Unit 5: Kinetic Theory of Gases and Radiation Physics: An Exam-Ready Study Guide

Unit 1: Kinetic Theory of Gases

Derivation of Maxwell’s Law of Distribution of Velocities and its Experimental Verification

The Kinetic Theory of Gases describes the macroscopic properties of gases in terms of the motion of their constituent molecules. A fundamental aspect is understanding how the velocities of gas molecules are distributed.

Maxwell’s Law of Distribution of Velocities: This law describes the statistical distribution of speeds of molecules in an ideal gas at a specific temperature. It states that not all molecules in a gas move at the same speed; instead, there is a range of speeds, and the number of molecules with a certain speed follows a specific probability distribution.

Conceptual Derivation Steps:

1. Consider a large number of gas molecules moving randomly in a container.
2. Assume elastic collisions between molecules and with container walls.
3. Apply statistical mechanics principles, often involving Boltzmann distribution for energy states, to determine the probability of a molecule having a certain velocity component (v_x, v_y, v_z).
4. Combine these probabilities to find the probability of a molecule having a resultant speed v = √(v_x² + v_y² + v_z²).
5. The resulting distribution function, F(v), gives the fraction of molecules having speeds between v and v + dv.

Maxwell-Boltzmann Speed Distribution Function:
d
N_v / N = 4π (m / (2πk_BT))^3/2 v² e^{(-mv2 / 2k_BT)} dv
Where:
N_v = Number of molecules with speeds between v and v + dv
N = Total number of molecules
m = Mass of a single molecule
k_B = Boltzmann constant
T = Absolute temperature
v = Molecular speed

Key Observations from Maxwell's Distribution Curve:

• The curve is asymmetric.
• There is a most probable speed (v_p), corresponding to the peak of the curve, where the maximum number of molecules are found.
• The curve shifts towards higher speeds and flattens out at higher temperatures, indicating a broader distribution of speeds and a higher average speed.
• The area under the curve represents the total number of molecules and remains constant.

Experimental Verification:

One classic method for experimental verification is Stern's Experiment (or similar molecular beam experiments):

• A stream of gas molecules emerges from an oven into an evacuated chamber.
• The molecules pass through a series of slits, forming a narrow beam.
• This beam then strikes a rotating drum or a pair of rotating discs with slits.
• Due to the rotation, molecules with different speeds travel different distances on the drum before hitting it, or pass through the second slit at different angular positions.
• By analyzing the distribution of deposits on the drum or the intensity of molecules passing through the second slit, the speed distribution can be experimentally measured and compared with Maxwell's theoretical prediction.

Mean Free Path

Mean Free Path (λ): It is defined as the average distance a molecule travels between successive collisions with other molecules in a gas.

Factors Affecting Mean Free Path:

• Temperature (T): As temperature increases, molecular speeds increase, leading to more frequent collisions if density remains constant, but if pressure is constant, density decreases, so mean free path increases.
• Pressure (P) / Number Density (n): As pressure increases (or number density increases), molecules are closer together, leading to more frequent collisions and a shorter mean free path.
• Molecular Diameter (σ): Larger molecules have a larger collision cross-section, leading to more frequent collisions and a shorter mean free path.

Formula for Mean Free Path:
λ = 1 / (√2 π n σ²)
or, using the ideal gas law (P = nk_BT), n = P / k_BT:
λ = k_BT / (√2 π P σ²)
Where:
n = Number density of molecules
σ = Molecular diameter
k_B = Boltzmann constant
T = Absolute temperature
P = Pressure

Transport Phenomena: Viscosity, Thermal Conductivity and Diffusion

Transport phenomena refer to the non-equilibrium processes where mass, momentum, or energy are transferred from one region to another within a system due to gradients in concentration, velocity, or temperature, respectively. In gases, these processes are explained by the random motion and collisions of molecules.

Viscosity (Momentum Transport)

Viscosity (η): It is a measure of a fluid's resistance to flow. In gases, viscosity arises from the transfer of momentum between adjacent layers moving at different velocities.

Explanation: Consider two adjacent layers of gas moving parallel to each other at different speeds. Molecules from the faster layer move into the slower layer, carrying higher momentum, thus tending to speed up the slower layer. Conversely, molecules from the slower layer move into the faster layer, carrying lower momentum, thus tending to slow down the faster layer. This continuous exchange of momentum gives rise to an internal friction, which is the viscosity.

Coefficient of Viscosity (η):
η = (1/3) ρ c̄ λ
Where:
ρ = Density of the gas
c̄ = Average molecular speed
λ = Mean free path

Important Observation: For gases, viscosity generally increases with temperature (because c̄ increases) and is largely independent of pressure (as ρ and λ vary inversely with pressure, cancelling out). This is contrary to liquids where viscosity decreases with temperature.

Thermal Conductivity (Energy Transport)

Thermal Conductivity (K): It is a measure of a material's ability to conduct heat. In gases, thermal conductivity arises from the transfer of kinetic energy between molecules due to a temperature gradient.

Explanation: When a temperature gradient exists, molecules from a hotter region (higher average kinetic energy) move to a colder region, transferring their excess kinetic energy to the molecules there. Simultaneously, colder, less energetic molecules move into the hotter region, absorbing energy. This net transfer of kinetic energy from hotter to colder regions is what we perceive as heat conduction.

Coefficient of Thermal Conductivity (K):
K = (1/3) ρ c̄ λ c_V
Where:
ρ = Density of the gas
c̄ = Average molecular speed
λ = Mean free path
c_V = Specific heat capacity at constant volume per unit mass

Important Observation: Similar to viscosity, thermal conductivity of gases generally increases with temperature and is largely independent of pressure.

Diffusion (Mass Transport)

Diffusion (D): It 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.

Explanation: If there is a concentration gradient of a particular type of gas molecule (or component in a mixture of gases), molecules will move randomly. However, due to the higher number of molecules in the region of higher concentration, there will be a net flux of molecules from the higher concentration region to the lower concentration region, leading to diffusion.

Coefficient of Diffusion (D):
D = (1/3) c̄ λ
Where:
c̄ = Average molecular speed
λ = Mean free path

Important Observation: Diffusion coefficient for gases generally increases with temperature (due to c̄) and decreases with pressure (due to λ decreasing with increasing pressure).

Unit 2: Theory of Radiation

Black Body Radiation, Spectral Distribution, Concept of Energy Density

Black Body Radiation

Black Body: An ideal object that absorbs all electromagnetic radiation incident upon it, regardless of frequency or angle of incidence. It also emits radiation at all frequencies, and the spectrum of this emitted radiation depends only on its temperature, not on its composition or surface properties. Such an emitter is known as a black body radiator.

Characteristics of Black Body Radiation:

• It is a thermal radiation (emitted due to temperature).
• The radiation is isotropic (emitted equally in all directions).
• The spectrum of emitted radiation is continuous.
• At any given temperature, there is a peak wavelength (λ_max) at which the emission is most intense.

Spectral Distribution

The spectral distribution of black body radiation refers to how the intensity of emitted radiation varies with wavelength (or frequency) at a specific temperature. Experiments show the following characteristics:

• At a given temperature, the intensity of radiation increases with wavelength, reaches a maximum at a certain wavelength (λ_max), and then decreases for longer wavelengths.
• As the temperature of the black body increases:
  • The total energy radiated (area under the curve) increases significantly.
  • The peak wavelength (λ_max) shifts towards shorter wavelengths (i.e., from red to blue, then ultraviolet). This is described by Wien's Displacement Law.

The classical theories (Rayleigh-Jeans law) failed to explain the full spectral distribution, particularly at shorter wavelengths (the