Physics (9702) | Kinetic theory of gases

Kinetic Theory of Gases: Bridging the Micro and Macro World

Welcome to one of the most exciting topics in Physics: the Kinetic Theory of Gases! Don't worry if this chapter involves some unfamiliar formulas. At its heart, this theory is simply a fantastic model that allows us to explain the everyday properties of gases—like pressure and temperature—by looking at what their tiny, invisible molecules are doing.

You've already studied the Ideal Gas Equation (\( PV = nRT \)). Now, we are going deeper to understand *why* that equation works, linking the motion of individual atoms to the bulk behaviour we observe in a lab. You've got this!

1. Basic Assumptions of the Kinetic Theory Model

To mathematically model how billions of gas molecules behave, physicists use the idea of an Ideal Gas. This means we make a few simplifying assumptions about the molecules themselves. These assumptions are key knowledge for the exam:

Key Assumptions (The Ideal Gas Rules):

• Random Motion: The molecules are in constant, random motion. They move in straight lines until they collide.
• Elastic Collisions: All collisions (between molecules, and between molecules and the container walls) are perfectly elastic. This means no kinetic energy is lost during the collision, only transferred.
• Negligible Volume: The volume occupied by the molecules themselves is negligible compared to the volume of the container. (Think of a few grains of sand scattered inside a football stadium.)
• Negligible Intermolecular Forces: There are no forces of attraction or repulsion between the molecules, except during collisions.
• Short Collision Time: The time duration of any collision is negligible compared to the time between collisions.

Quick Review: Why do we use these assumptions? Because real gases only follow \( PV = nRT \) perfectly at very low pressures and high temperatures. Under these conditions, the molecules are far apart, making the "negligible volume" and "negligible force" assumptions accurate. Our model helps us understand the fundamental physics.

2. Explaining Pressure through Molecular Movement

How does a gas exert pressure on the walls of its container? It's all about collisions!

The macroscopic quantity we call pressure (\( P \)) is simply the force exerted per unit area, caused by countless microscopic impacts.

Step-by-Step: How Pressure is Generated

1. A gas molecule, travelling at velocity \( v \), approaches a container wall.
2. It collides with the wall and bounces back. Since the collision is assumed to be perfectly elastic, the molecule reverses its direction but keeps the same speed.
3. Momentum Change: Because velocity is a vector, there is a change in the molecule's momentum. If the mass is \( m \) and the initial velocity component perpendicular to the wall is \( v_x \), the change in momentum is \( \Delta p = m(-v_x) - m(v_x) = -2mv_x \).
4. Force on the Molecule: According to Newton’s Second Law (Force = Rate of change of momentum), the wall exerts a force on the molecule.
5. Force on the Wall: By Newton’s Third Law, the molecule exerts an equal and opposite force on the wall: \( F = \frac{\Delta p}{\Delta t} \).
6. Total Pressure: Since billions of molecules hit the walls every second, the total average force is constant, leading to the measurable pressure \( P = \frac{\text{Total Force}}{\text{Area}} \).

Analogy: Imagine holding a paddle in a severe hailstorm. Each hailstone (molecule) hits the paddle (wall) and imparts a tiny force. The collective, continuous barrage of impacts is the pressure you feel.

Key Takeaway: Pressure is directly proportional to the rate of momentum change due to elastic collisions with the walls.

3. The Kinetic Theory Equation (KTE)

By applying the assumptions of the ideal gas model and calculating the total force exerted by \( N \) molecules in a container, we arrive at the fundamental equation of the kinetic theory:

$$ PV = \frac{1}{3} Nm \langle c^2 \rangle $$

Understanding the Terms:

• \( P \): Pressure (Pa or \( \text{N m}^{-2} \))
• \( V \): Volume of the container (\( \text{m}^3 \))
• \( N \): Total number of molecules in the gas.
• \( m \): Mass of a single molecule (kg).
• \( \langle c^2 \rangle \): The mean-square speed.

Don't worry about memorizing the complex derivation—the syllabus focuses on *using* this relationship and understanding the concepts within it.

Understanding Mean-Square Speed (\( \langle c^2 \rangle \))

Why do we use the *square* of the speed? Not all molecules travel at the same speed. They have a range of speeds. If we simply calculated the mean speed (\( \langle c \rangle \)), this wouldn't accurately reflect the energy or momentum involved in collisions because momentum and kinetic energy depend on velocity squared or speed squared.

The term \( \langle c^2 \rangle \) is calculated by:

1. Squaring the speed of every molecule (\( c_1^2, c_2^2, c_3^2, ... \)).
2. Finding the mean (average) of these squared speeds.

Root-Mean-Square Speed (\( c_{\text{r.m.s.}} \))

Because \( \langle c^2 \rangle \) is in units of \( (\text{m/s})^2 \), we often use the Root-Mean-Square speed, \( c_{\text{r.m.s.}} \), which has the standard units of speed (\( \text{m/s} \)):

$$ c_{\text{r.m.s.}} = \sqrt{\langle c^2 \rangle} $$

This is a specific type of average speed that is most relevant in kinetic theory calculations.

Key Takeaway: The KTE, \( PV = \frac{1}{3} Nm \langle c^2 \rangle \), directly links the macroscopic properties (\( P, V \)) to the microscopic properties (molecular mass \( m \) and molecular speed \( \langle c^2 \rangle \)).

4. Connecting Temperature to Kinetic Energy

This is the most crucial part of the kinetic theory: establishing the physical meaning of temperature.

Comparing the Two Gas Equations

We have two valid equations for an ideal gas:

1. The Ideal Gas Equation (from previous section, 15.2): \( PV = NkT \)
2. The Kinetic Theory Equation (KTE): \( PV = \frac{1}{3} Nm \langle c^2 \rangle \)

Since both equations equal \( PV \), we can set them equal to each other:

$$ NkT = \frac{1}{3} Nm \langle c^2 \rangle $$

We can cancel the number of molecules \( N \) from both sides:

$$ kT = \frac{1}{3} m \langle c^2 \rangle $$

This expression relates temperature \( T \) directly to the molecular mass \( m \) and mean-square speed \( \langle c^2 \rangle \).

The Average Translational Kinetic Energy

Recall the formula for kinetic energy of a single molecule: \( E_K = \frac{1}{2} m c^2 \). Therefore, the average translational kinetic energy, \( \langle E_K \rangle \), is:

$$ \langle E_K \rangle = \frac{1}{2} m \langle c^2 \rangle $$

Let's rearrange our derived equation \( kT = \frac{1}{3} m \langle c^2 \rangle \) to isolate the \( \frac{1}{2} m \langle c^2 \rangle \) term:

$$ kT = \frac{2}{3} \left( \frac{1}{2} m \langle c^2 \rangle \right) $$

Substituting \( \langle E_K \rangle \):

$$ kT = \frac{2}{3} \langle E_K \rangle $$

Finally, we derive the critical relationship:

$$ \langle E_K \rangle = \frac{3}{2} kT $$

What does this mean?

This derived expression is arguably the most important conceptual result of the Kinetic Theory:

The average translational kinetic energy of a molecule is directly proportional only to the absolute temperature (\( T \)) of the gas.

This means:

• If two different gases (e.g., Helium and Xenon) are at the same temperature, their molecules have the *same average kinetic energy*.
• Since \( \langle E_K \rangle = \frac{1}{2} m \langle c^2 \rangle \), and Xenon has a much larger mass (\( m \)) than Helium, the Helium atoms must be moving much faster to achieve the same average kinetic energy.

Did you know? Temperature is defined at the A-Level level as a measure of the average translational kinetic energy of the molecules in a substance. This is the physical foundation of the absolute temperature scale (Kelvin).

Important Constants (Recap from Syllabus 15.2)

• Boltzmann Constant (\( k \)): This relates the energy (or kinetic energy) of individual molecules to the absolute temperature \( T \). $$ k = \frac{R}{N_A} $$ Where \( R \) is the Molar Gas Constant and \( N_A \) is the Avogadro Constant.
• Units Reminder: When using \( PV = NkT \) or \( \langle E_K \rangle = \frac{3}{2} kT \), the temperature \( T \) must be in Kelvin (K).

Common Mistake to Avoid: Ensure you distinguish between \( N \) (number of molecules) used with \( k \), and \( n \) (number of moles) used with \( R \). The Kinetic Theory equation \( PV = \frac{1}{3} Nm \langle c^2 \rangle \) always uses \( N \), the total molecule count.

Key Takeaway: Temperature is simply a scaled measurement of the average kinetic energy of gas molecules, shown by \( \langle E_K \rangle = \frac{3}{2} kT \).

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Chapter Summary: Kinetic Theory of Gases

The Kinetic Theory provides a microscopic explanation for the macroscopic properties of ideal gases. Remember these essential links:

1. Pressure (\( P \)) is caused by the rate of change of momentum during elastic molecular collisions with the container walls.

2. The Kinetic Theory Equation (KTE) connects pressure and volume to molecular motion:

$$ PV = \frac{1}{3} Nm \langle c^2 \rangle $$

3. Temperature (\( T \)) (in Kelvin) is directly proportional to the average translational kinetic energy of the molecules:

$$ \langle E_K \rangle = \frac{3}{2} kT $$

If you can recall the basic assumptions and these two critical equations, you will be well prepared to tackle problems on the Kinetic Theory of Gases!