Its specific heats are:

Now, we substitute these values into our master equation. Let's calculate the numerator (total $C_p$) first:

Next, we calculate the denominator (total $C_V$):

Finally, we divide the numerator by the denominator to find ${\gamma }_{mix}$:

Notice how beautifully the $nR$ terms and the factor of $2$ in the denominators cancel out. We are left with a clean, elegant fraction:

This result perfectly matches option (c). As a pro-tip, you can also solve this by first finding the equivalent degrees of freedom for the mixture using $f_{mix}=\frac{n_1{f}_1+n_2{f}_2}{n_1+n_2}$, and then applying the relation ${\gamma }_{mix}=1+\frac{2}{f_{mix}}$. Both paths lead to the same beautiful truth of thermodynamics!