Kinetic Theory of Gases

Kinetic Theory explains the behavior of gases in terms of molecular motion and provides a microscopic understanding of temperature and pressure.

Overview

graph TD
    A[Kinetic Theory] --> B[Ideal Gas]
    A --> C[Molecular Motion]
    A --> D[Degrees of Freedom]
    B --> B1[Gas Laws]
    B --> B2[Equation of State]
    C --> C1[Molecular Speeds]
    C --> C2[Pressure]
    D --> D1[Equipartition]
    D --> D2[Specific Heats]

Ideal Gas

Assumptions

1. Gas consists of large number of molecules in random motion
2. Molecules are point particles (negligible size)
3. Collisions are perfectly elastic
4. No intermolecular forces except during collision
5. Time of collision « time between collisions

Equation of State

$$\boxed{PV = nRT = NkT}$$

where:

• n = number of moles
• R = 8.314 J/mol·K (gas constant)
• N = number of molecules
• k = 1.38 × 10⁻²³ J/K (Boltzmann constant)

Gas Laws

Boyle’s Law (constant T):

$$PV = \text{constant}$$

Charles’s Law (constant P):

$$V \propto T$$

Gay-Lussac’s Law (constant V):

$$P \propto T$$

Pressure from Kinetic Theory

$$\boxed{P = \frac{1}{3}\rho \overline{v^2} = \frac{1}{3}\frac{Nm\overline{v^2}}{V}}$$

Kinetic Interpretation of Temperature

$$\frac{1}{2}m\overline{v^2} = \frac{3}{2}kT$$

Average kinetic energy per molecule is proportional to temperature.

For one mole:

$$\text{Total KE} = \frac{3}{2}RT$$

Molecular Speeds

Mean Speed

$$\overline{v} = \sqrt{\frac{8kT}{\pi m}} = \sqrt{\frac{8RT}{\pi M}}$$

RMS Speed

$$v_{rms} = \sqrt{\overline{v^2}} = \sqrt{\frac{3kT}{m}} = \sqrt{\frac{3RT}{M}}$$

Most Probable Speed

$$v_{mp} = \sqrt{\frac{2kT}{m}} = \sqrt{\frac{2RT}{M}}$$

Relationship

$$v_{mp} : \overline{v} : v_{rms} = 1 : 1.128 : 1.224$$ $$v_{mp} : \overline{v} : v_{rms} = \sqrt{2} : \sqrt{8/\pi} : \sqrt{3}$$

Degrees of Freedom

Number of independent ways a molecule can store energy.

Gas Type	Example	f
Monoatomic	He, Ar	3
Diatomic (rigid)	H₂, O₂	5
Diatomic (with vibration)	At high T	7
Polyatomic (linear)	CO₂	5
Polyatomic (non-linear)	H₂O	6

Equipartition of Energy

Each degree of freedom contributes $\frac{1}{2}kT$ energy per molecule.

Total energy per molecule:

$$E = \frac{f}{2}kT$$

Total energy per mole:

$$U = \frac{f}{2}RT$$

Specific Heat Capacities

For ideal gas:

$$C_V = \frac{f}{2}R$$ $$C_P = C_V + R = \frac{f+2}{2}R$$

Ratio:

$$\boxed{\gamma = \frac{C_P}{C_V} = \frac{f+2}{f} = 1 + \frac{2}{f}}$$

Gas Type	f	γ
Monoatomic	3	5/3 = 1.67
Diatomic	5	7/5 = 1.4
Polyatomic	6	8/6 = 1.33

Mean Free Path

Average distance traveled between collisions:

$$\lambda = \frac{1}{\sqrt{2}\pi d^2 n}$$

where n = number density, d = molecular diameter

Practice Problems

1. Find the RMS speed of oxygen molecules at 27°C. (M = 32 g/mol)

2. A container has 1 mole of helium and 1 mole of argon. Find the ratio of RMS speeds.

3. Calculate the specific heat at constant volume for a diatomic gas.

Further Reading

• Thermodynamics - Laws of thermodynamics
• Properties of Solids and Liquids - States of matter