Mean Free Path and Non-Ideal Gas: Notes & Key Points

Mean Free Path

• Definition: The average distance traveled by a molecule between successive collisions.

Key Points:

• Inversely proportional to pressure and directly proportional to temperature
• Relates to the collision frequency (number of collisions per second) by the equation: Collision Frequency = $${\bar v \over \lambda}$$ where $$\bar v$$ is the average speed of the molecules and (\lambda) is the mean free path.
• Factors affecting mean free path include temperature (higher temperature means more collisions and shorter mean free path), pressure (higher pressure means more molecules and shorter mean free path), particle size (larger particles have a shorter mean free path due to more collisions), and shape of the particles

NCERT References:

• Class 11 Physics, Chapter 13: Kinetic Theory, Section 13.1
• Class 12 Physics, Chapter 5: Thermal Properties of Matter, Section 5.2

Non-Ideal Gas:

Key Points:

• Real gases deviate from ideal gas behavior due to intermolecular forces and finite molecular volume.
• The Van der Waals equation is a modified version of the ideal gas equation that accounts for these deviations: $${\left( P+{a\over V^2}\right)}\left( V-b\right) = RT$$ where a and b are Van der Waals constants that depend on the gas.

For Class 11 and Class 12 NCERT text books, a and b are not defined.*

• a is a measure of the strength of intermolecular forces and b is a measure of the excluded volume occupied by the molecules.

• Real gases show non-ideal behavior at high pressure and low temperature. At low temperatures, the attractive forces between molecules become significant, leading to deviations from ideal gas behavior. At high pressures, the excluded volume becomes significant, again leading to deviations.
• Compressibility factor (Z) is used to quantify deviations from ideal gas behavior: $$Z =\frac{\text{Observed pressure}}{\text{Predicted pressure (using the ideal gas equation)}}$$
• Critical temperature, pressure, and volume are significant parameters that define the phase behavior of gases.
• The Virial equation of state is a more refined equation of state that takes into account higher-order interactions between molecules.

NCERT References:

• Class 11 Physics, Chapter 13: Kinetic Theory, Section 13.10

Collisions in Gases

Key Points

• Elastic collisions: Collisions where kinetic energy and momentum are conserved.
• Inelastic collisions: Collisions where some kinetic energy is lost.
• Collision cross-section: The effective area presented by particles for collisions
• Collision probability: The probability of two molecules colliding.
• Collision frequency: The number of collisions per second, related to the cross-section and molecular velocity.
• Transport phenomena: Viscosity, thermal conductivity, and diffusion are all affected by mean free path. Viscosity is the resistance to flow, thermal conductivity is the ability to transfer heat, and diffusion is the movement of particles from high to low concentration. All these phenomena are directly proportional to the mean free path.

NCERT References

• Class 11 Physics, Chapter 13: Kinetic Theory, Section 13.5
• Class 12 Physics, Chapter 5: Thermal Properties of Matter, Section 5.3

Gas Mixtures

Key Points:

• Dalton’s law of partial pressures: The total pressure of a gas mixture is equal to the sum of partial pressures of individual components.
• Partial pressure: The pressure exerted by an individual gas component in a mixture.
• Mole fraction: The mole fraction of a gas component is the ratio of its partial pressure to the total pressure.
• Graham’s law of diffusion: The rate of diffusion of a gas is inversely proportional to the square root of its molar mass.
• Graham’s law applies to both effusion (the flow of a gas through a tiny opening) and diffusion (the movement of a gas from high concentration to low concentration), the rate of effusion being inversely proportional to the square root of the molecular mass.
• Applications: Determining molar mass, separating gases, enrichment of isotopes.

NCERT References:

• Class 11 Physics, Chapter 13: Kinetic Theory, Section 13.7
• Class 12 Physics, Chapter 5: Thermal Properties of Matter, Section 5.4

Maxwell’s Distribution of Velocities

Key Points:

• Maxwell developed a mathematical expression to describe the distribution of molecular velocities in a gas, known as the Maxwell-Boltzmann distribution.
• Maxwell-Boltzmann distribution: Describes the probability of finding molecules with different velocities at a given temperature.
• Derivation: The distribution can be derived using statistical mechanics and the concept of kinetic energy partitioning among molecules.
• Significance: Provides valuable information about the average, most probable, and root-mean-square velocities of molecules.
• Average velocity ((\overline v)): Mean velocity of all gas molecules.
• Most probable velocity ((v_p)): The velocity at which the maximum number of molecules are found.
• Root-mean-square velocity ((v_{rms})): The velocity calculated from the square root of the mean of squared velocities, representing effective molecular speed.
• Relation between velocities: ( v_p ≈ 0.88 \overline v ) and ( v_{rms} ≈ 1.09 \overline v ).
• Average kinetic energy: Directly proportional to temperature and equal to (3/2)kT, where k is Boltzmann constant and T is the absolute temperature.

NCERT References

• Class 12 Physics, Chapter 5: Thermal Properties of Matter, Section 5.6

Additional Tips:

• Focus on understanding the underlying concepts rather than memorizing formulas.
• Practice solving a variety of problems related to mean free path, non-ideal gas behavior, and molecular velocities.
• Pay attention to units and ensure dimensional consistency while solving problems.
• Revise regularly and test yourself to identify areas for improvement.