The Maxwell-Boltzmann distribution is the probability distribution that describes the velocities distribution of gas particles in a container. The particles of a gas are identical but distinguishable.
In the 1880s, this theory was developed by James Clerk Maxwell and Ludwig Boltzmann. It is based on the kinetic theory of gases which states that gas molecules are constantly in motion and are colliding with each other at various energies.
The Maxwell-Boltzmann distribution of their energies at a given temperature can be graphed using a fraction of particles on the y-axis against kinetic energy on the x-axis. Moreover, the distribution can be used to calculate various statistical properties of a gas, such as the mean velocity, activation energy, and the effect of catalyst.
In the above graph, activation energy is labeled as Ea. It is the minimum amount of energy required for colliding particles to react. Vrms is the average root mean square velocity of all the particle present in the system.
What is Maxwell Boltzmann’s distribution law?
Maxwell Boltzman’s law states that the continuously colliding particles of an ideal gas are distributed in different fractions or groups, having a certain amount of energy. This statistical distribution of the particles is shown in the figure below:
Three postulates of Maxwell-Boltzman distribution law
- All particles present in the system are distinguished.
- Number particles are associated with each energy state.
- The total number of particles in the entire system is constant.
Factors affecting Maxwell Boltzmann distribution curve
The Boltzmann distribution curve displays the fraction of particles with motions. So, the changes in conditions may alter its representation. The most prominent effect is caused by two factors: temperature and catalyst.
Effect of temperature on Boltzmann distribution curve
The increase in temperature increases the kinetic energy, so the number of collisions raises the probability of successful collisions.
For instance, If we heat the gas to a higher temperature, the peak will shift towards the right due to an increase in kinetic energy. As the graph moves to the right, the height of the curve peak decrease. Since the area under the curve representing the total number of particles in the container is conserved, therefore upon cooling, the graph becomes taller and narrower while on heating it becomes wider and shorter.
In the graph mentioned below, the peak of the curve at T2 is slightly steeper than the T1. Moreover, the shaded area shows the particles with greater energy than the activation energy. Thus, increasing the temperature increases the rate of reaction.
The general graph which represents the curves obtained at various temperatures is given below:
Effect of catalyst
In a chemical reaction, the reactant particle requires energy to get converted to the product. This minimum energy is known as activation energy. So, the particles with energy greater than Ea can react. If a catalyst is added to the system, it provides an alternative pathway to the reactant with lower activation energy. Now the molecules under the shaded regions A and B can react to form a product at a given temperature.
Boltzmann emphasized the discrete nature of the distribution of energy levels. This was later verified by Max Plank’s quantum theory. After that, Maxwell Boltzmann’s distribution was subjected to verification due to its practical importance. Some of the experiments are explained below:
The Stern experiment is the most famous experiment for this purpose. The principle of this experiment is the heating of silver and the emission of atomic silver. The platinum wire is silver-doped is used in it. Upon heating, the emitted silver atoms whose speed has to measured is pass from the slit and the ranges of particle distribution to their respective velocity. However, this experiment has an accuracy of 85%.
Zartman KO experiment
Zartman and KO conduct the second experiment in 1930. The apparatus consists of an oval vessel with a narrow opening. A stream of bismuth atoms is passed through two parallel slits. Bismuth atoms collide with a drum-shaped screen. When the drum is stationary, the particles collide in the same spot, but when it moves quickly, the particles disperse throughout the drum. A microphotometer can be used to measure the thickness of deposited atoms, which is then used to calculate the velocity distribution. Furthermore, it confirms Maxwell Boltzmann’s distribution law with a 1% agreement of uncertainty.
Applications of Boltzmann law
Maxwell’s distribution law has many applications in thermodynamic systems of physics and chemical synthesis. It gives the basis for the authenticity of the kinetic molecular of gases. The Maxwell-Boltzmann velocity distribution function is valid even in non-ideal gases.
Note that, it is perfectly applicable where long-range interactions between the particles of the system are absent or feeble.
Some of its famous applications are mentioned below:
- The average kinetic energy of a system
- Activation energy
- Suitability of the catalyst
- Specific heat of solids and liquids
- Statistical thermodynamics in classical limits
What does the area under a Maxwell-Boltzmann distribution represent?
The area under the curve represents the number of molecules present in the sample. The molecules of the system remain unchanged during changes in temperature.
What is the Maxwell-Boltzmann distribution?
Maxwell-Boltzmann distribution is the law that describes the probability of finding the particles at different temperatures in a system.
What is James Clerk Maxwell most famous for?
James Clark Maxwell is famous for his work on energy distribution and the theory of electromagnetic radiation.
How does the Maxwell-Boltzmann Distribution work?
It works as the population or ratio of particles over the total number of particles. The graph can describe the distribution of molecules at various temperatures. Moreover, The effect of temperature and the addition of a catalyst can be assumed by using the Maxwell-Boltzmann distribution law.
What are the properties of Maxwell Boltzmann’s statistics?
The maxwell Boltzmann statistics gives insight into the distribution of particles at several states. The statistical approach is only applicable at the classical level because here, the particle density is not negligible.
What is the Maxwell-Boltzmann distribution curve?
It is the representation of particle distribution at different temperatures. It is used to derive the average kinetic energy of the particle, root means square value and the effect of the catalyst on activation energy.
What are the limitations of the Maxwell-Boltzmann distribution law?
This law is only applicable when the particle size is within classical limits. It does explain the photoelectric effect and black body radiation.
Why is the Boltzmann distribution important?
It describes the distribution of ideal gas particles at different temperatures in a given state. Moreover, This distribution is further used to calculate the energy of activation and temperature ranges required for a chemical reaction.
What is a good explanation of the Maxwell-Boltzmann distribution?
It is simply explained as a graph plotted between the number of particles at the y-axis against temperature ranges at the x-axis.
- Alternative Derivation of the Maxwell Distribution of Speeds(acs.org)
- Maxwell-Boltzmann Distribution in Solids By Pirooz Mohazzabi, Siva P. Shankar Department of Mathematics and Physics, University of Wisconsin-Parkside, Kenosha, WI, USA