Kinetic theory and the gas laws (QCE Chemistry Unit 2)

What this dot point is asking

QCAA wants you to describe gases using the kinetic theory of matter and predict the effect on a gas of changing pressure, volume or temperature using the named gas laws. The dot point sets up the ideal gas equation and gas stoichiometry that follow.

The answer

The kinetic theory of gases models gas particles as small, hard, freely moving objects whose macroscopic properties (pressure, volume, temperature) follow from the average behaviour of huge numbers of collisions. The gas laws (Boyle, Charles, Gay-Lussac, combined) are quantitative summaries of how these macroscopic properties trade off when one is held constant.

Kinetic theory of an ideal gas

Five assumptions:

1. Gas particles are in continuous, random, straight-line motion.
2. The particles have negligible volume compared with the volume of the container.
3. The particles exert no intermolecular forces on each other except during instantaneous elastic collisions.
4. Collisions between particles and with the walls are perfectly elastic (no kinetic energy lost).
5. The average kinetic energy of the particles is directly proportional to the absolute (Kelvin) temperature.

Real gases approximate ideal behaviour best at low pressure and high temperature, where the particles are far apart and moving fast enough that intermolecular forces are negligible. They deviate from ideal behaviour at high pressure and low temperature, where particle volume and intermolecular attractions matter.

Pressure from the particle picture

Pressure is the total force exerted by particle-wall collisions divided by the wall area. Three things can change pressure:

• More particles in the same volume: more collisions per second, higher pressure.
• Smaller volume at the same particle count: collisions concentrated on a smaller area, higher pressure.
• Higher temperature: particles move faster, collide harder and more often.

This intuition underwrites every gas law.

Boyle's law: P and V at constant T and n

At constant temperature, pressure is inversely proportional to volume for a fixed amount of gas.

Compressing a gas to half its volume doubles the pressure (twice as many wall collisions per second). A P-V graph is a hyperbola; P against 1/V is a straight line through the origin.

Charles's law: V and T at constant P and n

At constant pressure, volume is directly proportional to absolute temperature.

T must be in Kelvin (K = degrees C + 273.15). A V-T graph extrapolates to zero volume at 0 K (absolute zero), which is the experimental basis for the Kelvin scale.

Gay-Lussac's law: P and T at constant V and n

At constant volume, pressure is directly proportional to absolute temperature.

A sealed container heated from 300 K to 600 K doubles its internal pressure. This is why aerosol cans warn "do not heat".

Combined gas law

When more than one variable changes simultaneously (and n is fixed):

Use this when a gas sample moves between two different sets of conditions. T must always be in Kelvin; P and V can use any units provided they are consistent on both sides.

Worked example: combined gas law

A 2.50 L sample of air at 20 degrees C and 100 kPa is heated to 80 degrees C and compressed to 1.00 L. Find the new pressure.

Convert: T_1 = 293 K, T_2 = 353 K.

Sanity check: compression should increase pressure (it did, factor of 2.5), heating should also increase pressure (it did, factor of 1.20). Final factor is about 3.0.

Units and conventions used in QCE

• Pressure: kPa preferred (or Pa, atm, mmHg). 1 atm = 101.325 kPa = 760 mmHg.
• Volume: L or mL. Convert mL to L by dividing by 1000.
• Temperature: must be in Kelvin for gas-law calculations. K = degrees C + 273 (273.15 for exact).
• Amount: mol.

QCAA's data sheet lists R = 8.314 J/(mol K) for use with P in kPa and V in L (yielding J units). Get used to the unit set early.

Examples in context

Example 1. Hot-air ballooning over the Atherton Tablelands. Cairns operators inflate balloon envelopes by burning propane at the burner, heating internal air from $20^{\circ}\text{C}$ to $120^{\circ}\text{C}$. Applying Charles's law at constant pressure, $V_1/T_1 = V_2/T_2$. Ratio $V_2/V_1 = 393.15 / 293.15 = 1.34$. The $34\%$ volume increase pushes excess air out the open throat, reducing internal density (now $0.92 \, \text{kg m}^{-3}$ vs cold $1.21 \, \text{kg m}^{-3}$); the buoyant force from the displaced cooler outside air lifts the balloon. Kinetic theory frames the result: hotter molecules collide more energetically with the envelope walls, requiring greater volume to maintain ambient pressure.

Example 2. Compressed-air storage at Mount Isa mines. Mount Isa underground compressed-air systems run at $700 \, \text{kPa}$ for pneumatic rock drills. When operators bleed air into a $50 \, \text{L}$ surface tank at $25^{\circ}\text{C}$ and then the tank rolls into $45^{\circ}\text{C}$ summer sun, Gay-Lussac's law at constant volume gives $p_2/T_2 = p_1/T_1$, so $p_2 = 700 \times 318/298 = 747 \, \text{kPa}$. Engineering tolerance must cover this. Boyle's law also bites: if the same air leaks into a wider $100 \, \text{L}$ chamber, pressure halves to $350 \, \text{kPa}$. Kinetic theory accounts for all three laws via molecular collisions with walls.

Try this

Q1. State Boyle's law and explain it using kinetic molecular theory. [3 marks]

• Cue. At constant $T$, $pV =$ constant. Smaller volume means more frequent wall collisions per unit area $\rightarrow$ higher pressure.

Q2. A $4.00 \, \text{L}$ vessel at $300 \, \text{K}$ contains gas at $150 \, \text{kPa}$. The gas is compressed to $1.00 \, \text{L}$ and warmed to $400 \, \text{K}$. Calculate the new pressure. [4 marks]

• Cue. Combined gas law: $p_2 = 150 \times (4.00/1.00) \times (400/300) = 800 \, \text{kPa}$.

Q3. Kinetic theory assumes (i) negligible molecular volume and (ii) no intermolecular forces. (a) State a condition where each assumption breaks down. (b) Predict whether real $\text{NH}_3$ at $25^{\circ}\text{C}$ deviates above or below ideal. (c) Justify. [2+1+3 marks]

• Cue. (a) High $p$, low $T$. (b) Below: H-bonding pulls molecules together so observed $V < V_{ideal}$. ISMG analysis + judgement.