Library › Kinetic Theory & Ideal Gases

Physics (IAL)

Kinetic Theory & Ideal Gases

Revise Kinetic Theory & Ideal Gases for Physics (IAL) — revision notes and instant AI marking. Free to start.

📖 Revision notes · preview

Edexcel IAL Physics · Revision Guide

Kinetic Theory & Ideal Gases

The Big Idea: Gases are just huge swarms of tiny particles zooming around randomly — and by tracking their energy, speed, and collisions, we can predict exactly how a gas's pressure, volume, and temperature relate to each other.

Quick Overview

Internal energy = sum of the kinetic + potential energy of every molecule in a substance.
Absolute zero (0 K) is the coldest possible temperature — where molecules have zero kinetic energy.
The Kelvin and Celsius scales have equal-sized divisions — a change of 1°C = a change of 1 K.
The ideal gas equation pV = NkT links pressure, volume, molecule number, and temperature.
Boyle's Law, Charles's Law and the Pressure Law are special cases of the ideal gas equation where one variable is held constant.
Core Practical 14 uses a syringe + weights to verify Boyle's Law experimentally.
The average molecular kinetic energy of a gas molecule is E_k = ³⁄₂kT — directly proportional to temperature.

1. Internal Energy

Every substance — solid, liquid, or gas — is made of molecules that are constantly jiggling, vibrating, or zooming around. Each of those molecules carries two kinds of energy:

Kinetic energy — because the molecule is moving (vibrating in a solid, sliding past neighbours in a liquid, or flying freely in a gas). This is what gives a substance its temperature.
Potential energy — because of the molecule's position relative to its neighbours, and the intermolecular forces pulling or pushing on it.

Add up every single molecule's kinetic and potential energy in a given mass of substance, and you get its internal energy.

Definition

Internal energy (U) = ΣKE + ΣPE of all molecules in a substance

Units: Joules (J). Symbol: U.

Analogy: Picture a glass of water. Every water molecule is a tiny bouncing ball with its own personal speed (kinetic energy) and its own personal "stretchiness" of the bonds holding it to its neighbours (potential energy). Internal energy is just the total scorecard if you added up the KE and PE of every single ball in the glass.

What determines internal energy?

Temperature — higher temperature means higher average kinetic energy.
Random motion of molecules — the more chaotic/energetic the motion, the higher the KE contribution.
Phase of matter — gases have the highest internal energy (particles are free and fast), solids have the lowest (particles are locked in place, just vibrating).
Intermolecular forces — stronger forces between particles mean higher potential energy (this ties directly to the phase of matter).

How internal energy changes

Internal energy increases when...	Internal energy decreases when...
Work is done on the substance	Heat is lost to the surroundings
Heat is added to the substance	The substance changes state from gas → liquid or liquid → solid

Common Mix-Up

Students often think a fast-moving object (like a thrown egg) has "high internal energy" because it's moving fast. Wrong! The egg's kinetic energy from flying through the air is separate from its internal energy. Internal energy only counts the vibration/motion of molecules inside the egg — not the movement of the whole object.

Practice Question 1

Explain, in terms of kinetic and potential energy, why a gas has a higher internal energy than a solid of the same substance and mass.

2. Temperature & Absolute Zero

What is absolute zero?

On the thermodynamic (Kelvin) scale, absolute zero is the lowest temperature theoretically possible: 0 K, which equals −273.15 °C.

Definition

Absolute zero = the temperature at which molecules have zero kinetic energy

Because it's impossible to have less than zero kinetic energy, temperature in Kelvin can never be negative. Once a system reaches 0 K, there's no more energy left to remove from it — that's why it's called "absolute" zero. Interestingly, even the emptiness of outer space isn't quite at absolute zero — it sits at roughly 2.7 K, thanks to leftover radiation from the Big Bang. No lab has ever actually reached exactly 0 K either — it remains a theoretical limit.

Converting between Celsius and Kelvin

Conversion Equations

θ / °C = T / K − 273.15

T / K = θ / °C + 273.15

Memory Trick

Remember: 0 °C = 273.15 K. Since the alphabet goes C before K, going from Celsius → Kelvin means you add. Going backwards from K → C, you subtract. ("C comes first, so add to move forward.")

One more important fact: the size of one division is the same on both scales. A change of 1°C is exactly equal to a change of 1 K. This matters because in equations involving a temperature difference (like ΔE = mcΔθ for specific heat capacity), it doesn't matter whether you use Celsius or Kelvin — the difference comes out the same either way.

Practice Question 2

Room temperature is often taken as 300 K in ideal gas problems. What is this in °C?

Kinetic energy & temperature — the physical picture

Heating an object supplies energy to its molecules, which they absorb as kinetic energy — meaning they move faster (or, in a solid, vibrate more vigorously). This kinetic energy is what determines the temperature of a substance — temperature is essentially a measure of average molecular kinetic energy.

State	Density	Arrangement	Movement	Energy
Solid	High	Regular pattern	Vibrate around fixed position	Low
Liquid	Medium	Randomly arranged	Move around each other	Greater
Gas	Low	Randomly arranged	Move quickly in all directions	Highest

3. The Ideal Gas Equation

An "ideal gas" is a simplified model physicists use where gas molecules are treated as tiny points that don't attract or repel each other except during collisions. Under this assumption, three separate experimental gas laws (Boyle's, Charles's, and the Pressure Law) can all be combined into one master equation.

The Ideal Gas Equation

pV = NkT

p = pressure (Pa)
V = volume (m³)
N = number of molecules
k = the Boltzmann constant
T = temperature (K) — must be in Kelvin, never Celsius

The Boltzmann constant, k

The Boltzmann constant is the bridge between the microscopic world (energy of individual molecules) and the macroscopic world (temperature we can measure with a thermometer). It's defined as:

k = R / N_A

R = molar gas constant (8.31 J K⁻¹ mol⁻¹)
N_A = Avogadro's constant (6.02 × 10²³ mol⁻¹)

k = 8.31 / (6.02 × 10²³) = 1.38 × 10⁻²³ J K⁻¹

Its value is tiny because the increase in kinetic energy of just one molecule for every 1 K rise in temperature is incredibly small — you need huge numbers of molecules (like N_A) before the effects become noticeable on a human scale.

Watch Out

Whenever you use pV = NkT, temperature must be in Kelvin. Plugging in Celsius will give you completely wrong (and often negative or nonsensical) answers. Always convert first!

The Three Gas Laws (each holds one variable constant)

Boyle's Law — temperature constant, pressure and volume vary:

p ∝ 1/V → p₁V₁ = p₂V₂

Squeeze a gas into a smaller space (at the same temperature) and its pressure goes up — pressure and volume are inversely proportional.

Charles's Law — pressure constant, volume and temperature vary:

V ∝ T → V₁/T₁ = V₂/T₂

Heat a gas at constant pressure and it expands — volume is directly proportional to temperature (in Kelvin!).

Pressure Law — volume constant, pressure and temperature vary:

p ∝ T → p₁/T₁ = p₂/T₂

Heat a sealed, rigid container of gas and the pressure rises — molecules move faster and slam into the walls more often and more forcefully.

Analogy: Think of gas molecules as a swarm of angry bees in a box.
• Boyle's Law: Shrink the box (same bee energy) → more bee-wall collisions per second → higher pressure.
• Charles's Law: Heat the bees up (fixed "roof pressure") → they fly faster and need more room → box expands.
• Pressure Law: Heat the bees in a sealed box (fixed size) → they hit the walls harder and more often → pressure spikes.

Practice Question 3

A storage cylinder of an ideal gas has a volume of 8.3 × 10³ cm³. The gas is at a temperature of 15 °C and a pressure of 4.5 × 10⁷ Pa. Calculate the number of molecules of gas in the cylinder.

Practice Question 4

A fixed mass of gas is sealed in a rigid container at a pressure of 1.2 × 10⁵ Pa and a temperature of 20 °C. The gas is heated until the temperature reaches 100 °C. Calculate the new pressure.

4. Core Practical 14: Investigating Gas Pressure & Volume

This practical is designed to experimentally verify Boyle's Law — that pressure and volume are inversely proportional at constant temperature.

Variables

Independent variable: Mass, m (kg) — added to the plunger
Dependent variable: Volume, V (m³) — read off the syringe
Control variables: Temperature; cross-sectional area of the syringe

Setup & Method (simplified)

Measure the syringe's internal diameter with a Vernier caliper (average of 3 readings).
Fit rubber tubing over the syringe nozzle, sealed with a pinch clip, to trap the air inside.
Clamp the syringe vertically to a stand (secured with a counterweight or G-clamp).
Push the plunger up to the smallest visible volume and record it (no masses attached).
Hang a 100 g mass holder from a string loop at the base of the plunger. Wait a few seconds for temperature to settle, then record the new volume.
Repeat, adding 100 g each time, for up to 10 readings.

Turning the results into data for Boyle's Law

The masses pull the plunger down, which reduces the actual gas pressure below atmospheric pressure — so you have to calculate the true gas pressure, not just read it off a gauge:

Exerted pressure, p = F/A

where F = weight of masses = mg, and A = πd²/4 (cross-sectional area of the syringe)

Gas pressure = Atmospheric pressure − Exerted pressure

Atmospheric pressure ≈ 101 kPa

Once you have gas pressure (p) and volume (V) for each mass added, plot a graph of p against 1/V. A straight line through (roughly) the origin confirms Boyle's Law, since it shows p ∝ 1/V, meaning pV = constant.

Why plot p vs 1/V and not p vs V?

If pressure and volume are inversely proportional (pV = constant), a graph of p against V would be a curve (a hyperbola) — hard to check for a "proper" proportional relationship by eye. But plotting p against 1/V turns that same relationship into a straight line, which is much easier to verify and to extract data from (e.g. checking the gradient).

Sources of error

Type	Source & Fix
Systematic	Friction in the syringe plunger resists motion — use a low-friction/lubricated syringe so the only force acting is from the hanging weights.
Random	Take the volume reading a few seconds after adding each mass, to let the temperature re-equilibrate (compressing/expanding a gas can briefly change its temperature). Keep room temperature constant throughout.

Safety

Use a counterweight or G-clamp to stop the clamp stand toppling over under the weight of the masses — especially important if the bench surface isn't perfectly flat.

Practice Question 5

In this experiment, why is a graph of p against 1/V expected to not pass exactly through the origin, and what does this indicate?

5. Average Molecular Kinetic Energy

We can combine two versions of the ideal gas equation to work out something remarkable: the exact relationship between a gas molecule's average kinetic energy and the temperature of the gas.

Start with the ideal gas equation (in terms of number of molecules) and the equation linking pressure to the mean square speed of molecules:

pV = NkT

pV = ⅓Nm(c_rms)²

Since both equal pV, we can set them equal to each other. The N's cancel, and after rearranging, we get the key result:

Average Molecular Kinetic Energy (per molecule)

E_k = ½m(c_rms)² = 3/2 kT

E_k = average kinetic energy of one molecule (J)
m = mass of one molecule (kg)
(c_rms)² = mean square speed of the molecules (m² s⁻²)
k = Boltzmann constant
T = temperature (K)

The single most important takeaway from this equation: E_k ∝ T. The average kinetic energy of a gas molecule is directly proportional to the thermodynamic (Kelvin) temperature — nothing else. Double the Kelvin temperature, and you exactly double the average kinetic energy per molecule.

Memory Rhyme

"Average K.E. is three-halves kT." Say it a few times — it sticks.

If you want the total kinetic energy for all N molecules in the gas (not just one), multiply both sides by N:

E_k(total) = ½Nm(c_rms)² = 3/2 NkT

You can also substitute k = R/N_A to rewrite the per-molecule equation using the molar gas constant R:

E_k = ½m(c_rms)² = 3/2 kT = 3RT / 2N_A

Useful Trick for "Change in Speed" Questions

Since (c_rms)² ∝ T for a fixed gas (m and k don't change), you get c_rms ∝ √T. This means you can solve "find the new rms speed at a new temperature" problems without ever calculating the molecule's mass — just use ratios!

Practice Question 6

Helium can be treated as an ideal gas. Helium molecules have an r.m.s. speed of 720 m s⁻¹ at 45 °C. Calculate the r.m.s. speed of the molecules at 80 °C.

Practice Question 7

A gas is heated so that its Kelvin temperature triples. What happens to (a) the average kinetic energy per molecule, and (b) the r.m.s. speed of the molecules?

What to Memorise

Internal Energy

U = total KE + total PE of all molecules in a substance. Increases with heat/work in; decreases with heat lost or condensing/freezing.

Absolute Zero

0 K = −273.15 °C. The temperature at which molecules have zero kinetic energy. Never achieved in a lab.

Kelvin Conversion

T/K = θ/°C + 273.15. Divisions on both scales are equal in size.

Ideal Gas Equation

pV = NkT — always use T in Kelvin.

Boltzmann Constant

k = R/N_A = 1.38 × 10⁻²³ J K⁻¹

Boyle's Law

p ∝ 1/V (T constant) → p₁V₁ = p₂V₂

Charles's Law

V ∝ T (p constant) → V₁/T₁ = V₂/T₂

Pressure Law

p ∝ T (V constant) → p₁/T₁ = p₂/T₂

Molecular KE (per molecule)

E_k = ½m(c_rms)² = 3/2 kT — directly proportional to T

Pressure–Speed Link

pV = ⅓Nm(c_rms)²

Concepts Checklist

Tick off each idea once you can explain it out loud without looking at your notes.

I can define internal energy and explain what determines it.
I can explain why a moving object's KE is different from its internal energy.
I can define absolute zero and state its value in K and °C.
I can convert confidently between Celsius and Kelvin, both directions.
I understand why temperature differences (Δθ) are the same in K or °C.
I can state and use the ideal gas equation pV = NkT.
I know what the Boltzmann constant represents and can calculate it from R and N_A.
I can state Boyle's Law, Charles's Law, and the Pressure Law, and identify which variable is held constant in each.
I can describe the Core Practical 14 setup, method, and how to calculate true gas pressure.
I understand why plotting p against 1/V (not p against V) verifies Boyle's Law.
I can identify systematic vs random errors in the Boyle's Law practical.
I can derive/state E_k = 3/2 kT and explain why E_k ∝ T.
I can use c_rms ∝ √T to solve "new temperature, new speed" problems without knowing molecular mass.

Exam Tips & Common Mistakes

Trap #1: Forgetting to convert to Kelvin

Almost every gas law calculation requires T in Kelvin — if a question gives you °C, convert first. This is the single most common way marks are lost in this topic.

Trap #2: Confusing c_rms with (c_rms)²

E_k ∝ T means (c_rms)² ∝ T, NOT c_rms ∝ T. If temperature doubles, speed increases by a factor of √2, not 2. Always take the square root at the right stage!

Trap #3: Sanity-checking your answer

After any gas-law calculation, ask: "does this make physical sense?" If you heat a gas at constant volume, the final pressure must be higher than the initial pressure. If your answer says otherwise, you've made an arithmetic or setup error.

Trap #4: Internal energy vs. bulk kinetic energy

Examiners love testing whether you understand that a fast-moving object's overall KE (e.g., a thrown ball) is completely separate from the internal energy of the vibrating molecules inside it. Don't mix the two up in explanation questions.

What examiners actually look for

Clear, labelled substitution of values into the correct equation — show your working, not just the final number.
Correct unit conversions (cm³ → m³, °C → K, g → kg) done before substitution, not after.
Using proportionality reasoning (∝) for "explain what happens if..." questions rather than jumping straight to numbers.
In practical-based questions, naming specific sources of error (not vague ones like "human error") and giving a specific fix.
Correct significant figures matching the data given in the question.

🔓 Read the full Kinetic Theory & Ideal Gases note — free You're seeing the preview · free account, no card needed

Also in the full note

2. Temperature & Absolute Zero
4. Core Practical 14: Investigating Gas Pressure & Volume
Exam Tips & Common Mistakes
Kinetic energy & temperature — the physical picture
Setup & Method (simplified)

What's inside

📖 Revision notes 🎯 Learn mode ✦ AI flashcards ✓ Instant AI marking 🧊 3D explorers 🧪 Experiments & simulations 📈 Progress tracking

Read the full Kinetic Theory & Ideal Gases notes free

That's the preview — create a free account to read the rest, plus flashcards and practice questions with instant AI marking. No credit card.

Unlock the full notes free →