Characteristic Speeds - 3D gas

When discussing gas molecules, we are often concerned with the magnitude of their motion, not the specific direction each molecule is moving in. Since velocity includes direction and gas molecules are moving in random directions, focusing on some characteristic speeds gives a clearer picture in the context of kinetic theory, where we usually deal with statistical averages of how fast particles are moving rather than the exact directions of their movement.

The Maxwell-Boltzmann distribution allows us to derive several important characteristic speeds in a gas. These include:

  • Most Probable Speed (\(v_{mp}\))
  • Average Speed (\(v_{avg}\))
  • Root-Mean-Squared Speed (\(v_{rms}\))

1. Most Probable Speed (\(v_{mp}\))

The most probable speed is the speed at which the greatest number of particles in a gas are moving. It corresponds to the maximum of the Maxwell-Boltzmann distribution. To find it, we set the derivative of the distribution function \( f(v) \) with respect to \( v \) equal to zero and solve for \( v \).

Derivation:

Starting with the Maxwell-Boltzmann distribution function:

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

We take the derivative of \( f(v) \) with respect to \( v \):

\[ \frac{df(v)}{dv} = 4\pi \left( \frac{m}{2\pi k_B T} \right)^{3/2} \left( 2v \exp\left( -\frac{m v^2}{2k_B T} \right) + v^2 \cdot \frac{-mv}{k_B T} \exp\left( -\frac{m v^2}{2k_B T} \right) \right) \]

Set this derivative equal to zero to find the most probable speed:

\[ v_{mp} = \sqrt{\frac{2k_B T}{m}} \]

This is the speed where the largest number of particles are found in the system.

 

2. Average Speed (\(v_{avg}\))

The average speed is calculated by integrating the product of speed \( v \) and the probability distribution function \( f(v) \), which gives a weighted average of the speed.

Derivation:

To find the average speed, we evaluate the following integral:

\[ v_{avg} = \int_0^\infty v f(v) dv \]

Substituting the Maxwell-Boltzmann distribution into the equation:

\[ v_{avg} = 4\pi \left( \frac{m}{2\pi k_B T} \right)^{3/2} \int_0^\infty v^3 \exp\left( -\frac{m v^2}{2k_B T} \right) dv \]

Using the standard result for Gaussian integrals:

\[ \int_0^\infty x^3 \exp(-ax^2) dx = \frac{1}{2a^2} \]
we find:

\[ v_{avg} = \sqrt{\frac{8k_B T}{\pi m}} \]

This is the average speed of particles in the system, and it is slightly higher than the most probable speed due to the skewness of the distribution.

 

3. Root-Mean-Squared Speed (\(v_{rms}\))

The root-mean-squared speed represents the square root of the mean of the squares of all particle speeds. It is a measure of the "typical" speed of particles in the gas.

Derivation:

To find \( v_{rms} \), we calculate the following integral:

\[ v_{rms} = \sqrt{\int_0^\infty v^2 f(v)} \]

Substituting the Maxwell-Boltzmann distribution:

\[ v_{rms} = \sqrt{4\pi \left( \frac{m}{2\pi k_B T} \right)^{3/2} \int_0^\infty v^4 \exp\left( -\frac{m v^2}{2k_B T} \right) dv} \]

Using the standard result for Gaussian integrals:

\[ \int_0^\infty x^4 \exp(-ax^2) dx = \frac{3}{4a^3} \]
the root-mean-squared speed is:

\[ v_{rms} = \sqrt{\frac{3k_B T}{m}} \]

The root-mean-squared speed is higher than both the most probable speed and the average speed due to the emphasis on higher velocities in the calculation.

 

4. Mean Relative Speed

The **mean relative speed** is a key concept in the kinetic theory of gases. It is, important when estimating the frequency of collisions between particles, which determine reaction rates, diffusion, viscosity, and thermal conductivity.

In a system of gas molecules moving in different directions and at various speeds, the relative velocity of any two particles is the vector sum of their individual vector velocities.

\[ \vec{v}_{rel} = \vec{v}_1 - \vec{v}_2 \]

The magnitude of the relative velocity (the relative speed) is the scalar product of \( \mathbf{v_{rel}} \) with itself:

\[ v_{rel} = \sqrt{\mathbf{v_{rel}} \cdot \mathbf{v_{rel}}} \]

Substituting the definition of \( \mathbf{v_{rel}} \):

\[ v_{rel} = \sqrt{(\mathbf{v_1} - \mathbf{v_2}) \cdot (\mathbf{v_1} - \mathbf{v_2})} \]

Expanding the dot product:

\[ v_{rel} = \sqrt{\mathbf{v_1} \cdot \mathbf{v_1} - 2 \mathbf{v_1} \cdot \mathbf{v_2} + \mathbf{v_2} \cdot \mathbf{v_2}} \]

Since the velocities \( \mathbf{v_1} \) and \( \mathbf{v_2} \) are random and uncorrelated, the cross-term \( \mathbf{v_1} \cdot \mathbf{v_2} \) averages to zero:

\[ v_{rel} = \sqrt{\mathbf{v_1} \cdot \mathbf{v_1} + \mathbf{v_2} \cdot \mathbf{v_2}} \]

This reduces to:

\[ v_{rel} = \sqrt{v_1^2 + v_2^2} \]

In a gas, the average speed \( \langle v_1 \rangle \) is equal to the average speed \( \langle v_2 \rangle \), as all particles are identical and subject to the same Maxwell-Boltzmann distribution. Therefore, we write:

\[ \langle v_1^2 \rangle = \langle v_2^2 \rangle = v_{avg}^2 \]

Now, the average relative speed \( \langle v_{rel} \rangle \) becomes:

\[ \langle v_{rel} \rangle = \sqrt{2} \cdot v_{avg} \]

The relative motion between molecules determines how frequently they collide, and the result \( \langle v_{rel} \rangle = \sqrt{2} v_{avg} \) reflects the fact that molecules approach each other at a combined speed greater than their individual speeds.