Mean free path in Chemistry

It is the average distance an object will go between collisions. The total distance a particle, such as a molecule in a gas, will travel before colliding, known as the free path, cannot be calculated without knowing the route of every particle in the region.

You may use this blog post to learn more about the concept and calculations of the mean free path. If you wish to learn it from a private online tutor, access our online chemistry tutor page for more info or directly contact us.

Significance of mean free path

The energy or direction of the traveling particle, or any other particle property, is altered due to successive collisions. A molecule, an atom, or a photon are all examples of moving particles.

The pencil beam of mono-energetic photons in radiography must have a mean free path. Furthermore, it is the average distance traveled by a photon between collisions with the atoms of the target material. This also implies that the free path is affected by the material and the photons’ energy.

Significance in electronics

In electronics, the electrical mobility of a charge carrier is proportional to the mean free path of electrons in a metal. Additionally, electrical mobility is a metric that is proportional to electrical conductivity.

Its values determine the greatest thickness permissible for a high-resolution Cd1xZnxTe gamma-ray spectrometer. The efficacy of a Cd1xZnxTe detector in blocking incident x-rays and gamma rays is thus another essential material attribute. If the product of a detector’s linear absorption coefficient and thickness is significantly less than unity, the detector’s sensitivity for monitoring incident energetic photons is inadequate.

Butler et al. (1993a) determined the linear absorption coefficient of Cd0.8Zn0.2Te, and the results are presented in Figure 3. The density of Cd0.8Zn0.2Te was calculated to be 5.811 g/cm3 using the lattice parameter values for CdTe and ZnTe (S. M. Johnson et al., 1990). The stopping power of Cd0.8Zn0.2Te for x-rays and gamma-rays in the 0.01-10 MeV range can be estimated using these computed values and the detector thickness. The linear absorption coefficient of a gamma-ray with an energy of 100 keV, for example, is roughly 10 cm^-1, and a detector with a thickness larger than 0.1 cm is preferred.

The unit of the mean free path is a meter (unit of length).

How to calculate mean free path – Examples

1. An oxygen molecule travels in air at 300 K and 1 atm, and the diameter of an oxygen molecule is 1.2 × 10 ^-10 m. Calculate the mean free path of the oxygen molecule.

Solution:

we are provided the temperature T=300K

Pressure =1atm= 1.01×10⁵

Diameter of oxygen molecule d = 1.2×10^-10 m

Boltzmann constant k= 1.38×10^-23

The formula for the mean free path is λ = kT/(√2πd^(2 ) P)

By substituting all the values in the formula, we get the value of the mean free path is = 76.9×10^-13 m

2. V_rel=671 m/s for N₂ molecules at 25℃ and collision frequency z= 7.4×10⁹ s^-1, P=1 atm, under same conditions. Calculate the mean free path of the N₂

Solution:

we are provided the relative velocity of nitrogen molecules v_rel = 671 m/s

Collision frequency of nitrogen molecules z= 7.4×10⁹ s^-1

The formula for mean free path λ = v_rel/Z

λ = (671 m/s)/(7.4×10⁹ s^-1)= 9.1×10^-8 m

Mean free path of Electrons, Gases, Air, and water

A. Electrons

The mean free path of an electron in a gas under normal conditions is 10^-5 cm, and the size of an atom with which an electron collides is 1000 times smaller or 10^-8 cm. Check this research paper for more info on the same.

B. Gases

The circle would sweep out the volume represented in time t, and the number of collisions could be approximated based on the number of gas molecules present. The path length divided by the number of collisions yielded the mean free path.

C. Air

The mean free path is established because one molecule is washed out by a molecular route within the “collision tube.” The criterion is λ= (N/V) r2 ≈1, where r is the molecule’s radius.

D. Water

Water vapor has the same d value as air. Absolute temperature is inversely related to numerical density.

therefore n = 2.7×1025×273373 = 2×1025 cm^-3

The mean free path is roughly 100 times the computed interatomic distance of 40 = 4109m. This immense mean free path value is what causes the characteristic gaseous behavior. Without a container, gases cannot be contained. The bulk observable qualities of viscosity, heat conductivity and diffusion can be connected to microscopic quantities like molecular size using the kinetic theory of gases. The molecular sizes were initially approximated using such relationships.

Scattering mean free path

In a medium, the scattering mean free path is the inverse of the scattering coefficient, the absorption mean free path is the inverse of the absorption coefficient, and the total mean free path is the inverse of the sum of the scattering and absorption coefficients.

Elastic mean free path

Mean free path in chemistry

The mean free path of a particle, such as a molecule, in the kinetic theory of gases, is the average distance the particle travels between collisions with other moving particles.

Equations/formula

Mean free path in physics

The average distance an object will go between collisions is the mean free path. The distance a particle, such as a molecule in a gas, will travel before colliding, known as the free path, cannot be calculated without knowing the route of every particle in the region.

Formula

Consider the movement of a gas molecule within an ideal gas. Furthermore, a typical molecule in an ideal gas will abruptly shift its orientation and speed. This is because it collides with other molecules in the same gas in an elastic manner.

Between impacts, the molecule must travel in a straight line at a constant speed. Most importantly, this is true for all gas molecules.

It’s tough to measure the random mobility of gas molecules. As a result, one must attempt to calculate its mean free path.

The symbol represents a molecule’s average distance between collisions. Furthermore, one can expect to fluctuate inversely with N/V, which is the number of molecules per unit volume or the molecular density.

The mean free path can be represented mathematically in the following way:

λ = 1/√2πd ² (N/V)

λ= represents the mean free path

d= density

N= number of molecules

V= volume of the gas

Derivation

Certain assumptions will be used in the equation’s derivation. Assume that the molecule is spherical. Furthermore, a collision occurs when one molecule collides with another. Furthermore, the focus is on the moving molecule while the others remain stationary.

Assume that a single molecule has a diameter of d. Consider the single molecule’s movement through the gas.

As a result, a short cylinder will be swept out. Furthermore, this short cylinder’s cross-section area is πd2.

It will move a distance of vt for time t between successive collisions. The molecule’s velocity is represented by v. Most notably, sweeping this cylinder would result in a volume of πd2*vt.

As a result, the number of collisions a molecule may have is determined by the number of point molecules contained within this container.

Since N/V is the number of molecules per unit volume, the number of molecules in the cylinder will be N/V multiplied by the volume of the cylinder i.e.

Πd²

vt, the mean free path can be derived as follows,

λ = length of the path during time t/number of Collison in time t

λ ≈ vt/πd²vt(N/V)

We have approximated the equation because we assumed that all particles are stationary concerning the particle we are studying; however, all molecules move relative to each other. We have canceled two velocities in the above equation. Still, the v in the numerator is the average velocity, and the v in the denominator is relative velocity, so they both differ by a factor of √2.

Therefore, the final equation will be

Λ= 1/√2πd²(N/V)

The mean free path factors are as follows:

• Density
• Molecular radius
• Quantity of molecules
• Temperature, pressure, and other factors

Density: As the density of the molecules grows, they become closer to one another, increasing the number of collisions and decreasing the mean free path.

The number of molecules: As the number of molecules grows, the likelihood of collision grows, and the mean free path shrinks.

The radius of the molecule: As the molecule’s radius grows, the space between the molecules shrinks, increasing the number of collisions and lowering the mean free path.

The gas density is affected by pressure, temperature, and other physical parameters, which affect the mean free path.

To get a 1:1 private tutor who can help you understand the intricacies of chemistry, visit our Online Chemistry Tutor page or contact us on WhatsApp.