Kinetic Theory & Ideal Gases

What is Kinetic Theory?

The Kinetic Theory of Gases is a brilliant scientific model that explains the macroscopic properties of gases (like pressure, temperature, and volume) by considering their microscopic composition and motion. It bridges the gap between the invisible world of bouncing molecules and the tangible world of balloons expanding or tyres popping.

Instead of tracking every single atom (which is impossible, since $1\ \text{cm}^3$ of air contains $\sim 2.7 \times 10^{19}$ molecules), kinetic theory uses statistical mechanics to average out their chaotic behaviours.

Learning Goals: By the end of this guide, you should be able to:

List the fundamental assumptions of an Ideal Gas.

Explain how molecular collisions create macroscopic gas pressure.

Relate absolute temperature to the average kinetic energy of molecules.

Interpret the Maxwell-Boltzmann distribution curve.

The 5 Assumptions of an Ideal Gas (RAVED)

A real gas is too complicated to model perfectly. To make the math work, physicists invented the Ideal Gas — a theoretical perfect gas that obeys the following assumptions (often remembered by the acronym RAVED):

Random motion: Molecules move in continuous, random, straight-line motion at varying speeds.
Attraction is zero: There are zero intermolecular forces (no pull or push) between molecules, except during an actual collision. (This means internal energy is purely kinetic!)
Volume of particles is zero: The volume of the molecules themselves is completely negligible compared to the total volume of the container. They are considered mathematical "point masses".
Elastic collisions: All collisions (between molecules, and with the container walls) are perfectly elastic. No kinetic energy is lost to heat or sound.
Duration of collisions is zero: The time spent during a collision is negligible compared to the time spent travelling between collisions.

How Gases Cause Pressure

Pressure ( $P$ ) is defined as the force ( $F$ ) exerted per unit area ( $A$ ): $P = \frac{F}{A}$ .

But how does empty space full of floating molecules exert force?

An individual gas molecule bounds toward the container wall with a certain momentum ( $p = mv$ ).
It hits the wall and bounces back elastically. Its momentum changes ( $\Delta p = -mv - mv = -2mv$ ).
According to Newton's Second Law, the force exerted is the rate of change of momentum ( $F = \frac{\Delta p}{\Delta t}$ ). Therefore, the wall exerts a force on the molecule.
According to Newton's Third Law, the molecule exerts an equal and opposite force on the wall.
With billions of trillions of molecules colliding with the walls every microsecond, these tiny, discrete impacts blend into a smooth, continuous macroscopic outward force, which we call Pressure!

(If you decrease the volume of the container, the molecules have less distance to travel, so they hit the walls more frequently per second, resulting in a higher rate of momentum transfer and therefore higher pressure. This is Boyle's Law!)

Temperature = Average Kinetic Energy

In physics, temperature is not an abstract concept of "hot or cold". The absolute temperature ( $T$ , measured in Kelvin) of a gas is directly proportional to the average translational kinetic energy of its molecules.

E_k = \frac{3}{2} k_B T

(Where $k_B$ is the Boltzmann constant, $1.38 \times 10^{-23}\ \text{J/K}$ )

This equation is profound: it means if you heat a gas, you are literally making its molecules move faster. It also explains absolute zero ( $0\ \text{K}$ ): it is the theoretical temperature where all molecular kinetic translation stops completely.

The Maxwell-Boltzmann Distribution

Because gas molecules collide and exchange energy randomly trillions of times a second, they do not all move at the same speed. Their speeds follow a specific statistical shape known as the Maxwell-Boltzmann distribution curve.

x-axis: Speed of molecules.
y-axis: Number (or probability density) of molecules.
Total Area under the curve: Represents the total number of particles (which remains constant).

The Effect of Temperature on the Curve

When you heat a gas:

The peak of the curve lowers and shifts to the right (indicating a higher most-probable speed).
The curve gets broader and flatter.
The "tail" on the right side becomes thicker, meaning a significantly larger fraction of molecules possess very high kinetic energy (crucial for overcoming activation energy in chemical reactions!).

(Remember: The area under the curve must stay exactly the same because no gas particles were added or removed).

Worked Examples

Example 1: Root Mean Square (RMS) Speed

Question: Calculate the root mean square speed ( $v_{\text{rms}}$ ) of an oxygen molecule ( $O_2$ ) at room temperature ( $300\ \text{K}$ ). The mass of one $O_2$ molecule is roughly $5.32 \times 10^{-26}\ \text{kg}$ .

Step 1: Use energy equivalence. Average kinetic energy = $\frac{1}{2} m v_{\text{rms}}^2 = \frac{3}{2} k_B T$ Step 2: Rearrange to solve for $v_{\text{rms}}$ :

v_{\text{rms}} = \sqrt{\frac{3 k_B T}{m}}

Step 3: Substitute values:

v_{\text{rms}} = \sqrt{\frac{3 \times (1.38 \times 10^{-23}) \times 300}{5.32 \times 10^{-26}}} \approx \sqrt{\frac{1.242 \times 10^{-20}}{5.32 \times 10^{-26}}} \approx 483\ \text{m/s}

The oxygen molecules in your room are currently zipping around at over $1700\ \text{km/h}$ !

Common Mistakes

Using Celsius instead of Kelvin — All gas calculations must be done in Kelvin ( $K$ ). Add 273.15 to your Celsius value. If you use Celsius, you might end up with dividing by zero or calculating negative volumes/pressures, which are physically impossible.
Assuming "Ideal Gases" actually exist — They don't! The assumptions completely ignore reality. Real gas molecules do attract each other (Van der Waals forces) and do take up space. The ideal gas model usually works well at high temperatures and low pressures, but falls apart at low temperatures and high pressures when the gas is about to condense into a liquid.
Confusing average speed with average velocity — The average velocity of gas molecules in a stationary balloon is strictly zero. Because velocity is a vector, for every molecule flying left, another flies right, and they average out. Average speed (a scalar) is what we care about here.

Exam Tips (A-Level / AP / IB)

"Show that the pressure exerted by a gas..." The 6-step derivation of $pV = \frac{1}{3}Nmc^2$ from first principles (momentum of a single particle in a 3D box) is a classic 6-mark question. Practice deriving it from scratch!
When drawing Maxwell-Boltzmann curves at two different temperatures, ensure the peaks are in the correct places (hotter curve peak is lower and further right). Crucially, ensure the two curves overlap and cross, otherwise the total area isn't conserved. The hotter curve should consistently remain above the colder curve at the far-right high-energy tail.

Frequently Asked Questions

Why don't the heavy gas molecules just sink to the floor?

They try to! However, their kinetic thermal energy is so immense compared to the tiny force of gravity on their microscopic mass that the random collisions constantly kick them back upwards, overpowering gravity. This perfectly random mixing prevents the air in a room from stratifying purely by mass.

Are collisions really perfectly elastic?

For noble gases bumping into each other, almost yes. But for complex molecules like $CO_2$ , collisions can temporarily transfer translational kinetic energy into internal rotational or vibrational energy. However, in our theoretical ideal gas, we conveniently pretend this doesn't happen.

Collision Theory (Chemistry) — See how the high-energy "tail" of the Maxwell-Boltzmann distribution determines reaction rates.
Kinematics (Motion Graphs) — The foundation of momentum ( $mv$ ) change that allows the walls to experience macroscopic force.