Kinetic Molecular Theory

Overview

Kinetic Molecular Theory (KMT) explains the behavior of gases based on the motion and properties of gas particles. It provides a microscopic explanation for the macroscopic gas laws.

Postulates of KMT

1. Particle Nature

Gases consist of tiny particles (atoms or molecules) in constant, random motion.

2. Negligible Volume

The volume of gas particles is negligible compared to the total volume of the container.

3. No Intermolecular Forces

Gas particles do not attract or repel each other (no intermolecular forces).

4. Elastic Collisions

Collisions between particles and with walls are perfectly elastic (no energy loss).

5. Average Kinetic Energy

The average kinetic energy of gas particles is proportional to absolute temperature.

Kinetic Energy and Temperature

Average Kinetic Energy

$$KE_{avg} = \frac{3}{2}kT = \frac{3}{2}\frac{RT}{N_A}$$

Where:

• $k$ = Boltzmann constant ($1.38 \times 10^{-23}$ J/K)
• $T$ = temperature (Kelvin)
• $R$ = gas constant (8.314 J/mol·K)
• $N_A$ = Avogadro's number

Key Point

At the same temperature, all gases have the same average kinetic energy.

Molecular Speeds

Root Mean Square Speed ($u_{rms}$)

$$u_{rms} = \sqrt{\frac{3RT}{M}} = \sqrt{\frac{3kT}{m}}$$

Where:

• $R$ = 8.314 J/(mol·K)
• $T$ = temperature (K)
• $M$ = molar mass (kg/mol)
• $m$ = mass of one molecule (kg)

Types of Molecular Speed

Speed	Formula	Description
Most Probable ($u_{mp}$)	$\sqrt{\frac{2RT}{M}}$	Speed of most molecules
Average ($u_{avg}$)	$\sqrt{\frac{8RT}{\pi M}}$	Mean of all speeds
Root Mean Square ($u_{rms}$)	$\sqrt{\frac{3RT}{M}}$	$\sqrt{\text{average of } u^2}$

Relationship

$$u_{mp} < u_{avg} < u_{rms}$$ $$u_{mp} : u_{avg} : u_{rms} = 1 : 1.128 : 1.225$$

Maxwell-Boltzmann Distribution

Describes the distribution of molecular speeds in a gas.

Features

• Bell-shaped curve (asymmetric)
• Peak at most probable speed
• Higher T → curve shifts right and flattens
• Lower M → curve shifts right

Effect of Temperature

Higher T → Higher average speed → Broader distribution → More high-energy molecules

Effect of Molar Mass

Lower M → Higher average speed → Broader distribution

Examples

Example 1: Calculate $u_{rms}$

Find the root mean square speed of N₂ at 25°C.

$T = 298$ K, $M = 0.028$ kg/mol (convert from 28 g/mol)

$$u_{rms} = \sqrt{\frac{3RT}{M}} = \sqrt{\frac{3 \times 8.314 \times 298}{0.028}} = \sqrt{265,000} = 515 \text{ m/s}$$

Example 2: Compare Speeds

Compare $u_{rms}$ of H₂ and O₂ at the same temperature.

$$\frac{u_{rms}(\text{H}_2)}{u_{rms}(\text{O}_2)} = \sqrt{\frac{M(\text{O}_2)}{M(\text{H}_2)}} = \sqrt{\frac{32}{2}} = \sqrt{16} = 4$$

H₂ moves 4 times faster than O₂.

Explaining Gas Laws with KMT

Boyle's Law ($P \propto 1/V$ at constant T)

Smaller volume → more frequent collisions with walls → higher pressure

Charles's Law ($V \propto T$ at constant P)

Higher T → faster particles → more forceful collisions Volume must increase to maintain constant pressure

Avogadro's Law ($V \propto n$ at constant T, P)

More particles → more collisions → pressure increases unless volume increases

Dalton's Law ($P_{total} = \Sigma P_i$)

Each gas exerts pressure independently (no interactions) Total pressure = sum of individual pressures

Effusion and Diffusion

Diffusion

Movement of gas molecules from high to low concentration.

Effusion

Escape of gas molecules through a tiny hole.

Graham's Law

$$\frac{\text{Rate}_1}{\text{Rate}_2} = \sqrt{\frac{M_2}{M_1}}$$

Lighter gases effuse/diffuse faster.

Mean Free Path

Average distance traveled between collisions:

$$\lambda = \frac{V}{\sqrt{2} \cdot \pi \cdot d^2 \cdot N}$$

Where:

• $d$ = molecular diameter
• $N$ = number of molecules

Affected by:

• Lower pressure → longer mean free path
• Larger molecules → shorter mean free path

Real Gas Deviations

KMT assumes ideal behavior. Real gases deviate when:

High Pressure

• Molecular volume becomes significant
• Actual V > predicted V

Low Temperature

• Intermolecular forces become significant
• Actual P < predicted P
• Molecules can liquefy

Polar or Large Molecules

• Stronger intermolecular forces
• Greater deviation from ideal

Summary

Concept	Relationship
$KE_{avg}$	$\propto T$
$u_{rms}$	$\propto \sqrt{T/M}$
Collision frequency	$\propto \sqrt{T}$, $\propto 1/\sqrt{M}$
Rate of effusion	$\propto 1/\sqrt{M}$