Elementary Kinetic Theory

1. What is the Kinetic Theory?

Imagine a box filled with tiny, invisible particles (gas molecules) zipping around at incredibly high speeds. The Kinetic Theory is a simple model that helps us understand how this chaotic microscopic motion explains the macroscopic properties of a gas we can measure, like pressure and temperature. It’s a set of assumptions about how gas particles behave. A gas that perfectly follows these rules is called an Ideal Gas.

2. The Basic Assumptions of the Kinetic Theory

(i) The Particle Size Assumption:

• The gas consists of a very large number of extremely tiny particles (atoms or molecules).
• What it means: The actual volume of the gas molecules themselves is so small compared to the total volume of the container that we can consider it zero. Think of a few footballs (the molecules) in a huge stadium (the container). The space the footballs take up is negligible.

(ii) The Force Assumption:

• There are no forces of attraction or repulsion between the gas particles.
• What it means: The particles don’t stick to each other or push each other away. They are completely independent and ignore each other until they collide.

(iii) The Motion Assumption:

• The particles are in constant, random, and straight-line motion.
• What it means: The particles are always moving. They zoom around in straight lines until they hit something—either another particle or the wall of the container. This motion is random, meaning there’s no pattern to it.

(iv) The Collision Assumption:

• Collisions between particles and with the walls of the container are perfectly elastic.
• What it means: When two particles collide, they bounce off each other like perfect billiard balls. The total kinetic energy (energy of motion) remains the same before and after the collision.

(v) The Energy Assumption:

• The average kinetic energy of the gas particles is directly proportional to the absolute temperature (in Kelvin) of the gas.
• What it means: Temperature is just a measure of how fast the particles are moving on average.
  • High Temperature = Particles moving very fast (high kinetic energy).
  • Low Temperature = Particles moving slowly (low kinetic energy).

3. How Do Real Gases Deviate?

No gas is truly “ideal.” Real gases (like oxygen, nitrogen, CO₂) break these rules, especially when two conditions are met: (i) High Pressure, (ii) Low Temperature.

Deviation 1: Particles DO Have Volume.

• Assumption Broken: Particle Size
• What happens in real gases: Under high pressure, we are squeezing the gas particles into a much smaller space. Suddenly, the volume of the molecules themselves is no longer negligible. The space they can move in is actually less than the volume of the container.
• Effect: The measured volume of the gas is larger than what the Ideal Gas Law predicts. The particles can’t be compressed as much as we thought.

Deviation 2: Forces DO Exist.

• Assumption Broken: Forces between particles
• What happens in real gases: At low temperatures, the particles move much slower. The weak attractive forces between them (called intermolecular forces) now have a chance to pull particles together.
• Effect: This attraction makes the gas easier to compress. It also slightly reduces the pressure the gas exerts on the container walls (because the particles are pulled slightly away from the walls). So, the measured pressure is less than what the Ideal Gas Law predicts.

Summary of Deviations:

4. Pressure Exerted by an Ideal Gas

We want to find a mathematical formula for the pressure an ideal gas exerts on the walls of its container. From the kinetic theory, we know that pressure is caused by the constant bombardment of gas molecules on the walls. Each collision exerts a tiny force. The sum of all these forces per unit area is the pressure. To make this manageable, we use a simplified model with these assumptions:

• The container is a cube of side length L.
• The molecules are identical, point-like particles (zero volume).
• Collisions with the walls are perfectly elastic (no energy loss).
• The motion is random. To simplify the math, we assume that 1/3 of the molecules move along the x-axis, 1/3 along the y-axis, and 1/3 along the z-axis. (This is a common simplification to avoid using calculus. A full derivation with calculus gives the same result!)

Consider a single molecule with mass m moving in the x-direction with velocity v_x . It heads towards the right wall of the cube (the wall parallel to the YZ-plane). Before the collision, its momentum is p_{in}=mv_x (towards the right), and after an elastic collision, it rebounds with velocity p_{fi}=-mv_x (towards the left). Therefore, the change in momentum due to the collision is

\Delta {p_x} = {p_{fi}} - {p_{in}} = \left( { - m{v_x}} \right) - \left( {m{v_x}} \right) =  - 2m{v_x}

The magnitude of the change in momentum is 2mv_x . By Newton’s Third Law, this is also the magnitude of the momentum imparted by the molecule to the wall.

After hitting the right wall, the molecule rebounds, travels to the left wall (a distance of L ), rebounds again, and returns to the right wall. The total distance traveled between two consecutive collisions with the right wall is L + L = 2L. Time between two consecutive collisions with the right wall is

\Delta t = \frac{{{\rm{Distance}}}}{{{\rm{speed}}}} = \frac{{2L}}{{{v_x}}}

According to Newton’s second law, force is equal to the rate of change of momentum. Here, the rate of change of momentum of one molecule is

\frac{{\Delta p}}{{\Delta t}} = \frac{{2m{v_x}}}{{\left( {2L/{v_x}} \right)}} = \frac{{mv_x^2}}{L}

So, force exerted by one molecule on the right wall is

{F_x} = \frac{{\Delta p}}{{\Delta t}} = \frac{{mv_x^2}}{L}

Now, all molecules (suppose N ) moving along x-axis do not have the same speed. So, we can use the average (mean) of the square of their x-component velocities \left\langle {v_x^2} \right\rangle , therefore, force exerted by N molecules on the right wall is

{F_X} = \frac{{mN\left\langle {v_x^2} \right\rangle }}{L}

For any molecule, [{v^2} = v_x^2 + v_y^2 + v_z^2\;\;\; \Rightarrow \;\left\langle {{v^2}} \right\rangle = \left\langle {v_x^2} \right\rangle + \left\langle {v_y^2} \right\rangle + \left\langle {v_z^2} \right\rangle ] . Since gas molecules move entirely randomly and all directions of motion are equally probable, it means that gas molecules do not have any preferred direction of motion. Therefore,

\left\langle {v_x^2} \right\rangle  = \left\langle {v_y^2} \right\rangle  = \left\langle {v_z^2} \right\rangle  = \frac{1}{3}\;\left\langle {{v^2}} \right\rangle \;\;\; \Rightarrow \;\;{F_X} = \frac{1}{3}\frac{{mN\left\langle {{v^2}} \right\rangle }}{L}

Now, presuure exerted by N molecules on the right wall is

{P_X} = \frac{{{\rm{force}}}}{{{\rm{area}}\;{\rm{of}}\;{\rm{right}}\;{\rm{wall}}}}\; = \frac{{{F_X}}}{{{L^2}}}\; = \frac{1}{3}\frac{{mN\left\langle {{v^2}} \right\rangle }}{{{L^3}}}

Generalizing this result, the pressure exerted by an ideal gas on the walls of a container can be written as

P = \frac{1}{3}\frac{{mN\left\langle {{v^2}} \right\rangle }}{{{L^3}}} = \frac{1}{3}\frac{{mN\left\langle {{v^2}} \right\rangle }}{V} = \frac{1}{3}mn\left\langle {{v^2}} \right\rangle

where V=L^3 is the volume of the container, n=\frac{N}{V} is the number density of the gas and \left\langle {{v^2}} \right\rangle is the mean square velocity of the gas molecules.

The above expression for the pressure exerted by an ideal gas can also be written as

P = \frac{1}{3}mn\left\langle {{v^2}} \right\rangle  = \frac{1}{3}mnv_{rms}^2 = \frac{1}{3}\rho v_{rms}^2

where \rho = mn is the density of the gas and {v_{rms}} = \sqrt {\left\langle {{v^2}} \right\rangle } is root mean square velocity of the gas molecules.

Relation between Pressure and Kinetic Energy

P = \frac{1}{3}\rho v_{rms}^2 = \frac{2}{3} \times \frac{1}{2}\rho v_{rms}^2 = \frac{2}{3} \times \frac{1}{2}mnv_{rms}^2 = \frac{2}{3} \left\langle {{E_{k}}} \right\rangle

where \left\langle {{E_{k}}} \right\rangle = \frac{1}{2}mv_{rms}^2 is the average kinetic energy of a gas molecule