A gas mixture consists of 3 moles of oxygen and 5 moles of argon at temperature T. Considering only

Q: Why do we only consider translational and rotational modes?

At normal room temperatures, the thermal energy is not sufficient to excite the vibrational modes of molecules. Therefore, unless specified otherwise or if the temperature is very high, we only count translational and rotational degrees of freedom.

Q: Why is the degree of freedom for Oxygen 5?

Oxygen ($O_2$) is a diatomic molecule. It has 3 translational degrees of freedom (moving along x, y, z axes) and 2 rotational degrees of freedom (rotating about the two axes perpendicular to its bond). $3 + 2 = 5$.

Q: Why doesn't Argon have rotational degrees of freedom?

Argon is a monoatomic gas, meaning it exists as single atoms. An atom is considered a point mass in kinetic theory, and the moment of inertia of a point mass about its own center is negligible. Hence, rotational kinetic energy is zero.

Q: Could we have calculated an equivalent degree of freedom for the mixture first?

Yes! You can calculate $f_{mix} = \frac{n_1 f_1 + n_2 f_2}{n_1 + n_2}$ and then use $U = \frac{f_{mix}}{2} (n_1 + n_2) RT$. However, calculating individual energies and adding them is mathematically identical and often less prone to calculation errors.

Q: Does the temperature change when the gases are mixed?

The problem states the mixture is at temperature $T$, which implies the gases were either already at temperature $T$ before mixing, or they have reached thermal equilibrium at $T$. We don't need to worry about temperature changes for this specific calculation.

Question

A gas mixture consists of 3 moles of oxygen and 5 moles of argon at temperature $T$. Considering only translational and rotational modes, the total internal energy of the system is

Options:

(A) $12 RT$

(B) $15 RT$

(C) $20 RT$

(D) $4 RT$

Sigma Percentile · Accepted Answer

### The Anatomy of the Mixture

Imagine you have a sealed container holding a mixture of two distinct gases: Oxygen ($O_2$) and Argon ($Ar$). The entire system is in thermal equilibrium at a steady temperature $T$. When dealing with non-reacting ideal gas mixtures, a beautiful principle of physics comes to our rescue: **The Principle of Superposition**.

This principle states that the total internal energy of the mixture is simply the algebraic sum of the internal energies of its individual gaseous components. Mathematically, we can write this as:

$$U_{total} = U_1 + U_2$$

To find the total energy, we just need to analyze each gas separately as if the other wasn't even there, and then add their energies together.

### Degrees of Freedom: The Ways to Move

The internal energy of an ideal gas is a direct measure of the kinetic energy of its molecules. This energy depends on three factors: the number of moles ($n$), the absolute temperature ($T$), and the **degrees of freedom** ($f$). The formula connecting them is:

$$U = \frac{f}{2} nRT$$

Let's break down the degrees of freedom for our two gases. The problem explicitly asks us to consider only translational and rotational modes.

**1. Oxygen ($O_2$):**
Oxygen is a diatomic molecule, resembling a tiny dumbbell. It can move in 3 independent directions in space (3 translational degrees of freedom). Additionally, it can rotate about 2 independent axes perpendicular to the bond connecting the two oxygen atoms (2 rotational degrees of freedom). Therefore, its total degrees of freedom is $f_1 = 3 + 2 = 5$.

Given that we have $n_1 = 3$ moles of Oxygen, its internal energy is:

$$U_1 = \frac{5}{2} (3) RT = \frac{15}{2} RT$$

**2. Argon ($Ar$):**
Argon, on the other hand, is a noble gas and exists as single, isolated atoms (monoatomic). A point-like atom can only translate in 3 spatial dimensions; it has no meaningful rotational inertia. Thus, its degree of freedom is purely translational, meaning $f_2 = 3$.

With $n_2 = 5$ moles of Argon, its internal energy becomes:

$$U_2 = \frac{3}{2} (5) RT = \frac{15}{2} RT$$

### Calculating the Total Energy

Now that we have the internal energies of both components, the final step is a simple addition. We bring back our superposition principle:

$$U_{total} = U_1 + U_2$$

Substituting the values we just calculated:

$$U_{total} = \frac{15}{2} RT + \frac{15}{2} RT$$

$$U_{total} = \frac{30}{2} RT = 15 RT$$

And there we have it! The total internal energy of the system is exactly **$15 RT$**. By breaking the mixture down into its atomic and molecular constituents, a seemingly complex thermodynamic system becomes incredibly straightforward to solve.

Animated Solution for Physics - Thermodynamics: A gas mixture consists of 3 moles of oxygen and 5 moles of argon at temperature $T$ . Considering only translational and rotational modes, the total internal energy of the system is

Select Answer:

Visualized Solution

Visualizing the Mixture

Principle of Superposition

Internal Energy Formula

Energy of Oxygen

Energy of Argon

Total Internal Energy

Final Conclusion

The Sigma Insight: Kinetic Theory of Gases

The Anatomy of the Mixture

Degrees of Freedom

Calculating the Total Energy

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