Describe the pressure and the changes in pressure of a gas in terms of the motion of its particles and their collisions with a surface

2.1.2 Particle Model

Objective

Describe the pressure of a gas and explain how changes in pressure arise from the motion of its particles and their collisions with a surface.

1. Particle Structure of the Three States of Matter (Core)

• Solids: Particles are tightly packed in a regular arrangement, vibrate about fixed positions, and give the material a definite shape and definite volume.
• Liquids: Particles are close together but not fixed; they can slide past one another, giving liquids a definite volume only (no fixed shape).
• Gases: Particles are far apart and move freely in straight lines until they collide with each other or with the walls of their container, so gases have neither a definite shape nor a definite volume.

2. What Is Pressure? (Core)

Pressure (\(p\)) is the force (\(F\)) exerted per unit area (\(A\)) of a surface:

\( p = \dfrac{F}{A} \)

In a gas the force originates from the change in momentum of particles when they strike a surface.

3. How Particle Motion Produces Pressure (Core)

• When a gas particle hits a wall it exerts a force for a very short contact time.
• Each collision changes the particle’s momentum; the impulse (\(\Delta p\)) transferred to the wall is \( \Delta p = m\,\Delta v \).
• The cumulative effect of countless such collisions over the whole surface gives a measurable pressure.

4. Temperature, Kinetic Energy and Particle Speed (Core)

• Temperature is a measure of the average kinetic energy of the particles: \(\displaystyle \overline{KE} = \tfrac{3}{2}k_{\mathrm B}T\).
• Higher temperature → particles move faster on average → more energetic collisions.
• Because kinetic energy varies with the square of speed, a doubling of temperature (in kelvin) roughly increases the average speed by a factor of \(\sqrt{2}\) (≈ 1.41). This proportionality is often useful in exam questions.
• Absolute zero (‑273 °C or 0 K) is the theoretical temperature at which particle motion would cease.

5. Brownian Motion (Core)

Brownian motion is the random jittery movement of tiny particles (e.g., pollen) suspended in a fluid. Under a microscope the particles appear to move erratically because they are constantly bombarded by the surrounding gas or liquid molecules. This provides direct visual evidence that the molecules of the surrounding medium are in perpetual motion.

6. Core Quantitative Relationships

Core Example

A piston contains 0.5 mol of air at 300 K and 100 kPa. The gas is compressed isothermally until its volume is halved. Because \(pV = \text{constant}\), the pressure doubles to 200 kPa.

7. Supplementary Material (Extension)

• Ideal‑gas equation: \(\displaystyle pV = nRT\)

  
  When to use: calculations involving the number of moles (\(n\)) or when temperature is not held constant. Not required for the Core.

• Kinetic‑theory expression: \(\displaystyle p = \frac{1}{3}\,N\,m\,\overline{v^{2}}\;\frac{1}{V}\)

  
  where \(N\) = number of particles, \(m\) = mass of one particle, \(\overline{v^{2}}\) = mean‑square speed, \(V\) = container volume.

8. Factors That Influence Gas Pressure (Kinetic‑Theory View) (Core)

1. Number of particles (or amount of gas, \(n\)) – pressure ∝ \(n\).
2. Average kinetic energy (temperature, \(T\)) – pressure ∝ \(T\).
3. Volume of the container (\(V\)) – pressure ∝ \(1/V\).

9. Effects of Changing Variables (Core + Supplement)

10. Everyday Illustrations (Core)

• Hot soda bottle: A sealed glass bottle left in a hot car warms the gas inside. The particles move faster, collide more forcefully with the glass, and the internal pressure rises. If the pressure exceeds the strength of the glass, the bottle can burst.
• Winter tyre deflation: A car tyre filled with air at room temperature is taken into a cold garage. The temperature drops, particle speeds fall, collisions become less energetic, and the tyre pressure drops noticeably. This demonstrates the direct link between temperature and pressure.

11. Qualitative Summary (Core)

• Higher temperature → faster particles → more frequent & more forceful collisions → higher pressure.
• Smaller volume → particles hit walls more often → higher pressure.
• More gas (more particles) → greater number of collisions → higher pressure.
• Lower temperature or larger volume → opposite effects, reducing pressure.

12. Common Misconceptions (Core)

• Pressure is not caused by the weight of the gas; it is caused by particle collisions with surfaces.
• Temperature measures average kinetic energy, not how “hot” a gas feels.
• The size of individual gas particles is negligible compared with the distances between them.

13. Check Your Understanding (Core)

1. If a sealed container of gas is heated, explain why the pressure rises even though the number of particles does not change.
2. A syringe plunger is pushed in, reducing the volume of the trapped air. Describe what happens to the speed of the air molecules and the pressure inside the syringe.
3. Two balloons contain the same amount of gas at the same temperature, but one balloon is larger. Which balloon has the higher pressure? Why?
4. What would happen to the pressure of a gas if it were cooled to absolute zero? (Hint: think about particle motion.)