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kinetic theory of gases - maxwell speed and velocity distribution,comarison at two different temperatures,derivation of ideal gas equation,equipartion theorm,specific heat . Include examples and applications as well

Kinetic Theory of Gases: Complete Notes & Derivations

The Kinetic Theory of Gases bridges the gap between the microscopic world of colliding molecules and macroscopic properties like pressure and temperature. By treating gases as large collections of particles in random motion, we can derive the Ideal Gas Law and understand how energy is distributed within a system.

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1. Assumptions of Kinetic Theory

To mathematically model a gas, we assume an "Ideal Gas" scenario:

  • Random Motion: Molecules move in straight lines randomly until they collide.
  • Point Masses: The volume of individual molecules is negligible compared to the container's volume.
  • Elastic Collisions: No kinetic energy is lost during collisions between molecules or with walls.
  • No Forces: There are no intermolecular attractive or repulsive forces.

2. Derivation of the Ideal Gas Equation

Pressure Equation

Consider a gas molecule of mass \( m \) moving with velocity \( v_x \) inside a cubic container of length \( L \). When it strikes the wall, the change in momentum is:

\[ \Delta p = p_f - p_i = -mv_x - (mv_x) = -2mv_x \]

The force exerted on the wall is the rate of change of momentum. The time between two collisions with the same wall is \( \Delta t = \frac{2L}{v_x} \). Thus, force \( F \):

\[ F = \frac{|\Delta p|}{\Delta t} = \frac{2mv_x}{2L/v_x} = \frac{mv_x^2}{L} \]

Summing for \( N \) molecules and averaging over all three dimensions (since \( \langle v^2 \rangle = 3\langle v_x^2 \rangle \)), the Pressure \( P \) is:

\[ P = \frac{1}{3} \frac{Nm \langle v^2 \rangle}{V} \quad \text{or} \quad PV = \frac{1}{3} Nm v_{rms}^2 \]

Kinetic Energy and Temperature

Kinetic interpretation of temperature states that the average kinetic energy is proportional to absolute temperature \( T \):

\[ KE_{avg} = \frac{1}{2} m v_{rms}^2 = \frac{3}{2} k_B T \]

Substituting \( m v_{rms}^2 = 3k_B T \) into the pressure equation:

\[ PV = \frac{1}{3} N (3 k_B T) \implies PV = N k_B T \]

For \( n \) moles, this becomes the familiar \( PV = nRT \).

3. Maxwell-Boltzmann Speed Distribution

Molecules in a gas move at various speeds. The probability of finding a molecule with speed \( v \) is given by the Maxwell distribution:

\[ f(v) = 4\pi \left( \frac{m}{2\pi k_B T} \right)^{3/2} v^2 e^{-mv^2 / 2k_B T} \]

Comparison at Different Temperatures

If we plot the distribution for a gas at two temperatures \( T_1 \) and \( T_2 \) where \( T_2 > T_1 \):

  • At \( T_1 \) (Lower Temp): The curve is sharp and peaked. Most molecules have low kinetic energy.
  • At \( T_2 \) (Higher Temp): The curve flattens and the peak shifts to the right (higher speed). The distribution spreads out, indicating a wider range of molecular speeds.

Characteristic Speeds

Three key speed metrics are derived from this distribution:

  • Most Probable Speed (\( v_{mp} \)): \( \sqrt{2RT/M} \)
  • Average Speed (\( v_{avg} \)): \( \sqrt{8RT/\pi M} \)
  • RMS Speed (\( v_{rms} \)): \( \sqrt{3RT/M} \)

4. Law of Equipartition of Energy

This theorem states that for a system in thermal equilibrium, the total energy is divided equally among all available degrees of freedom. Each degree of freedom contributes \( \frac{1}{2}k_B T \) to the internal energy per molecule.

5. Specific Heat Capacities

The specific heat depends on the degrees of freedom (\( f \)). The internal energy for 1 mole is \( U = \frac{f}{2} RT \).

Atomicity Degrees of Freedom (\( f \)) \( C_V = \frac{f}{2}R \) \( C_P = (1+\frac{f}{2})R \) \( \gamma = C_P/C_V \)
Monatomic (He, Ar) 3 \( 1.5 R \) \( 2.5 R \) 1.67
Diatomic (N2, O2) 5 \( 2.5 R \) \( 3.5 R \) 1.40
Polyatomic 6 \( 3.0 R \) \( 4.0 R \) 1.33

6. Applications

  • Atmospheric Physics: Explains why lighter gases (Hydrogen, Helium) escape Earth's gravity more easily than heavier gases (Oxygen, Nitrogen) due to higher RMS speeds.
  • Thermodynamics: Forms the basis for calculating work done in heat engines and refrigerators.
  • Brownian Motion: The random movement of particles suspended in a fluid (pollen grains in water) is visual proof of the kinetic theory.