Unit 4: Kinetic Theory of Gases

Maxwell-Boltzmann Distribution of Velocities

The Maxwell-Boltzmann distribution law describes the statistical distribution of speeds for molecules in an ideal gas at a state of thermal equilibrium. In an ideal gas, molecules are in constant random motion, exchanging momentum and energy through collisions. Although individual molecular velocities change continuously, the overall distribution of speeds remains constant at a given temperature.

First-order differential form: dN = 4 * pi * N * (m / (2 * pi * k * T))^(3/2) * v^2 * e^(-m * v^2 / (2 * k * T)) * dv

Where:

• dN is the number of molecules with speed between v and v + dv.
• N is the total number of gas molecules in the system.
• m is the mass of a single gas molecule.
• k is the Boltzmann constant (k = R / N_A).
• T is the absolute temperature in Kelvin.
• v is the molecular velocity (speed).

Key Features of the Velocity Distribution Curve

• Zero Limit: The fraction of molecules with zero velocity or extremely high velocities approaches zero.
• The Peak: The peak of the curve corresponds to the most probable speed (v_mp), which is the speed possessed by the maximum number of molecules.
• Asymmetric Profile: The curve is not symmetric; it has a long, tailing end toward the higher-speed side because molecular speeds can theoretically extend to infinity.
• Temperature Dependence: As temperature T increases, the curve flattens and its peak shifts to the right. This indicates that at higher temperatures, a larger fraction of molecules possess higher speeds.

Experimental Verification of Velocity Distribution

The Maxwell-Boltzmann velocity distribution was experimentally validated by several physical setups. The most notable direct verification is Lammert's Experiment, which utilized rotating slotted discs to filter and measure molecular speeds.

Lammert's Slotted Disc Experiment

This experiment physically isolates molecules of specific speeds using mechanical selectors. The experiment comprises:

1. Molecular Source (Oven): An oven containing the test substance (e.g., bismuth or cesium vapor) heated to a high temperature to generate a stream of vaporized molecules.
2. Collimation Slits: Fine slits that filter the escaping molecules into a highly parallel, narrow molecular beam.
3. Rotating Wheels (Discs): Two metallic discs fixed on a shared axle separated by a distance L. Each disc contains narrow radial slots at its edge. Crucially, the slots of the second disc are angularly offset from the slots of the first disc by an angle theta.
4. Detector: A cold plate receiver that collects and counts the molecules that pass successfully through both discs.

Working Principle

When the axle is rotated at a constant angular speed (omega), a molecule passing through a slot on the first disc can only pass through the slot on the second disc if the time it takes to travel the distance L is exactly equal to the time it takes for the second slot to rotate into position.

Time of flight: t = L / v Time of rotation: t = theta / omega Equating both: v = (omega * L) / theta

By adjusting the rotation speed (omega), researchers select and measure the quantity of molecules at distinct velocities (v). The resulting intensity measurements perfectly match the profile predicted by the Maxwell-Boltzmann distribution law.

Mean, RMS, and Most Probable Speeds

Three unique molecular speeds are calculated from the Maxwell-Boltzmann distribution function to represent the statistical averages of gas behavior:

1. Most Probable Speed (v_mp)

The speed corresponding to the peak of the speed distribution curve, representing the speed held by the largest fraction of molecules.

v_mp = sqrt(2 * k * T / m) = sqrt(2 * R * T / M)

2. Mean (Average) Speed (v_mean)

The mathematical average of all individual speeds within the gas sample.

v_mean = sqrt(8 * k * T / (pi * m)) = sqrt(8 * R * T / (pi * M))

3. Root Mean Square Speed (v_rms)

The square root of the average of the squared molecular speeds. This value is directly proportional to the average translational kinetic energy of the gas.

v_rms = sqrt(3 * k * T / m) = sqrt(3 * R * T / M)

Comparison and Ratios

Because these speeds represent different mathematical averages of an asymmetric distribution, their values always follow a specific order:

v_mp < v_mean < v_rms

Their ratio is constant for any ideal gas regardless of temperature and mass:

v_mp : v_mean : v_rms = sqrt(2) : sqrt(8/pi) : sqrt(3) = 1 : 1.128 : 1.224

Degrees of Freedom

The degrees of freedom (f) of a molecule refer to the total number of independent coordinates or motion types (translational, rotational, or vibrational) needed to completely define the dynamic state of that molecule.

Note on Vibrational Degrees: Vibrational modes are generally frozen out at low or moderate room temperatures because their activation energy quantum is high. They only participate significantly at high temperatures.

Law of Equipartition of Energy

The Law of Equipartition of Energy is a fundamental principle of classical statistical mechanics.

For any dynamic system in thermal equilibrium at absolute temperature T, the total kinetic energy of the system is distributed equally among all its independent degrees of freedom, and the average energy associated with each degree of freedom is equal to (1/2) * k * T.

Thus, for a molecule with f degrees of freedom:

Average energy per molecule = (f / 2) * k * T

For one mole of an ideal gas (where N_A is Avogadro's number and R = N_A * k):

Total internal energy (U) = (f / 2) * R * T

Specific Heats of Gases

The molar specific heats at constant volume (Cv) and constant pressure (Cp) of an ideal gas can be determined directly from its degrees of freedom using the Equipartition of Energy law.

Theoretical Derivations

• Molar Specific Heat at Constant Volume (Cv): Defined as the rate of change of internal energy with respect to temperature:

Cv = dU/dT = (f / 2) * R

• Molar Specific Heat at Constant Pressure (Cp): Obtained via Mayer's Relation (Cp - Cv = R):

Cp = Cv + R = (f / 2 + 1) * R

• Ratio of Specific Heats (gamma):

gamma = Cp / Cv = (f + 2) / f = 1 + 2/f

Calculated Values for Different Atomicity

Important Observation: As the complexity of the molecule increases (more atoms, higher f), the value of the specific heat ratio gamma decreases toward 1.

Mean Free Path and Collision Probability

Concept of Mean Free Path

Gas molecules are in constant motion and collide with each other frequently. The distance traveled by a molecule between two consecutive collisions is called a free path. Because these individual free paths vary greatly, we define the Mean Free Path (lambda) as the average distance traveled by a molecule between successive collisions.

lambda = 1 / (sqrt(2) * pi * d^2 * n)

Where:

• d is the collision diameter of the molecule (effective size).
• n is the number density of the gas (number of molecules per unit volume, n = N/V = P / (k * T)).

Using the ideal gas law, lambda can be rewritten as:

lambda = (k * T) / (sqrt(2) * pi * d^2 * P)

Key Dependencies of Mean Free Path

• Pressure (P): Inversely proportional to pressure (lambda proportional to 1/P).
• Temperature (T): Directly proportional to temperature at constant pressure (lambda proportional to T).
• Molecular size (d): Inversely proportional to the square of molecular diameter.

Collision Probability

The probability that a molecule will travel a distance x without undergoing a single collision is given by the statistical survival formula:

P(x) = e^(-x / lambda)

This means the number of molecules that travel a distance x without colliding decreases exponentially with distance.

Transport Phenomena in Ideal Gases

When a gas is not in a uniform state, physical properties vary across its spatial coordinates. The collisions of molecules drive the transport of physical properties to restore equilibrium. These processes are called transport phenomena.

1. Viscosity (Transport of Momentum)

If adjacent parallel layers of gas are moving with different flow velocities, a velocity gradient (dv/dz) is established. Molecules crossing between layers transport momentum, creating internal friction.

Coefficient of Viscosity: eta = (1/3) * rho * lambda * v_mean

Where rho is the mass density of the gas (rho = m * n). Substituting the mean free path:

eta = (1/3) * (m * n) * (1 / (sqrt(2) * pi * d^2 * n)) * v_mean = (m * v_mean) / (3 * sqrt(2) * pi * d^2)

Important Exam Observation: Because the number density terms (n) cancel out, the coefficient of viscosity (eta) is completely independent of gas pressure at a given temperature. It depends only on the square root of the absolute temperature (eta proportional to sqrt(T)).

2. Thermal Conductivity (Transport of Thermal Energy)

If there is a temperature gradient (dT/dz) in the gas, molecules moving from hotter regions transfer higher kinetic energy to colder regions.

Coefficient of Thermal Conductivity: K = (1/3) * rho * lambda * v_mean * c_v

Where c_v is the specific heat capacity at constant volume per unit mass (c_v = Cv / M). Like viscosity, thermal conductivity is independent of pressure.

K = eta * c_v

3. Diffusion (Transport of Mass)

If there is a concentration gradient (dn/dz) of molecules, mass transport occurs. Molecules diffuse from high-concentration areas to low-concentration areas.

Coefficient of Self-Diffusion: D = (1/3) * lambda * v_mean

Since lambda is inversely proportional to density/pressure, the diffusion coefficient is inversely proportional to pressure (D proportional to 1/P). It increases with temperature as T^(3/2).

Summary Table of Transport Coefficients

Common Exam Mistake: Confusing the pressure dependency of Viscosity and Diffusion. Although both contain the mean free path (lambda), the density term in viscosity cancels the density term in the mean free path, making Viscosity independent of pressure. Diffusion (D) has no density factor in the numerator, so it remains dependent on 1/P.

Einstein’s Theory of Translational Brownian Motion

Brownian motion is the continuous, rapid, and highly irregular random motion of microscopic suspended particles (such as pollen grains or smoke particles) in a fluid. In 1905, Albert Einstein formulated a mathematical theory of this phenomenon, providing definitive proof of the molecular-kinetic theory of heat and the discrete atomic nature of matter.

Einstein's Theoretical Approach

Einstein approached the problem by combining two concepts:

1. Diffusion Process: The random motion of suspended particles is mathematically modeled as a macroscopic diffusion process. Under a concentration gradient, they diffuse with a diffusion coefficient D.
2. Viscous Drag: The suspended particles experience a viscous drag force as they move through the fluid, which is modeled by Stokes' Law:

F = 6 * pi * eta * r * v

Equating the osmotic driving force to the viscous resistance, Einstein derived the Einstein-Smoluchowski relation for the diffusion coefficient:

D = (R * T) / (N_A * 6 * pi * eta * r)

Where:

• R is the universal gas constant.
• T is the absolute temperature.
• eta is the viscosity of the fluid.
• r is the radius of the spherical Brownian particle.
• N_A is the Avogadro number.

Mean Square Displacement

Because a Brownian particle moves in a highly jagged, unpredictable path, its net displacement at any instant average is zero. Instead, Einstein measured the average of the square of the displacement along a single axis (x-axis) over a time interval t:

[x^2]_avg = 2 * D * t

Substituting the expression for the diffusion coefficient D:

[x^2]_avg = (R * T * t) / (3 * pi * eta * r * N_A)

Experimental Significance

• Proof of Atomicity: This equation links macroscopic, observable quantities (like particle radius r, viscosity eta, time t, and mean square displacement) to a sub-microscopic constant (Avogadro's number, N_A).
• Jean Perrin's Verification: In 1908, Jean Perrin conducted precise experimental trials on suspensions of gum-mastic and gamboge particles. By measuring the mean square displacement of these particles, he calculated N_A to be approximately 6.02 * 10^23, verifying Einstein's theory and proving the existence of atoms. Perrin was awarded the Nobel Prize in Physics in 1926 for this work.