FSc Notes Punjab Board: 2nd Year Physics Thermal Physics

Thermal Physics explains hot and cold with tiny moving molecules. Gas pressure comes from molecules hitting the wall. Temperature is the average kinetic energy of those molecules. Learn three formulas and five assumptions, and the whole chapter opens up.

After a long drive on the motorway, the tyre pressure reads higher. Nobody pumped in any air. The same air is in the same tyre. So what changed? The air got hotter. Hotter air means faster molecules. Faster molecules hit the tyre wall harder and more often, so the pressure rises.

This note follows our lessons on Brownian motion, kinetic theory, gas pressure and kinetic energy. They follow the new PECTAA Physics 12 book (2026-27), Thermal Physics. Keep your own book open too.

What to know

Brownian motion. Robert Brown, a botanist, looked at tiny pollen grains in water under a microscope. The grains never stopped moving. They jerked left, then right, then back, with no pattern. Nothing was stirring the water. Water molecules, far too small to see, were hitting each grain from every side. At any moment a few more hit one side than the other, so the grain jumps.

• It is a random, zig-zag motion caused by molecular collisions.
• Lighter particles get greater speeds from the same kicks, because v = Δp ÷ m.
• Lower viscosity means faster Brownian movement. Pollen jiggles more in water than in honey. Viscosity is the internal friction of a fluid, its resistance to flow.
• Einstein explained it in 1905. Perrin confirmed it by experiment in 1908. That earned him the 1926 Nobel Prize in physics.
• Dust shows the same motion, so the pollen is not moving because it is alive.

Kinetic theory: the five assumptions. A gas that follows all five is an ideal gas.

1. A gas has a very large number of tiny particles, far apart compared with their size. Their own volume is negligible.
2. The particles are in constant, rapid motion in random directions.
3. Collisions with the walls are perfectly elastic. No kinetic energy is lost.
4. There are no forces of attraction or repulsion between particles, except when they collide. Between collisions each particle moves in a straight line.
5. The average kinetic energy depends on the temperature of the gas.

Real gases come closest to ideal at high temperature and low pressure. The mean free path is the average distance a particle covers before a collision that really changes its motion.

Key points

Pressure from one molecule in a cube of side L.

• Change in momentum: Δp = −mvₓ − mvₓ = −2mvₓ.
• Time between hits on the same wall: Δt = 2L ÷ vₓ.
• Force: F = 2mvₓ ÷ (2L/vₓ) = mvₓ²/L.
• Pressure: P = mvₓ²/L³.

All molecules. Motion is random, so x, y and z share the speed equally: ⟨vₓ²⟩ = ⅓⟨v²⟩. Then:

• PV = ⅓Nm⟨v²⟩
• P = ⅓ρ⟨v²⟩, where ρ = Nm/V

Why square the speeds? A molecule hits at +v and leaves at −v. The plain average of velocity is zero. So square first, average, then take the root. That gives the rms speed.

Temperature and energy.

• ⟨K.E.⟩ = (3/2)k_BT, where k_B = 1.38 × 10⁻²³ J K⁻¹.
• v_rms = √(3k_BT/m).
• At 300 K, one air molecule carries 6.21 × 10⁻²¹ J on average.
• At the same temperature, all gases have the same average kinetic energy. Lighter molecules move faster.
• Always use kelvin.

Worked examples

Example 1. Oxygen at 0 °C has density 1.44 kg m⁻³ and rms speed 456.4 m s⁻¹. Find the pressure.

1. P = ⅓ρv²_rms
2. P = ⅓ × 1.44 × (456.4)²
3. P ≈ 99,984 Pa, which is about 1.0 × 10⁵ Pa.

Example 2. A gas has density 4.5 kg m⁻³ at pressure 9.3 × 10⁵ Pa. Find the rms speed.

1. From P = ⅓ρ⟨v²⟩ we get ⟨v²⟩ = 3P ÷ ρ.
2. ⟨v²⟩ = 3 × 9.3 × 10⁵ ÷ 4.5 = 6.2 × 10⁵ m² s⁻².
3. v_rms = √(6.2 × 10⁵) ≈ 787 m s⁻¹.

Example 3. The rms speed of a gas at 27 °C is v. What is it at 327 °C?

1. T₁ = 27 + 273 = 300 K. T₂ = 327 + 273 = 600 K.
2. Speed goes as √T. So v₂ = v × √(600/300) = √2 × v.

Example 4. Why does the pressure rise when a gas at constant temperature is squeezed into a smaller volume?

1. The temperature is constant, so the average speed is the same.
2. A smaller box means shorter trips and more wall hits per second.
3. More force per unit area means higher pressure.

Common mistakes in the exam

1. Writing Δp = 0. The speed is the same but the direction reverses. So Δp = −2mvₓ.
2. Using the average velocity. It is zero. Use the mean-square speed.
3. Forgetting the ⅓. The formula is PV = ⅓Nm⟨v²⟩.
4. Using Celsius in formulas. Add 273 first. With Celsius, 327 °C looks 12 times hotter than 27 °C. In kelvin it is only twice.
5. Saying heavier molecules have more kinetic energy at the same temperature. They do not. They move slower.
6. Choosing "inelastic collisions" as an assumption. The collisions are perfectly elastic.
7. Writing "low temperature and low pressure" for ideal behaviour. It is high temperature and low pressure.
8. Copying the year 1927 for Brown. It is 1827. Brown came before Einstein's 1905.

Exam technique

• Write the formula first, then substitute, then give the answer with units.
• For "explain on the basis of kinetic theory" questions, use two steps: what the molecules do, then what happens to the pressure.
• For ratio questions, do not calculate everything. Use proportions. Speed goes as √T. Energy goes as T.
• In derivations, write each step on its own line. Examiners give a mark for each step.

Practice questions

1. Which scientist explained Brownian motion in 1905? Answer: Albert Einstein.
2. A gas has pressure 3.2 × 10⁵ N m⁻² and density 6 kg m⁻³. Find the rms speed of its particles. Answer: v = √(3P/ρ) = √(3 × 3.2 × 10⁵ ÷ 6) = 400 m s⁻¹.
3. The kelvin temperature of a gas becomes 4 times larger. What happens to its rms speed? Answer: It becomes 2 times larger, because speed goes as √T.
4. A gas at 300 K must reach three times its rms speed. At what temperature? Answer: 2700 K. The temperature must be 9 times larger, and 300 × 9 = 2700.
5. Why does pressure rise when a gas at constant temperature is squeezed into a smaller volume? Answer: The molecules make shorter trips, so they hit the walls more often. More force per unit area means higher pressure.
6. If the density of a gas is doubled and nothing else changes, what happens to its pressure? Answer: It doubles, because P = ⅓ρ⟨v²⟩.
7. Why is Brownian motion slower in honey than in water? Answer: Honey has a much higher viscosity. Higher viscosity means more internal friction and slower movement.

Quick revision

• Brown 1827, Einstein 1905, Perrin 1908.
• Five assumptions. Ideal behaviour: high temperature, low pressure.
• P = ⅓ρ⟨v²⟩. ⟨K.E.⟩ = (3/2)k_BT.
• Kelvin only.

Learn it step by step in the lessons Brownian Motion (/learn/fsc-2/physics/brownian-motion), Kinetic Theory of Gases (/learn/fsc-2/physics/kinetic-theory-of-gases), Pressure in a Gas (/learn/fsc-2/physics/pressure-of-a-gas) and Average Translational Kinetic Energy of a Gas (/learn/fsc-2/physics/kinetic-energy-temperature).

Ready to practise this? Continue in 2nd Year Physics.