⚛Molecular Physics Unit 13 – Kinetic Theory and Gas Transport

Key Concepts and Definitions

• Kinetic theory describes the behavior of gases based on the motion of their constituent molecules or atoms
• Gas particles are in constant random motion, colliding with each other and the walls of their container
• Temperature is a measure of the average kinetic energy of the gas particles
• Pressure results from the force exerted by gas particles colliding with the walls of the container
• Volume is the amount of space occupied by the gas
• Ideal gases are hypothetical gases that perfectly follow the assumptions of kinetic theory
• Real gases deviate from ideal behavior due to intermolecular forces and the finite volume of gas particles
• Mean free path is the average distance a particle travels between collisions

Fundamental Assumptions of Kinetic Theory

• Gas particles are treated as point masses with negligible volume compared to the total volume of the gas
• Collisions between gas particles are perfectly elastic, meaning kinetic energy is conserved
• Gas particles move in straight lines between collisions
• The average kinetic energy of gas particles is proportional to the absolute temperature
• Gas particles do not interact with each other except during collisions (no intermolecular forces)
• The time of a collision is negligible compared to the time between collisions
• The distribution of particle velocities follows a Maxwell-Boltzmann distribution

Ideal Gas Laws and Their Limitations

• The ideal gas law relates pressure ($P$), volume ($V$), number of moles ($n$), and temperature ($T$) as $PV = nRT$, where $R$ is the ideal gas constant
• Boyle's law states that pressure and volume are inversely proportional at constant temperature and number of moles: $P_1V_1 = P_2V_2$
• Charles's law states that volume and temperature are directly proportional at constant pressure and number of moles: $\frac{V_1}{T_1} = \frac{V_2}{T_2}$
• Gay-Lussac's law states that pressure and temperature are directly proportional at constant volume and number of moles: $\frac{P_1}{T_1} = \frac{P_2}{T_2}$
• Avogadro's law states that volume and number of moles are directly proportional at constant pressure and temperature: $\frac{V_1}{n_1} = \frac{V_2}{n_2}$
• The ideal gas law assumes gas particles have no volume and no intermolecular forces, which is not true for real gases
• Deviations from ideal behavior become more significant at high pressures and low temperatures

Molecular Velocity Distributions

• The Maxwell-Boltzmann distribution describes the probability distribution of particle velocities in a gas at a given temperature
• The distribution is characterized by a peak at the most probable velocity and a long tail at high velocities
• The root mean square (RMS) velocity is the square root of the average of the squares of the velocities: $v_{rms} = \sqrt{\frac{3kT}{m}}$, where $k$ is the Boltzmann constant and $m$ is the mass of a gas particle
• The average velocity is related to the RMS velocity by $v_{avg} = \sqrt{\frac{8kT}{\pi m}}$
• Higher temperatures result in a broader velocity distribution and higher average velocities
• The velocity distribution is important for understanding gas diffusion, effusion, and transport properties

Mean Free Path and Collision Rates

• The mean free path ($\lambda$) is the average distance a particle travels between collisions: $\lambda = \frac{1}{\sqrt{2} \pi d^2 n}$, where $d$ is the particle diameter and $n$ is the number density of particles
• The collision frequency ($z$) is the average number of collisions per particle per unit time: $z = \frac{1}{\tau} = \frac{v_{avg}}{\lambda}$, where $\tau$ is the average time between collisions
• The mean free path decreases with increasing pressure and particle size
• The collision frequency increases with increasing temperature and pressure
• The mean free path and collision frequency are important for understanding gas transport properties and reaction rates

Transport Phenomena in Gases

• Diffusion is the net movement of particles from regions of high concentration to regions of low concentration
  • Fick's first law relates the diffusive flux to the concentration gradient: $J = -D \frac{dC}{dx}$, where $J$ is the flux, $D$ is the diffusion coefficient, and $\frac{dC}{dx}$ is the concentration gradient
• Thermal conductivity is the transport of heat through a gas due to temperature gradients
  • Fourier's law relates the heat flux to the temperature gradient: $q = -k \frac{dT}{dx}$, where $q$ is the heat flux, $k$ is the thermal conductivity, and $\frac{dT}{dx}$ is the temperature gradient
• Viscosity is the resistance of a gas to shear stress and is responsible for the transfer of momentum between layers of the gas
  • Newton's law of viscosity relates the shear stress to the velocity gradient: $\tau = \mu \frac{dv}{dy}$, where $\tau$ is the shear stress, $\mu$ is the viscosity, and $\frac{dv}{dy}$ is the velocity gradient
• The transport properties of gases depend on the mean free path and collision frequency
• Gases with longer mean free paths and lower collision frequencies generally have higher diffusion coefficients, thermal conductivities, and lower viscosities

Real Gases and Deviations from Ideal Behavior

• Real gases deviate from ideal behavior due to the finite volume of gas particles and the presence of intermolecular forces (van der Waals forces)
• The van der Waals equation modifies the ideal gas law to account for these deviations: $(P + \frac{an^2}{V^2})(V - nb) = nRT$, where $a$ and $b$ are constants specific to the gas
  • The term $\frac{an^2}{V^2}$ accounts for attractive intermolecular forces
  • The term $nb$ accounts for the finite volume of gas particles
• The compressibility factor ($Z$) is the ratio of the actual volume of a gas to the volume predicted by the ideal gas law: $Z = \frac{PV}{nRT}$
  • For ideal gases, $Z = 1$
  • For real gases, $Z$ deviates from 1, especially at high pressures and low temperatures
• The critical point is the temperature and pressure at which the liquid and vapor phases of a substance become indistinguishable
  • At temperatures and pressures above the critical point, the gas cannot be liquefied by compression alone
• The Joule-Thomson effect describes the temperature change of a gas when it expands through a porous plug or valve without exchanging heat with its surroundings
  • The Joule-Thomson coefficient $(\mu_{JT})$ is defined as $\mu_{JT} = (\frac{\partial T}{\partial P})_H$, where $H$ is the enthalpy

Applications and Experimental Techniques

• Kinetic theory is used to explain the behavior of gases in various applications, such as:
  • Atmospheric science (weather forecasting, climate modeling)
  • Combustion and propulsion systems (engines, turbines)
  • Gas separation and purification (membranes, adsorption)
  • Vacuum technology (pumps, gauges)
• Experimental techniques for studying gases include:
  • Pressure measurements (manometers, pressure transducers)
  • Temperature measurements (thermocouples, resistance temperature detectors)
  • Flow measurements (pitot tubes, hot-wire anemometers)
  • Spectroscopic techniques (absorption, emission, Raman spectroscopy)
• Molecular beam experiments are used to study gas-phase reactions and intermolecular forces by creating collimated beams of gas particles in a vacuum
• Knudsen cells are used to study the vapor pressure and evaporation rates of materials by effusion through a small orifice
• Gas chromatography is a technique for separating and analyzing mixtures of gases based on their different affinities for a stationary phase and their diffusion rates through a mobile phase