Kinetic Particle Model of Matter: Question 4

Worked solution

Part (a): Explaining the pressure increase in particle terms

The temperature of the gas does not change, so the average kinetic energy, and therefore the average speed, of the gas particles stays the same.

What does change is the volume: the same number of gas particles are now confined to a smaller space. Each particle has less distance to travel between the syringe walls, so it collides with the walls more frequently in a given time. More frequent collisions in the same wall area means a greater total force per unit area, so the pressure increases. (The pressure rise is due to more frequent collisions, not to particles hitting the walls harder. Their speed hasn’t changed.)

Part (b): Calculating the new pressure

For a fixed mass of gas at constant temperature:

$p_1 V_1 = p_2 V_2$

Rearranging for $p_2$:

$p_2 = \frac{p_1 V_1}{V_2}$

Substituting the values:

$p_2 = \frac{1.2\times10^{5} \times 60}{40}$

$p_2 = \frac{7.2\times10^{6}}{40}$

$p_2 = \boxed{1.8\times10^{5}}\text{ Pa}$

Part (c): The pressure–volume relationship

Since $pV = \text{constant}$ for this fixed mass of gas at constant temperature, if $V$ decreases then $p$ must increase by the same factor so that the product $pV$ stays the same, and if $V$ increases, $p$ decreases by the same factor. This is exactly what an inverse proportionality means: $p \propto \dfrac{1}{V}$.

(Check: $1.2\times10^5 \times 60 = 7.2\times10^6$, and $1.8\times10^5 \times 40 = 7.2\times10^6$. The same product, as expected.)

Final answers

• (a) Same particle speed (constant temperature), but more frequent collisions with the walls in the smaller volume, so pressure increases.
• (b) New pressure $=$ $1.8\times10^5\text{ Pa}$
• (c) Pressure is inversely proportional to volume at constant temperature ($p \propto 1/V$).