A gas mixture consists of 3 moles of oxygen and 5 moles of argon at temperature T. Assuming the gase

Q: Why can we just add the internal energies of the two gases together?

For ideal gases, we assume there are no intermolecular forces between the different types of molecules. Because they don't interact, the total internal energy of the mixture is simply the sum of the internal energies of the individual non-interacting components.

Q: What does 'rigid bond' mean for the oxygen molecule?

A 'rigid bond' means the distance between the two oxygen atoms is fixed. Because it cannot stretch or compress, the molecule has no vibrational kinetic or potential energy. This restricts its degrees of freedom to just $3$ translational and $2$ rotational, making $f=5$.

Q: Why is argon considered monoatomic?

Argon is a noble gas (Group 18 of the periodic table). Noble gases have full valence electron shells, making them highly stable and unreactive. Therefore, they exist as single, isolated atoms rather than forming molecules.

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

The problem states the mixture is at a temperature $T$. This implies the gases are already in thermal equilibrium with each other and the container. We don't need to worry about heat transfer between them.

Q: What if the temperature was very high and the bonds weren't rigid?

At very high temperatures, the collisions between molecules become energetic enough to excite vibrational modes. A diatomic molecule would gain $1$ degree of freedom for vibrational kinetic energy and $1$ for vibrational potential energy, making its total $f = 7$.

Question

A gas mixture consists of $3$ moles of oxygen and $5$ moles of argon at temperature $T$. Assuming the gases to be ideal and the oxygen bond to be rigid, the total internal energy (in units of $RT$) of the mixture is

Options:

(A) 15

(B) 13

(C) 11

(D) 20

Sigma Percentile · Accepted Answer

### Visualizing the Gas Mixture

Imagine you are looking inside a sealed container maintained at a constant temperature $T$. Inside this container, a microscopic dance is happening. We have a mixture of two distinct ideal gases: $3$ moles of oxygen ($O_2$) and $5$ moles of argon ($Ar$).

Our goal is to find the **total internal energy** of this entire system. Because these are ideal gases, they don't interact with each other. This means we can simply calculate the internal energy of the oxygen, calculate the internal energy of the argon, and add them together.

### The Master Equation: Degrees of Freedom

The internal energy $U$ of an ideal gas is intimately tied to how its molecules can move—what physicists call its **degrees of freedom** ($f$). The master equation for the internal energy of $n$ moles of an ideal gas is:

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

To use this equation, we must carefully determine the degrees of freedom for both gases in our mixture.

Let's start with oxygen ($O_2$). Oxygen is a **diatomic** molecule, meaning it looks like a tiny dumbbell. The problem explicitly states that the oxygen bond is **rigid**. This is a crucial keyword! A rigid diatomic molecule can move in three spatial directions (translation) and rotate around two independent axes (rotation). It cannot vibrate. Therefore, its degrees of freedom are:

$$f_{O_2} = 3 	ext{ (translational)} + 2 	ext{ (rotational)} = 5$$

Next, let's look at argon ($Ar$). Argon is a noble gas, meaning it exists as single, isolated atoms. It is **monoatomic**. A single atom can only move through space in three directions; it has no meaningful rotation or vibration. Therefore, its degrees of freedom are:

$$f_{Ar} = 3 	ext{ (translational)} = 3$$

### Calculating the Energies

Now that we have our degrees of freedom, the rest is smooth sailing. We just need to substitute our values into the master equation.

For the $3$ moles of oxygen:

$$U_{O_2} = \frac{f_{O_2}}{2} n_{O_2} RT$$

$$U_{O_2} = \frac{5}{2} (3) RT = \frac{15}{2} RT$$

For the $5$ moles of argon:

$$U_{Ar} = \frac{f_{Ar}}{2} n_{Ar} RT$$

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

### The Final Summation

To find the total internal energy of the mixture, we simply add the two individual energies together:

$$U_{	ext{total}} = U_{O_2} + U_{Ar}$$

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

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

And there we have it! The total internal energy of the mixture is exactly **$15 RT$**.

*A quick thought experiment for the road:* What if the problem had said the temperature was extremely high, and the oxygen bonds were *not* rigid? In that case, vibrational modes would activate, adding $2$ more degrees of freedom to oxygen ($f=7$). Always pay close attention to the word "rigid" in thermodynamics problems!

Animated Solution for Physics - Thermodynamics: A gas mixture consists of $3$ moles of oxygen and $5$ moles of argon at temperature $T$ . Assuming the gases to be ideal and the oxygen bond to be rigid, the total internal energy (in units of $R T$ ) of the mixture is

Select Answer:

Visualized Solution

System Setup

Internal Energy Formula

Degrees of Freedom

Internal Energy of Oxygen

Internal Energy of Argon

Total Internal Energy

The Way Forward

The Sigma Insight: Kinetic Theory of Gases

Visualizing the Gas Mixture

The Master Equation

Calculating the Energies

The Final Summation

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