Kinetic Molecular Theory, Graham’s Law of Effusion, and Gas Particle Velocities

Kinetic Molecular Theory of Gases

Introduction to Kinetic Molecular Theory

The Kinetic Molecular Theory (KMT) provides a molecular-level explanation for the behavior of ideal gases. It is based on several key postulates that describe the motion and interactions of gas particles.

• Postulate 1: Gases consist of large numbers of molecules (atoms) that are in continuous, random motion.

• Postulate 2: The combined volume of the gas molecules is negligible compared to the total volume in which the gas is contained.

• Postulate 3: Attractive and repulsive forces between gas molecules are negligible.

• Postulate 4: Collisions between gas molecules and with the walls of the container are perfectly elastic (no energy is lost).

• Postulate 5: The average kinetic energy of the molecules is proportional to the absolute temperature (in Kelvin). All gases at the same temperature have the same average kinetic energy.

Application: Explaining Gas-Phase Behavior

The KMT can be used to explain qualitative observations about gases, such as changes in pressure, volume, and temperature.

• Temperature Decrease: When temperature decreases, the average kinetic energy of gas particles decreases.

• Effect on Collisions: Lower kinetic energy results in fewer and less forceful collisions with container walls, leading to a decrease in gas pressure if the volume is constant.

• Example: A balloon shrinks when cooled because the gas pressure inside drops as particle kinetic energy decreases.

Key Equation:

$KE = \frac{1}{2}mv^2$

where KE is kinetic energy, m is mass, and v is velocity.

Graham’s Law of Effusion

Derivation from Kinetic Energy

Graham’s Law describes the rate at which gases effuse (escape through a small hole) and is rooted in the kinetic energy of gas particles.

• At the same temperature, different gases have the same average kinetic energy:

$KE_1 = KE_2$

$\frac{1}{2}m_1v_1^2 = \frac{1}{2}m_2v_2^2$

Solving for the ratio of velocities (rates of effusion):

$\frac{v_1}{v_2} = \sqrt{\frac{m_2}{m_1}}$

Since the rate of effusion is proportional to particle velocity:

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

where M is the molar mass of the gas.

• Example Calculation: If helium (He) and oxygen (O2) are compared, helium effuses faster due to its lower molar mass.

Gas Particle Velocity Distributions

Effect of Mass and Temperature on Velocity

The velocity of gas particles is not uniform; it is distributed over a range of values, described by the Maxwell-Boltzmann distribution.

• For Equal Amounts of Different Gases at the Same Temperature: Lighter gases (lower molar mass) have higher average velocities than heavier gases.

• For the Same Gas at Different Temperatures: As temperature increases, the average velocity of gas particles increases, and the distribution broadens.

Graphical Representation:

• Graphs show that at a given temperature, H2 (hydrogen) has a higher most probable speed than O2 (oxygen).

• For a single gas, the distribution shifts to higher velocities as temperature increases (e.g., from 100 K to 2000 K).

Key Equation for Root Mean Square Velocity:

$v_{rms} = \sqrt{\frac{3RT}{M}}$

where R is the gas constant, T is temperature in Kelvin, and M is molar mass in kg/mol.

Summary Table: Graham’s Law of Effusion

Gas	Molar Mass (g/mol)	Relative Rate of Effusion
He	4.00	Fastest (compared to O2)
O2	32.00	Slower

Additional info: The table illustrates that lighter gases effuse more rapidly than heavier gases, as predicted by Graham’s Law.