Kinetic Theory of Gases
Maxwell’s Law of Distribution of Velocities
Maxwell’s law describes how the velocities of molecules are distributed in an ideal gas at thermal equilibrium. It states that not all molecules move at the same speed; rather, their speeds are distributed over a wide range following a specific probability distribution curve.
Assumptions:
- The gas consists of a large number of identical molecules moving randomly.
- Molecules are point masses and undergo perfectly elastic collisions.
- No intermolecular forces exist except during collisions.
- The gas is in thermal equilibrium.
Derivation (Conceptual Overview)
The derivation involves statistical methods to determine the probability that a molecule has a certain velocity component (u, v, w) along the x, y, z axes. Assuming the distribution of velocities is isotropic (same in all directions) and independent for each component, and that the energy distribution follows a Boltzmann factor, one arrives at the speed distribution function.
f(v) dv = 4π (m / 2πkT)3/2 v2 e-mv2 / 2kT dv
Where:
- f(v) dv is the fraction of molecules with speeds between v and v+dv.
- m is the mass of a single molecule.
- k is Boltzmann’s constant.
- T is the absolute temperature of the gas.
- e is the base of the natural logarithm.
Important Observations:
- The curve is asymmetric, starting from zero, rising to a maximum (most probable speed), and then decreasing asymptotically towards zero.
- As temperature increases, the peak of the curve shifts to higher speeds, and the curve broadens, indicating a wider range of velocities and higher average speeds.
- Three characteristic speeds are derived from this distribution: Most probable speed (vp), Average speed (vavg), and Root Mean Square speed (vrms).
Experimental Verification
The Maxwell-Boltzmann distribution has been experimentally verified by various methods, most notably by molecular beam experiments. In such experiments (e.g., those by Otto Stern, Zartman, Ko), a beam of molecules is passed through slits and then across a rotating drum or disk. The deflection and deposition pattern of the molecules on the drum/disk, which depends on their speed, matches the distribution predicted by Maxwell's law, confirming its validity.
Mean Free Path
λ = 1 / (√2 π d2 n)
The mean free path (λ) is defined as the average distance a molecule travels between two successive collisions with other molecules in a gas.
Factors affecting Mean Free Path:
- Molecular diameter (d): A larger molecular diameter leads to a shorter mean free path because molecules collide more frequently.
- Number density (n): Higher number density (more molecules per unit volume) results in a shorter mean free path due to increased collision probability.
- Pressure: At constant temperature, higher pressure means higher number density, leading to a shorter mean free path.
- Temperature: At constant pressure, higher temperature means lower number density (gas expands), leading to a longer mean free path.
Transport Phenomena
Transport phenomena in gases refer to the processes involving the transfer of momentum, energy, or mass due to the random motion of molecules across a gradient. These phenomena are explained by the kinetic theory of gases.
Viscosity (Momentum Transport)
η ≈ (1/3) ρ λ <v>
Definition: Viscosity is a measure of a fluid's resistance to flow. In gases, it arises from the transfer of momentum between layers of gas moving at different velocities.
Kinetic Explanation: When gas layers move past each other, faster-moving molecules from a higher-velocity layer collide with slower-moving molecules in an adjacent layer, imparting momentum. Conversely, slower-moving molecules transfer momentum to the faster layer. This net exchange of momentum acts as an internal friction, opposing the relative motion of the layers.
Observation: Unlike liquids, the viscosity of gases generally increases with temperature because molecular speeds (<v>) and mean free path (λ) increase, leading to more effective momentum transfer.
Thermal Conductivity (Energy Transport)
K ≈ (1/3) ρ cv λ <v>
Definition: Thermal conductivity is a measure of a material's ability to conduct heat. In gases, it is the process of energy transfer across a temperature gradient.
Kinetic Explanation: In a gas with a temperature gradient, molecules from a hotter region (higher average kinetic energy) move into a colder region, transferring their excess kinetic energy through collisions. Simultaneously, colder molecules move into the hotter region, absorbing energy. This net transfer of kinetic energy from hot to cold regions constitutes heat flow.
Observation: Similar to viscosity, the thermal conductivity of gases increases with temperature because molecular speeds and collision efficiency increase.
Diffusion (Mass Transport)
D ≈ (1/3) λ <v>
Definition: Diffusion is the net movement of particles from a region of higher concentration to a region of lower concentration, resulting in the uniform distribution of substances.
Kinetic Explanation: When there is a concentration gradient of a particular gas species, molecules will, by their random thermal motion, move from regions where they are more numerous to regions where they are less numerous. Although individual molecular motion is random, there is a net statistical flow down the concentration gradient until equilibrium is reached.
Observation: Diffusion is crucial in many natural and industrial processes, such as the mixing of gases and the transport of nutrients in biological systems.
Theory of Radiation
Black Body Radiation
A black body is an idealized physical body that absorbs all incident electromagnetic radiation, regardless of frequency or angle of incidence. Due to its perfect absorption, it is also the best possible emitter of thermal radiation, emitting energy at every frequency in a characteristic spectrum that depends only on its absolute temperature.
Characteristics:
- It absorbs 100% of the incident radiation.
- It emits radiation uniformly in all directions (isotropic).
- The spectrum of emitted radiation depends solely on its temperature, not on its material composition.
- Examples: A small hole in a cavity behaves as a near-perfect black body, as any radiation entering the hole is highly likely to be absorbed after multiple reflections within the cavity. Stars also approximate black bodies.
Spectral Distribution
The spectral distribution of black body radiation describes how the intensity or energy density of the emitted radiation varies with wavelength or frequency for a given absolute temperature. When plotted as a graph of intensity versus wavelength:
- The curves show a distinct peak, indicating a wavelength at which the emitted intensity is maximum.
- As the temperature of the black body increases, the total energy emitted (area under the curve) increases dramatically.
- The peak of the emission curve shifts towards shorter wavelengths (or higher frequencies) as the temperature increases. This is known as Wien's displacement law.
Classical physics (Rayleigh-Jeans law) failed to accurately describe this observed spectral distribution, especially at short wavelengths, leading to the