Animated Solution for Physics - Kinetic Theory: A gaseous mixture consists of $16\text{ g}$ of helium and $16\text{ g}$ of oxygen. The ratio $C_p/C_V$ of the mixture is

Two Gases, Two Personalities

Imagine you are in a laboratory, and you have two distinct containers in front of you.

In the first container, you have $16\text{ g}$ of Helium. Helium is a noble, monoatomic gas. It likes to keep to itself, bouncing around as single atoms.

In the second container, you have $16\text{ g}$ of Oxygen. Oxygen is a diatomic gas, meaning its atoms travel in pairs, like tiny dumbbells.

Our mission is to mix these two gases together and determine the specific heat ratio, $\gamma =\frac{C_p}{C_V}$, of the resulting mixture.

Before we can mix them, we need to know exactly how much of each gas we have in terms of moles. Mass alone isn't enough because thermodynamics cares about the number of particles!

For Helium, the molar mass is $M_1=4\text{ g/mol}$. Using the formula $n=\frac{m}{M}$, we find the number of moles:

For Oxygen, the molar mass is $M_2=32\text{ g/mol}$. Calculating its moles gives us:

Degrees of Freedom

Now, let's talk about how these gases store heat energy. This is governed by their degrees of freedom ($f$).

Helium, being monoatomic, can only move in three directions (x, y, z). So, $f_1=3$. Its molar heat capacity at constant volume is:

Oxygen, being diatomic, can move in three directions and rotate in two independent axes. So, $f_2=5$. Its molar heat capacity at constant volume is:

The Weighted Average

When we mix the gases, the total internal energy is conserved. This leads to a beautiful weighted average formula for the mixture's heat capacity:

Let's carefully substitute our values into this master equation:

Now, we perform the atomic computation. The numerator becomes:

The denominator is simply the total number of moles:

Dividing the numerator by the denominator:

Mayer's Relation to the Rescue

We are almost there! The question asks for the ratio of specific heats, ${\gamma }_{mix}=\frac{C_{p_{mix}}}{C_{V_{mix}}}$.

Instead of calculating $C_{p_{mix}}$ from scratch, we can use a brilliant shortcut derived from Mayer's Relation ($C_p-C_V=R$):

Let's plug in our mixture's $C_V$:

The universal gas constant $R$ elegantly cancels out:

Finally, converting this fraction into a decimal gives us:

And there we have it! By understanding the fundamental properties of monoatomic and diatomic gases, and applying the principle of energy conservation, we've successfully decoded the thermodynamics of the mixture.