A gas mixture consists of 2 moles of oxygen and 4 moles of argon at temperature T. Neglecting all vi

Q: Why do we take $f=5$ for oxygen and not $7$?

A diatomic gas has $3$ translational, $2$ rotational, and $2$ vibrational degrees of freedom. However, vibrational modes are usually only active at very high temperatures. The problem explicitly tells us to neglect them, so we only count the $5$ translational and rotational modes.

Q: Why is Argon considered monoatomic?

Argon is a noble gas (Group 18). Noble gases have a full valence electron shell, making them highly stable and unreactive. They exist as single, independent atoms rather than forming molecules.

Q: Can we use the formula $U = n C_v T$ instead?

Yes, absolutely! The molar heat capacity at constant volume is $C_v = \frac{f}{2} R$. So, $U = n C_v T$ is mathematically identical to $U = \frac{f}{2} nRT$.

Q: Does the internal energy depend on the pressure or volume of the container?

For an ideal gas, the internal energy depends solely on its temperature and the number of moles. It is completely independent of pressure and volume, as long as the gas behaves ideally.

Q: What if the gases react with each other?

If the gases react, the number of moles and the types of molecules would change, altering the total degrees of freedom. However, at standard temperatures, oxygen and argon do not react, so we can treat them as an ideal non-reacting mixture.

Question

A gas mixture consists of $2$ moles of oxygen and $4$ moles of argon at temperature $T$. Neglecting all vibrational modes, the total internal energy of the system is

Options:

(A) $4 RT$

(B) $15 RT$

(C) $9 RT$

(D) $11 RT$

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Imagine a closed container holding a mixture of two different gases at a constant temperature $T$.

We are given $2$ moles of oxygen ($O_2$) and $4$ moles of argon (Ar).

Our goal is to find the total internal energy of this gaseous system.

The problem explicitly states to neglect all vibrational modes, which is a crucial hint for determining the degrees of freedom.

### The Master Equation

The internal energy $U$ of an ideal gas is directly related to its temperature and its degrees of freedom.

The formula is given by:

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

Here, $f$ represents the degrees of freedom, $n$ is the number of moles, $R$ is the universal gas constant, and $T$ is the absolute temperature.

Since internal energy is an extensive property, the total internal energy of a mixture is simply the algebraic sum of the internal energies of its constituent gases.

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

### Energy of Oxygen

Let's evaluate the internal energy for oxygen first.

Oxygen ($O_2$) is a diatomic molecule.

A rigid diatomic molecule has $3$ translational and $2$ rotational degrees of freedom.

Since we are neglecting vibrational modes, its total degrees of freedom $f_{O_2} = 5$.

Substituting $n = 2$ and $f = 5$ into our master equation:

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

### Energy of Argon

Next, we look at argon.

Argon (Ar) is a noble gas, which means it exists as single, unbonded atoms (monoatomic).

A monoatomic gas only has translational kinetic energy, giving it exactly $3$ degrees of freedom ($f_{	ext{Ar}} = 3$).

Substituting $n = 4$ and $f = 3$ into the equation:

$$U_{	ext{Ar}} = \frac{3}{2} (4) RT = 6 RT$$

### Final Calculation

Now, we just need to add the individual energies together to find the total internal energy of the system.

$$U_{	ext{total}} = 5 RT + 6 RT$$

$$U_{	ext{total}} = 11 RT$$

**The total internal energy of the gas mixture is $11 RT$.**

Animated Solution for Physics - Thermodynamics: A gas mixture consists of $2$ moles of oxygen and $4$ moles of argon at temperature $T$ . Neglecting all vibrational modes, the total internal energy of the system is

Select Answer:

Visualized Solution

$Total Internal Energy$

$Internal Energy Formula$

$Energy of Oxygen$

$Energy of Argon$

$Total Energy$

The Sigma Insight: Kinetic Theory of Gases

Analyzing the Setup

The Master Equation

Energy of Oxygen

Energy of Argon

Final Calculation

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