Animated Solution for Physics - Thermodynamics: One mole of a monoatomic ideal gas is mixed with one mole of a diatomic ideal gas. The molar specific heat of the mixture at constant volume is …… .

Mixing Gases

Imagine you are a chef in a molecular kitchen. You have two distinct ingredients: a monoatomic gas (like Helium or Neon) and a diatomic gas (like Oxygen or Nitrogen). Each of these gases has its own unique way of storing thermal energy, which is quantified by its molar specific heat at constant volume, $C_V$.

When you mix these two gases together, the resulting mixture will have a new, combined specific heat. But how do we calculate it? We can't just add them up, and we can't always just take a simple average. We need a method that accounts for how much of each gas is present.

Weighted Averages

The internal energy of an ideal gas depends entirely on its temperature and its specific heat. When we mix gases, the total internal energy of the mixture is simply the sum of the internal energies of the individual components. From this fundamental principle of energy conservation, we derive the formula for the molar specific heat of a mixture:

Notice the structure of this equation. It is a weighted average. The specific heat of each gas is multiplied by its number of moles (its "weight"), and then we divide by the total number of moles. This ensures that a gas present in a larger quantity has a greater influence on the final specific heat of the mixture.

Before we can plug numbers into our master equation, we need to know the $C_V$ values for our two gases. This requires a quick trip into the microscopic world of molecules.

For a monoatomic gas, the atoms are like tiny, independent spheres. They can only move in three directions (x, y, and z). These are 3 translational degrees of freedom. According to the equipartition theorem, each degree of freedom contributes $\frac{1}{2}R$ to the specific heat. Thus:

For a diatomic gas, the molecules look like tiny dumbbells. Not only can they translate in 3 directions, but they can also rotate around 2 independent axes perpendicular to the bond connecting the atoms. This gives them $3+2=5$ degrees of freedom at room temperature. Thus:

The problem states that we are mixing exactly one mole of the monoatomic gas with one mole of the diatomic gas. This makes our weights equal: $n_1=1$ and $n_2=1$.

Let's substitute these values into our weighted average formula:

Now, we just need to perform the arithmetic. Let's simplify the numerator first:

Since the denominators of the fractions in the numerator are the same, we can simply add the numerators:

simplifies perfectly to . So our equation becomes:

Dividing by 2 gives us our final, elegant result:

We successfully found the specific heat at constant volume. But what if the question had asked for the specific heat at constant pressure, $C_{P_{mix}}$?

You have two powerful tools at your disposal. You could use the exact same weighted average formula, but swap the $C_V$ values for $C_P$ values ($C_{P_1}=\frac{5}{2}R$ and $C_{P_2}=\frac{7}{2}R$).

Alternatively, you could use Mayer's Relation, which holds true even for gas mixtures: $C_{P_{mix}}=C_{V_{mix}}+R$. Since we already found $C_{V_{mix}}=2R$, we can instantly see that $C_{P_{mix}}=2R+R=3R$. Physics is beautifully consistent!