P11: Particle Motion and Pressure
Brownian motion, gas pressure, temperature and pressure, volume and pressure (Boyle's law), absolute zero and the Kelvin scale.
Brownian Motion
Brownian motion is the random, zigzag movement of tiny particles suspended in a fluid. It provides evidence for the particle model of matter.
What is Brownian Motion?
- In 1827, Robert Brown observed pollen grains in water moving in random, jerky paths
- This motion is caused by the uneven bombardment of the larger visible particles by much smaller, invisible fluid particles
- The larger particles are hit from random directions by many smaller particles at once, causing the jerky movement
- Brownian motion provides direct evidence that matter is made of tiny particles in constant random motion
Brownian Motion in Smoke and Pollen
- Smoke particles in air show Brownian motion when viewed under a microscope
- Each smoke particle is hit from different sides by different numbers of air molecules
- The imbalance of forces causes the smoke particle to change direction randomly
- At higher temperatures, the motion is more vigorous because the air molecules move faster
When explaining Brownian motion, always describe the smaller particles hitting the larger ones from random directions. The key point is that it is the smaller, invisible particles that cause the motion of the larger, visible ones.
Gas Pressure
A gas exerts pressure on the walls of its container due to the collisions of its particles with the walls.
How Gas Pressure Arises
- Gas particles move rapidly in random directions
- When they collide with the walls of the container, they exert a force on the wall
- The total force from many collisions per unit area creates the gas pressure
- Pressure = force / area
- Pressure is measured in pascals (Pa) where 1 Pa = 1 N/m²
Factors Affecting Gas Pressure
- Temperature — higher temperature means particles move faster, hitting the walls harder and more often, increasing pressure
- Volume — smaller volume means particles hit the walls more often because they have less distance to travel between collisions, increasing pressure
- Number of particles — more particles mean more collisions per second with the walls, increasing pressure
Temperature and Pressure
For a fixed mass of gas at constant volume, the pressure increases as the temperature increases.
How Temperature Affects Pressure
- Heating a gas increases the kinetic energy of its particles
- The particles move faster on average
- They hit the walls of the container with greater force and more frequently
- This increases the pressure
- Cooling a gas reduces the kinetic energy, slowing the particles and reducing the pressure
Pressure and Temperature (Constant Volume)
p1 / T1 = p2 / T2
p = pressure in pascals (Pa)
T = temperature in kelvin (K)
This relationship only works when temperature is in kelvin.
Worked Example
A gas is at a pressure of 100 kPa at 300 K. The temperature is increased to 450 K while the volume stays the same. Calculate the new pressure.
p1 / T1 = p2 / T2
100 / 300 = p2 / 450
p2 = 100 × 450 / 300 = 150 kPa
Always convert temperatures to kelvin before using the gas laws. To convert from Celsius to kelvin, add 273. For example, 20 °C = 293 K.
Volume and Pressure (Boyle's Law)
For a fixed mass of gas at constant temperature, the pressure increases as the volume decreases.
How Volume Affects Pressure
- Reducing the volume of a container forces the same number of particles into a smaller space
- The particles travel a shorter distance between collisions with the walls
- This means more collisions per second, so the pressure increases
- If the volume is doubled, the pressure halves (and vice versa)
- Pressure is inversely proportional to volume at constant temperature
Boyle's Law (pV = constant)
p1V1 = p2V2
p = pressure in pascals (Pa)
V = volume in metres cubed (m³)
Temperature and mass of gas must remain constant.
Worked Example
A gas has a volume of 0.5 m³ at a pressure of 200 kPa. The gas is compressed to a volume of 0.2 m³ at the same temperature. Calculate the new pressure.
p1V1 = p2V2
200 × 0.5 = p2 × 0.2
100 = 0.2 p2
p2 = 100 / 0.2 = 500 kPa
Worked Example
A sealed balloon contains gas at a pressure of 101 kPa and has a volume of 2.0 m³. If the pressure increases to 202 kPa at the same temperature, what is the new volume?
p1V1 = p2V2
101 × 2.0 = 202 × V2
202 = 202 V2
V2 = 1.0 m³
Doubling the pressure halves the volume, as expected.
Absolute Zero and the Kelvin Scale
Absolute zero is the lowest possible temperature. The Kelvin scale starts from absolute zero.
What is Absolute Zero?
- Absolute zero is −273 °C (or 0 K)
- At absolute zero, particles have the minimum possible kinetic energy — they are essentially stationary
- It is impossible to cool anything below absolute zero
- At absolute zero, the pressure of an ideal gas would be zero because the particles would not be moving to collide with the walls
The Kelvin Scale
- The Kelvin scale starts at absolute zero (0 K = −273 °C)
- Each kelvin is the same size as each degree Celsius
- To convert from Celsius to kelvin: T(K) = θ(°C) + 273
- To convert from kelvin to Celsius: θ(°C) = T(K) − 273
- Kelvin is the SI unit of temperature and must be used in gas law calculations
Worked Example
Convert the following temperatures to kelvin: (a) 0 °C, (b) 100 °C, (c) −273 °C
(a) T = 0 + 273 = 273 K
(b) T = 100 + 273 = 373 K
(c) T = −273 + 273 = 0 K (absolute zero)
Worked Example
Convert the following temperatures to Celsius: (a) 300 K, (b) 77 K
(a) θ = 300 − 273 = 27 °C
(b) θ = 77 − 273 = −196 °C (the boiling point of liquid nitrogen)
Doing Work on a Gas
When you compress a gas, you do work on it and its internal energy increases, which can raise its temperature.
Work Done on a Gas
- Compressing a gas means applying a force through a distance — this does work on the gas
- The work done transfers energy to the gas particles, increasing their internal energy
- If no heat escapes, the temperature of the gas increases
- This is why a bicycle pump gets warm when you use it to inflate a tyre
- Conversely, when a gas expands, it does work on its surroundings and its temperature decreases
Real-World Example
When you pump up a bicycle tyre, each stroke of the pump compresses the air inside. The work you do on the air transfers energy to the air particles, increasing their internal energy. The air inside the pump becomes noticeably warmer. This is why the pump feels warm after extended use.
Combined Gas Law
The relationships between pressure, volume and temperature can be combined into a single equation.
Combined Gas Equation
p1V1 / T1 = p2V2 / T2
This combines Boyle's law and the pressure-temperature relationship.
Remember: temperature must always be in kelvin.
Worked Example
A gas has a pressure of 100 kPa, a volume of 0.3 m³ and a temperature of 300 K. The gas is heated to 450 K and compressed to a volume of 0.2 m³. Calculate the new pressure.
p1V1 / T1 = p2V2 / T2
(100 × 0.3) / 300 = (p2 × 0.2) / 450
0.1 = 0.2 p2 / 450
p2 = 0.1 × 450 / 0.2 = 225 kPa
Particles in Liquids and Gases
The particle model also explains why gases are easier to compress than liquids and why gases exert pressure in all directions.
Gases vs Liquids Under Pressure
- Gases are easily compressed because there are large spaces between the particles
- Liquids are very difficult to compress because the particles are already close together
- This is why hydraulic systems use liquid (usually oil) — it does not compress, so force is transmitted effectively
- Gas pressure acts equally in all directions because particles move randomly in all directions
| Property | Liquid | Gas |
|---|---|---|
| Compressibility | Very difficult to compress | Easily compressed |
| Spacing of particles | Close together | Far apart |
| Effect of increased temperature | Slight expansion, small pressure increase in sealed container | Significant expansion or large pressure increase in sealed container |
| Effect of decreased volume | Negligible compression | Significant pressure increase |
Practice Questions
1. Explain how Brownian motion provides evidence for the particle model of matter.
Brownian motion shows that larger visible particles move in random, jerky paths. This is best explained by the smaller, invisible particles colliding with them from random directions. This supports the idea that matter is made of tiny particles in constant random motion.
2. A gas has a volume of 0.8 m³ at a pressure of 150 kPa. The volume is reduced to 0.3 m³ at constant temperature. Calculate the new pressure.
p1V1 = p2V2; 150 × 0.8 = p2 × 0.3; 120 = 0.3 p2; p2 = 400 kPa
3. A sealed container of gas is at 20 °C and 120 kPa. The temperature is raised to 80 °C. Calculate the new pressure. (Volume is constant.)
T1 = 20 + 273 = 293 K; T2 = 80 + 273 = 353 K
p1/T1 = p2/T2; 120/293 = p2/353; p2 = 120 × 353/293 = 144.6 kPa
4. Explain why a bicycle pump gets warm when you use it.
When you compress the air in the pump, you do work on the gas. This work transfers energy to the air particles, increasing their internal energy. The temperature of the air rises, and this thermal energy is conducted to the pump body, making it feel warm.
5. Convert (a) 37 °C to kelvin and (b) 0 K to Celsius. What is significant about 0 K?
(a) 37 + 273 = 310 K; (b) 0 − 273 = −273 °C. 0 K is absolute zero, the lowest possible temperature, where particles have minimum kinetic energy and are essentially stationary.
Maths Skills
Boyle's Law: pV = constant
For a fixed mass of gas at constant temperature, pressure and volume are inversely proportional. If you double the pressure, the volume halves. Use p1V1 = p2V2 to solve problems where one variable changes and another is unknown.
Worked Example
A gas cylinder has a volume of 0.02 m³ at a pressure of 300 kPa. If the gas is released until the pressure drops to 100 kPa at the same temperature, what is the new volume?
p1V1 = p2V2 → 300 × 0.02 = 100 × V2 → V2 = 6 / 100 = 0.06 m³
Tripling the volume at a third of the pressure confirms the inverse relationship.
Kelvin and Celsius Conversions
All gas law calculations require temperature in kelvin. To convert: T(K) = θ(°C) + 273. A common error is using Celsius values directly in p/T calculations, which gives incorrect answers.
Worked Example
A sealed container of gas is at 27 °C and 120 kPa. It is heated to 127 °C at constant volume. Find the new pressure.
T1 = 27 + 273 = 300 K, T2 = 127 + 273 = 400 K
p1/T1 = p2/T2 → 120/300 = p2/400 → p2 = 120 × 400/300 = 160 kPa
Proportional Reasoning
Pressure is directly proportional to temperature in kelvin (at constant volume). If temperature in kelvin increases by a factor of 3/2, pressure increases by the same factor. Pressure is inversely proportional to volume (at constant temperature). If volume is reduced to a quarter, pressure increases fourfold.
Common Misconceptions
Gas Pressure and Particle Collisions
Gas pressure is caused by particles pushing on each other. Gas pressure is caused by particles colliding with the walls of the container. The force from billions of particle-wall collisions per second, divided by the wall area, gives the pressure. Particle-particle collisions are random and do not contribute to the measured pressure on the container.
Absolute Zero
At absolute zero, particles stop moving completely. At absolute zero, particles have the minimum possible kinetic energy. Due to quantum effects, they still possess some residual motion (zero-point energy). Absolute zero means the particles have the lowest energy state possible, not that all motion ceases.
6-Mark Extended Question
Explain, in terms of particle motion, why the pressure of a gas increases when its temperature increases at constant volume. [6 marks]
When the temperature of a gas increases, thermal energy is transferred to the gas particles, increasing their average kinetic energy (1). The particles move faster on average (1). Because the volume is constant, the particles travel the same distance between collisions with the walls (1). However, since they are moving faster, they collide with the walls more frequently (1) and each collision exerts a greater force on the wall because the change in momentum per collision is larger (1). The total force on the walls per unit area therefore increases, which means the pressure increases (1).
AO3: Analyse and Evaluate
A student collects the following data for a fixed mass of gas at constant temperature:
| Pressure (kPa) | Volume (m³) | p × V (kPa·m³) |
|---|---|---|
| 100 | 0.050 | 5.0 |
| 125 | 0.040 | 5.0 |
| 200 | 0.025 | 5.0 |
| 250 | 0.019 | 4.75 |
| 400 | 0.011 | 4.40 |
(a) Does the data follow Boyle's law throughout? Justify your answer. (b) Suggest a reason why the last two rows deviate from the expected pattern.
Evaluation
(a) For the first three rows, pV is constant at 5.0 kPa·m³, confirming Boyle's law. However, for the last two rows, pV decreases (4.75 and 4.40), showing that Boyle's law is no longer being followed.
(b) At very high pressures, real gases deviate from ideal gas behaviour because the particles are forced so close together that intermolecular forces become significant and the volume of the particles themselves is no longer negligible compared to the container volume. This causes the gas to be more compressible than an ideal gas, so the volume is less than predicted and pV drops.
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