A gas mixture consists of 2.0 moles of oxygen and 4.0 moles of neon at temperature $T$. Neglecting all vibrational modes, calculate the total internal energy of the system. (Oxygen has two rotational modes.)

A gas mixture consists of 2.0 moles of oxygen and 4.0 moles of neon at temperature T. Neglecting all vibration

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Concept understanding — Kinetic Theory of Gases

Kinetic Theory of Gases

Imagine you're sitting in a quiet room. The air around you feels still — but it isn't. Every second, billions of tiny particles (molecules of nitrogen, oxygen, and others) are zipping past you at hundreds of metres per second. They're constantly crashing into each other and into the walls, your skin, the furniture. You don't feel each individual hit because the molecules are so small and the collisions happen so fast. But collectively, those countless tiny impacts produce something you do feel: pressure.

That's the core intuition behind the kinetic theory of gases. It says: all the macroscopic properties of a gas — pressure, temperature, volume — can be explained by the motion of its molecules.

The Big Idea

Instead of treating a gas as a continuous, smooth substance (like a fluid), the kinetic theory treats it as a swarm of tiny, hard, perfectly elastic balls in constant, random motion. "Perfectly elastic" means that when two molecules collide, no kinetic energy is lost — they bounce off each other like ideal billiard balls, not like sticky clay.

From this simple picture, we can derive the gas laws (Boyle's, Charles's, Avogadro's) and even calculate things like the speed of sound in a gas.

The Five Assumptions (The Precise Statement)

For a gas to behave according to the kinetic theory in its simplest form, we make these assumptions:

A gas consists of a very large number of molecules.

The number is so huge that we can use statistics — individual molecules don't matter, only averages do.
The molecules are in constant, random motion.

They move in straight lines until they hit something (another molecule or a wall). There's no preferred direction.
The molecules are point masses.

Their actual size is negligible compared to the distance between them. In other words, the volume of the molecules themselves is tiny compared to the volume of the container.
Collisions are perfectly elastic.

No kinetic energy is lost when molecules collide with each other or with the walls. Total energy of the system stays constant.
There are no intermolecular forces.

The molecules don't attract or repel each other except during collisions. Between collisions, they move freely.

Important

These assumptions define an ideal gas. Real gases deviate from this behaviour at high pressure or low temperature, but the kinetic theory gives an excellent approximation for most everyday conditions.

How It Explains Pressure

Pressure is the force per unit area exerted by the gas on the walls of its container. In the kinetic picture:

A molecule moving toward a wall hits it and bounces back.
During the collision, the wall exerts a force on the molecule to reverse its momentum.
By Newton's third law, the molecule exerts an equal and opposite force on the wall.
Multiply that by the billions of collisions happening every second, and you get a steady, measurable pressure.

If you heat the gas, the molecules move faster. They hit the walls harder and more often — pressure increases. If you compress the gas into a smaller volume, molecules hit the walls more frequently — pressure increases again.

The Key Result: The Kinetic Equation

From these assumptions, we can derive a relationship between pressure $P$ , volume $V$ , and the average kinetic energy of the molecules. The result is:

P V = \frac{1}{3} N m \overline{v^2}

Where:

$N$ = number of molecules
$m$ = mass of one molecule
$\overline{v^2}$ = mean square speed of the molecules (average of the squares of their speeds)

Since the average kinetic energy of a molecule is $\overline{K} = \frac{1}{2} m \overline{v^2}$ , we can rewrite this as:

$P V = \frac{2}{3} N \overline{K}$ …

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The total internal energy of a gas mixture is the sum of the internal energies of its components, determined by their respective degrees of freedom and the equipartition theorem. For 2 moles of oxygen (5 degrees of freedom) and 4 moles of neon (3 degrees of freedom), the total internal energy is $\boxed{11RT}$ .

The internal energy of an ideal gas is the sum of the kinetic energies of its constituent molecules. This energy is distributed among the various ways a molecule can move or store energy, known as its degrees of freedom. The Equipartition Theorem provides a fundamental way to calculate this.

The theorem states that, for a system in thermal equilibrium, each degree of freedom contributes an average energy of $\frac{1}{2}kT$ per molecule, where $k$ is Boltzmann's constant and $T$ is the absolute temperature. For one mole of gas, this translates to $\frac{1}{2}RT$ per degree of freedom, where $R$ is the ideal gas constant.

The total internal energy ( $U$ ) for $n$ moles of a gas with $f$ degrees of freedom at temperature $T$ is given by:

U = n \left(f \times \frac{1}{2}RT\right) = \frac{f}{2} nRT

The number of degrees of freedom ( $f$ ) depends on the molecular structure and the temperature range:

Translational degrees of freedom: All molecules (monatomic, diatomic, polyatomic) have 3 translational degrees of freedom, corresponding to movement along the x, y, and z axes.
Rotational degrees of freedom:
- Monatomic gases (like Neon) have negligible moment of inertia and thus 0 rotational degrees of freedom.
- Diatomic gases (like Oxygen) and linear polyatomic gases have 2 rotational degrees of freedom (rotation about two axes perpendicular to the molecular axis).
- Non-linear polyatomic gases have 3 rotational degrees of freedom.
Vibrational degrees of freedom: These arise from the oscillation of atoms within a molecule. Each vibrational mode contributes 2 degrees of freedom (one for kinetic energy and one for potential energy). However, vibrational modes are typically excited at higher temperatures, and the problem explicitly states to neglect them.

Let's apply these concepts to the given gas mixture.

Determine the degrees of freedom for each gas:
- Neon ( $Ne$ ): Neon is a monatomic gas.
  - Translational degrees of freedom: 3
  - Rotational degrees of freedom: 0
  - Vibrational degrees of freedom: 0 (monatomic gases do not vibrate)
  - Total degrees of freedom for Neon, $f_{Ne} = 3$ .
- Oxygen ( $O_2$ ): Oxygen is a diatomic gas. …

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Q.A gas mixture consists of 2.0 moles of oxygen and 4.0 moles of neon at temperature TTT. Neglecting all vibrational modes, calculate the total internal energy of the system. (Oxygen has two rotational modes.)

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Q.A gas mixture consists of 2.0 moles of oxygen and 4.0 moles of neon at temperature $T$ . Neglecting all vibrational modes, calculate the total internal energy of the system. (Oxygen has two rotational modes.)

Unlock everything free for 14 days