Particle Motion and Gas Pressure

Where:

• $P$ = pressure in pascals (Pa) or newtons per metre squared (N/m²)
• $F$ = force in newtons (N)
• $A$ = area in metres squared (m²)

Worked Example: Calculating Pressure

A sealed gas cylinder has an internal surface area of $0.04\text{ m}^2$. The gas particles exert a total combined outward force of $6,000\text{ N}$ against these walls. Calculate the gas pressure inside the cylinder.

Step 1: Identify the known variables.

• $F = 6,000\text{ N}$
• $A = 0.04\text{ m}^2$

Step 2: Substitute the values into the pressure equation.

• $P = \frac{6,000}{0.04}$

Step 3: Calculate the final answer with units.

• $P = 150,000\text{ Pa}$ (or $150\text{ kPa}$)

Temperature and Particle Motion

• A hot cup of tea cools down, but what exactly is happening to the microscopic particles inside?
• Heating a substance increases its internal energy, which is the total energy stored by its particles.
• Specifically, the temperature of a gas is directly proportional to the average kinetic energy of its particles.
• Because kinetic energy depends on speed ($KE = \frac{1}{2}mv^2$), an increase in temperature means the gas particles move at a higher average speed.
• Note that individual particles within the gas travel at a huge variety of different speeds; temperature only tells us about the average energy.

The Causal Chain: How Temperature Increases Pressure

• Why does a car tyre look flatter on a freezing cold morning compared to a boiling hot summer day?
• If you heat a gas inside a fixed, rigid container (meaning the volume cannot change), the pressure will increase.
• OCR expects you to explain this using a specific four-step causal mechanism:
  1. Heating the gas increases the average kinetic energy and average speed of the particles.
  2. Because they are moving faster, the particles hit the container walls more frequently (more collisions every second).
  3. Because they are moving faster, each collision involves a greater change in momentum, so the particles hit the walls more forcefully.
  4. The combination of more frequent AND more forceful collisions increases the total force exerted per unit area, resulting in a higher pressure.

Absolute Zero and The Kelvin Scale

• There is a universal speed limit (the speed of light), but is there a universal temperature limit?
• Yes — it is called absolute zero, which occurs at $-273^\circ\text{C}$ (or $0\text{ K}$).
• At this theoretical temperature, particles have minimum or zero kinetic energy, meaning they stop moving entirely and exert zero gas pressure.
• In physics, we measure absolute temperature using the Kelvin scale. A change of $1\text{ K}$ is exactly the same size as a change of $1^\circ\text{C}$.
• To convert from Celsius to Kelvin, you add 273: $T(\text{K}) = \theta(^\circ\text{C}) + 273$.

The Pressure-Temperature Law

• For a fixed mass of gas at a constant volume, the pressure is directly proportional to its absolute temperature in Kelvin ($P \propto T$).
• This means if you double the temperature in Kelvin, the pressure also doubles.
• If you plot a graph of Pressure against Temperature in Kelvin, it forms a straight line passing perfectly through the origin $(0,0)$.
• If plotted in Celsius, the straight line will not pass through the origin, but will instead cross the x-axis at $-273^\circ\text{C}$.

Worked Example: Changing Temperature and Pressure

A sealed canister of gas has a pressure of $120\text{ kPa}$ at room temperature ($20^\circ\text{C}$). If the canister is left in the sun and heats up to $50^\circ\text{C}$, what will the new pressure be? Assume the volume remains constant.

Step 1: Convert BOTH temperatures from Celsius to Kelvin.

• Initial temperature ($T_1$) = $20 + 273 = 293\text{ K}$
• Final temperature ($T_2$) = $50 + 273 = 323\text{ K}$

Step 2: Set up the proportional relationship formula.

• $\frac{P_1}{T_1} = \frac{P_2}{T_2}$

Step 3: Substitute the known values and rearrange for $P_2$.

• $\frac{120}{293} = \frac{P_2}{323}$
• $P_2 = \frac{120 \times 323}{293}$

Step 4: Calculate the final answer.

• $P_2 = 132.3\text{ kPa}$

Doing Work on a Gas

• If you have ever rapidly pumped up a bicycle tyre using a hand pump, you might have noticed the metal cylinder getting noticeably warm.
• Compressing a gas requires a mechanical force to move a distance — this means work is being done on the gas.
• Doing work transfers energy to the gas particles, increasing their average kinetic energy.
• Because average kinetic energy is directly proportional to temperature, the gas heats up.