Kinetic Theory of Gases – Speed Distributions

The speed distribution of gas molecules provides information about how likely it is to find particles with specific speeds. This is derived from the Maxwell-Boltzmann distribution, which is a fundamental result of the kinetic theory of gases and statistical mechanics.

Boltzmann Distribution

The Boltzmann distribution describes the probability that a system occupies a state with energy \(E_i\) at a given temperature \(T\). For a system of particles, it is assumed that the particles are ideally distributed and the probability \(P_i\) that the system is in state \(i\) with energy \(E_i\) is:

\[ P_i = \frac{g_i \exp\left( -\frac{E_i}{k_B T} \right)}{q} \]

where:

  • \(P_i\) is the probability of the system being in state \(i\),
  • \(g_i\) is the degeneracy of the state (the number of ways that state can occur),
  • \(E_i\) is the energy of the state,
  • \(k_B\) is Boltzmann’s constant, and
  • \(Z\) is the partition function, which normalizes the distribution:

 

\[ q = \sum_i g_i \exp\left( -\frac{E_i}{k_B T} \right) \]

This distribution shows that states with lower energy are more likely to be occupied than those with higher energy. However, at higher temperatures, more energetic states become accessible.

 

Maxwell-Boltzmann Speed Distribution – 1D Case

In kinetic gas theory, we are interested in the distribution of molecular speeds. The speed distribution is related to the Boltzmann distribution, where the energy of a molecule is given by its kinetic energy:

\[ E_k = \frac{1}{2} m v^2 \]

Number of Particles with Specific Velocity

The number of particles \(N_i\) with velocity \(v_x\) in a gas is proportional to the Boltzmann factor. This factor reflects how likely it is to find a particle with a given energy:

\[ N_i = N \cdot \exp\left( -\frac{E_i}{k_B T} \right) = N \cdot \exp\left( -\frac{mv_x^2}{2k_B T} \right) \]

where:

  • \(N_i\) is the number of particles with velocity \(v_x\),
  • \(N\) is the total number of particles,
  • \(E_i = E_k = \frac{mv_x^2}{2}\) is the kinetic energy of the particles, and
  • \(k_B T\) is the thermal energy at temperature \(T\).

This shows that the number of particles with a certain velocity decreases exponentially with increasing velocity because higher velocity means higher energy, and higher energy states are less probable.

Connecting Distribution to Probability

We can express the probability that a particle has a velocity \(v_x\) by dividing the number of particles in that velocity range by the total number of particles. This is given by the **probability density function**:

\[ f(v_x) dv_x = \frac{dN}{N} = A \cdot \exp\left( -\frac{mv_x^2}{2k_B T} \right) dv_x \]

Here, \(A\) is the normalization factor that ensures the total probability across all velocities is 1. The probability that a particle has a velocity between \(v_x\) and \(v_x + dv_x\) is given by \(f(v_x) dv_x\).

Normalization of the Distribution

To ensure the distribution is normalized, we must have:

\[ \int_{-\infty}^{\infty} f(v_x) \, dv_x = 1 \]

This leads to the condition:

\[ A \int_{-\infty}^{\infty} \exp\left( -\frac{mv_x^2}{2k_B T} \right) dv_x = 1 \]

We solve for \(A\) by evaluating the Gaussian integral (e.g. integration by substitution and check a table integral):

  • Define the substitution:

\[ z = \sqrt{\frac{m}{2k_B T}} v_x \]

  • The differential becomes:

\[ dz = \sqrt{\frac{m}{2k_B T}} \, dv_x \]

Rewrite the distribution and perform the integration over the new variable \(z\), leading to the normalized result:

\[ A \cdot \sqrt{\frac{2\pi k_B T}{m}} = 1 \]

Thus, the normalization factor is:

\[ A = \sqrt{\frac{m}{2\pi k_B T}} \]

This ensures that the total probability is 1, meaning all possible velocities are considered.

The Maxwell-Boltzmann speed distribution in 1D gives the probability that a particle has a particular speed \(v_x\). The distribution is Gaussian in nature, meaning that most particles will have speed close to a given average speed, with fewer particles having very high or very low speed.

 

Maxwell-Boltzmann Speed Distribution – Transition to 2D Case

Now, we proceed with the 2D case, but first, we need to consider how the velocity components in the x and y directions contribute to the overall speed.

In 2D, the velocity components are \(v_x\) and \(v_y\), and the total speed \(v\) is related to these components by:

\[ v^2 = v_x^2 + v_y^2 \]

This is the velocity vector's magnitude in 2D, which can be broken down into its components along the x and y axes.

The velocity distribution function \(f(v_x, v_y)\) describes the probability that a particle has velocity components \(v_x\) and \(v_y\). In 2D, the distribution function can be separated as:

\[ f(v_x, v_y) dv_x dv_y = f(v) dv_x f(v_y) dv_y \]

This indicates that the x and y components of velocity are independent of each other, so the probability of a particle having specific values of \(v_x\) and \(v_y\) can be written as a product of the probabilities for each direction.

Coordinate Transformation from Cartesian to Polar Coordinates

To simplify the description of the speed in 2D, we perform a coordinate transformation from Cartesian coordinates \([v_x, v_y]\) to polar coordinates \([v, \theta]\), where \(v\) is the speed and \(\theta\) is the angle:

\[ v_x = v \cos(\theta), \quad v_y = v \sin(\theta) \]

This transformation allows us to express the velocity distribution in terms of the magnitude of the velocity (speed) \(v\) and the direction \(\theta\), which is often more convenient when considering the overall speed of the particles.

We start by transforming from Cartesian coordinates \( (v_x, v_y) \) to polar coordinates \( (v, \theta) \), where \( v \) is the speed and \( \theta \) is the angle. The total differentials for \( v_x \) and \( v_y \) are:

\[ dv_x = \left( \frac{\partial v_x}{\partial v} \right)_\theta dv + \left( \frac{\partial v_x}{\partial \theta} \right)_v d\theta \]

\[ dv_y = \left( \frac{\partial v_y}{\partial v} \right)_\theta dv + \left( \frac{\partial v_y}{\partial \theta} \right)_v d\theta \]

We know that:

\[ v_x = v \cos(\theta), \quad v_y = v \sin(\theta) \]

We now perform substitutions for the derivatives of \( v_x \) and \( v_y \) with respect to \( v \) and \( \theta \):

  • \[ \frac{\partial v_x}{\partial v} = \cos(\theta), \quad \frac{\partial v_x}{\partial \theta} = -v \sin(\theta) \]
  • \[ \frac{\partial v_y}{\partial v} = \sin(\theta), \quad \frac{\partial v_y}{\partial \theta} = v \cos(\theta) \]

These describe how the velocity components \( v_x \) and \( v_y \) change in the new polar coordinates.

Jacobian Determinant

Next, we calculate the Jacobian determinant to account for the transformation of area elements from Cartesian to polar coordinates. The Jacobian is given by:

\[ \begin{bmatrix} dv_x \\ dv_y \end{bmatrix} = \begin{bmatrix} \cos(\theta) & -v \sin(\theta) \\ \sin(\theta) & v \cos(\theta) \end{bmatrix} \begin{bmatrix} dv \\ d\theta \end{bmatrix} \]

The determinant of the Jacobian matrix is:

\[ J = \begin{vmatrix} \cos(\theta) & -v \sin(\theta) \\ \sin(\theta) & v \cos(\theta) \end{vmatrix} = v \cos^2(\theta) + v \sin^2(\theta) = v \]

Thus, the Jacobian determinant is \( J = v \).

Transformed Area Element

Finally, the differential area element \( dv_x dv_y \) transforms to polar coordinates as:

\[ dv_x dv_y = J dv d\theta = v dv d\theta \]

This transformation introduces an additional factor of \(v\), which accounts for the number of ways a particle can have a given speed in two dimensions. The probability density function for speed in 2D is then:

\[ f(v) = \frac{m}{k_B T} v \exp\left( -\frac{m v^2}{2k_B T} \right) \]

Deriving the 2D speed distribution formula

In the 2D case, the speed distribution is calculated by separating the velocity components \( v_x \) and \( v_y \). Since all directions are equivalent, we use the following relationship:

\[ f(v) dv = f(v_x) \cdot f(v_y) \cdot v dv d\theta \]

The total speed is given by:

\[ v^2 = v_x^2 + v_y^2 \]

We now substitute the expressions for \( f(v_x) \) and \( f(v_y) \) into the equation. Using the Maxwell-Boltzmann distribution, we have:

\[ f(v) dv = \frac{m}{2\pi k_B T} \cdot \exp\left( -\frac{m(v_x^2 + v_y^2)}{2k_B T} \right) v dv d\theta \]

This simplifies to:

\[ f(v) dv = \frac{m}{2\pi k_B T} \cdot v \cdot \exp\left( -\frac{m v^2}{2k_B T} \right) dv d\theta \]

Next, we integrate over all angles \( \theta \) from 0 to \( 2\pi \), since all directions are equivalent:

\[ f(v) dv = \left( \int_0^{2\pi} d\theta \right) \frac{m}{2\pi k_B T} \cdot v \cdot \exp\left( -\frac{m v^2}{2k_B T} \right) dv \]

The integral over \( \theta \) gives \( 2\pi \), so the final expression for the speed distribution in 2D is:

\[ f(v) = \frac{m}{k_B T} \cdot v \cdot \exp\left( -\frac{m v^2}{2k_B T} \right) \]

This is the 2D Maxwell-Boltzmann speed distribution, which shows how the probability of finding a particle with speed \( v \) depends on the particle’s mass and temperature.

 

Maxwell-Boltzmann Speed Distribution – 3D Case

In three dimensions, the velocity components in the x, y, and z directions are independent. The total speed is related to the velocity components by:

\[ v^2 = v_x^2 + v_y^2 + v_z^2 \]

To describe the distribution of speeds in 3D, we must transition from Cartesian coordinates to spherical coordinates. The transformation from Cartesian to spherical coordinates is given by:

\[ v_x = v \sin(\theta) \cos(\phi), \quad v_y = v \sin(\theta) \sin(\phi), \quad v_z = v \cos(\theta) \]

The velocity distribution function \( f(v_x, v_y, v_z) \) is given as the product of the distributions of the individual components:

\[ f(v_x, v_y, v_z) dv_x dv_y dv_z = f(v) dv f(\theta) d\theta f(\phi) d\phi \]

This allows us to separate the distribution into independent parts for each velocity component.

To express the distribution in terms of speed \( v \), we again perform a coordinate transformation from Cartesian coordinates \( (v_x, v_y, v_z) \) to spherical coordinates \( (v, \theta, \phi) \), where:

\[ dv_x dv_y dv_z = v^2 \sin(\theta) dv d\theta d\phi \]

Thus, the differential volume element in spherical coordinates is:

\[ dV = v^2 \sin(\theta) \, dv \, d\theta \, d\phi \]

The factor \(v^2\) accounts for the number of ways a particle can achieve a given speed in three dimensions.

This transformation introduces a Jacobian factor of \( v^2 \sin(\theta) \), and the distribution function becomes:

\[ f(v) dv = \int_0^\infty \int_0^\pi \int_0^{2\pi} \left( \frac{m}{2\pi k_B T} \right)^{3/2} v^2 \sin(\theta) \exp\left( -\frac{m v^2}{2k_B T} \right) dv d\theta d\phi \]

Integration Over Angles

We first integrate over the angles \( \theta \) and \( \phi \). For \( \phi \), the integral is straightforward:

\[ \int_0^{2\pi} d\phi = 2\pi \]

For \( \theta \), we use the integral of \( \sin(\theta) \):

\[ \int_0^\pi \sin(\theta) d\theta = 2 \]

Thus, the final expression for the Maxwell-Boltzmann speed distribution in 3D is:

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

This describes the probability of finding a particle with speed \( v \) in three dimensions, depending on the temperature \( T \) and the particle mass \( m \).

 

Interpretation of the Maxwell-Boltzmann Distributions (3D Case)

The Maxwell-Boltzmann speed distribution in 3D describes the probability of finding a gas molecule with a specific speed at a given temperature. The general form of the distribution is:

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

This formula shows that the distribution depends on the mass \(m\) of the gas molecules and the temperature \(T\) of the system. The term \(v^2\) reflects the increasing number of possible velocities with higher speed, while the exponential term accounts for the decreasing probability of finding very high speeds.

Graphs of the Distribution

The plots in the figure illustrate the behavior of the Maxwell-Boltzmann distribution for different temperatures and molecular masses:

  • Left Graph: The distribution is plotted for three different temperatures \( T_1 < T_2 < T_3 \). As the temperature increases, the peak of the distribution shifts to higher speeds, and the curve broadens, indicating a wider range of molecular speeds.
  • Right Graph: The distribution is plotted for molecules with four different masses \( m_1 < m_2 < m_3 < m_4 \). Lighter molecules (smaller \(m\)) have higher peak speeds, while heavier molecules (larger \(m\)) have lower speeds and a narrower distribution.

Understanding the Behavior

  • Gaussian function → decaying exponential: The distribution follows a Gaussian shape, where the exponential decay (\( e^{-x^2} \)) ensures that only a small fraction of molecules have very high speeds.
  • For large \( m \), the factor
    \[ \exp\left( -\frac{mv^2}{2k_B T} \right) \]
    decays quickly to zero. This implies that heavier molecules are less likely to have high speeds.
  • For large \( T \), the factor
    \[ \exp\left( -\frac{mv^2}{2k_B T} \right) \]
    decays more slowly. This means more molecules are likely to have high speeds as the temperature increases.

 

Verification of the Maxwell-Boltzmann Distribution Law – The Miller-Kusch Experiment

The Miller-Kusch experiment was designed to measure and verify the velocity distributions of gas molecules, providing experimental support for the Maxwell-Boltzmann distribution. The setup consists of an oven that emits gas molecules, a rotating slit, and a velocity detector.

Experimental Setup

1. Oven: Gas molecules are emitted from the oven and pass through a slit in a spinning disk.

2. Spinning Slit: Only particles that pass through the slit at a specific angle \( \theta \) can reach the velocity detector.

3. Velocity Detector: The velocity \( v \) of the particles is measured after passing through the slit.

Velocity Determination

The velocity of the particles is related to the distance traveled \( x \) and the angular velocity \( \omega \) of the spinning slit. The time \( t \) taken for a particle to reach the detector is:

\[ v = \frac{x}{t}, \quad t = \frac{\theta}{\omega} \]

Therefore, the velocity of the particles is given by:

\[ v = \frac{x \cdot \omega}{\theta} \]

where:

  • \( v \) is the velocity of the particles,
  • \( x \) is the distance traveled by the particles,
  • \( \omega \) is the angular velocity of the spinning slit, and
  • \( \theta \) is the angular position of the slit at the time the particles pass through.

 

Apparatus Diagram

The diagram on the right side shows a detailed schematic of the apparatus used in the experiment, including the oven, spinning disk, and velocity detector.

Phys. Rev., 99 (1955) 1314