Recall and use the equation p V = constant for a fixed mass of gas at constant temperature, including a graphical representation of this relationship

2.1.2 Particle Model of Gases

Learning Objective

Recall and apply the equation p V = constant for a fixed mass of gas at constant temperature, and represent this relationship both algebraically and graphically. In addition, understand how temperature relates to particle motion and how pressure is defined quantitatively.

1. Particle Model – Key Features

• Arrangement: Gas particles are extremely small compared with the distances between them; the empty space dominates the volume.
• Separation: The average distance between particles is large, so a gas expands to fill any container.
• Motion: Particles move in straight lines at high speed until they collide with another particle or with the walls of the container. Collisions are perfectly elastic – kinetic energy is conserved.
• Temperature & kinetic energy: The absolute temperature (in kelvin) measures the average kinetic energy of the particles:

  
  ⟨Eₖ⟩ = \tfrac{3}{2}\,k_{\mathrm B}T.

  
  At 0 K (absolute zero) the average kinetic energy would be zero and particle motion would cease.

• Brownian motion (optional): The erratic jitter of microscopic particles (e.g., pollen grains) suspended in a fluid provides experimental evidence for the kinetic particle model of gases.

2. Pressure – Quantitative Definition

Pressure is the force exerted by gas particles on a surface per unit area:

p = F/A

The force F arises from countless particle‑wall collisions. More frequent or more energetic collisions give a larger force and therefore a higher pressure.

3. Boyle’s Law – p V = constant

For a fixed amount of gas, if the temperature remains unchanged, the product of its pressure (p) and volume (V) stays the same.

Mathematical form

\$pV = k\$

where k is a constant that depends on the amount of gas and the temperature.

In practice the two‑state form is used:

\$p{1}V{1}=p{2}V{2}\$

4. Why the Product Remains Constant (Qualitative Reasoning)

1. The average kinetic energy of the particles (and therefore their speed) does not change because the temperature is constant.
2. If the volume is reduced, the same number of particles are confined to a smaller space, so they strike the walls more often → pressure rises.
3. If the volume is increased, collisions become less frequent → pressure falls.
4. Because the kinetic energy (speed) is unchanged, the change in collision frequency exactly compensates for the change in volume, keeping p V constant.

5. Using the Equation

To find an unknown pressure or volume when temperature is constant, rearrange p₁V₁ = p₂V₂:

• Find pressure: p₂ = (p₁ V₁) / V₂
• Find volume: V₂ = (p₁ V₁) / p₂

Example

Initial state: p₁ = 100 kPa, V₁ = 2.0 L.

Final volume: V₂ = 3.0 L (temperature unchanged).

Calculate the final pressure:

\$p{2}= \frac{p{1}V{1}}{V{2}} = \frac{100 \times 2.0}{3.0}= 66.7\ \text{kPa}\$

6. Temperature–Pressure Relationship at Constant Volume

When the volume is held fixed, pressure is directly proportional to absolute temperature (Gay‑Lussac’s law):

\$\frac{p{1}}{T{1}} = \frac{p{2}}{T{2}}\$

To use this relationship, temperatures must be in kelvin:

Conversion formula: K = °C + 273 and °C = K – 273.

7. Graphical Representation

7.1 p vs V (Boyle’s law)

Plotting pressure (vertical) against volume (horizontal) at constant temperature gives a hyperbola:

• The curve passes through every (V, p) pair that satisfies p V = k.
• It approaches the axes asymptotically – infinite pressure at zero volume and zero pressure at infinite volume.

7.2 p vs T (constant V)

At constant volume the p–T graph is a straight line through the origin, illustrating the direct proportionality p ∝ T (Kelvin).

8. Common Mistakes to Avoid

• Ignoring temperature: p V = constant applies only when temperature is unchanged. Use the combined gas law otherwise.
• Unit inconsistency: Convert all volumes to the same unit (L or m³) and all pressures to the same unit (kPa, Pa, or atm) before using the equation.
• Assuming a universal constant: The constant k depends on the amount of gas and the temperature; it is different for different samples.
• Treating the p–V graph as linear: The relationship is hyperbolic, not a straight line. A straight line would imply p ∝ V, which contradicts Boyle’s law.
• Forgetting the Kelvin scale: When relating pressure to temperature, always use kelvin, not degrees Celsius.

9. Practice Questions

1. A gas occupies 1.5 L at a pressure of 120 kPa. If the volume is increased to 3.0 L at the same temperature, what is the new pressure?
2. At constant temperature, a gas has a pressure of 80 kPa when its volume is 5.0 L. What volume corresponds to a pressure of 200 kPa?
3. Explain qualitatively why the p–V graph is a hyperbola rather than a straight line.
4. In a sealed syringe, the pressure is 150 kPa when the plunger is 30 mL from the tip. If the plunger is pushed in until the volume is 15 mL (temperature unchanged), calculate the new pressure.
5. State two experimental observations that support Boyle’s law.
6. At constant volume, a gas has a pressure of 250 kPa at 27 °C. What will its pressure be at 127 °C? (Give your answer in kPa.)

10. Summary

For a fixed mass of gas kept at a constant temperature, the product of pressure and volume does not change (p V = constant). This is Boyle’s law and follows from the kinetic particle model: temperature fixes the average kinetic energy, while changing the container size alters the frequency of particle‑wall collisions. The law is used algebraically with p₁V₁ = p₂V₂ and visualised as a hyperbolic curve on a pressure‑volume graph. When volume is fixed, pressure varies directly with absolute temperature, requiring the use of kelvin for any temperature‑pressure calculations. Mastery of these relationships enables accurate prediction of gas behaviour under compression or expansion without a temperature change.