The Maxwell-Boltzmann Distribution
Claimed by Sai Srinivas
The Main Idea
In the context of the Kinetic Molecular Theory of Gases, a gas has a large number of particles moving around with varying speeds, colliding with each other, causing changes in the speeds and directions of the particles. A good understanding of the properties of a gas requires the knowledge of the distribution of particles speeds. Named after James Clerk Maxwell and Ludwig Boltzmann, the Maxwell-Boltzmann Distribution describes particles speeds in an idealized gas, in which the particles rarely interact with each other except for the brief collisions where energy and momentum are affected. The distribution of a particular gas depends on certain parameters, such as temperature of the system and mass of the gas particles. Certain properties of real gases inhibit their ability to be modeled by the Maxwell-Boltzmann Distribution so this distribution is best suited for application to ideal gases and certain rarefied gases at normal temperatures. Knowledge of particle speeds given by this distribution is important to scientists performing reactions because for a reaction to take place, particles must collide with sufficient energy to induce a transition state. This usually pertains to faster particles, so if the Maxwell-Boltzmann Distribution tells us how many particles have energies or speeds above a certain threshold, this is considered valuable information.
A Mathematical Model
The Maxwell–Boltzmann distribution is the function
- [math]\displaystyle{ f(v) = \sqrt{\left(\frac{m}{2 \pi kT}\right)^3}\, 4\pi v^2 e^{- \frac{mv^2}{2kT}}, }[/math]
where [math]\displaystyle{ m }[/math] is the particle mass and [math]\displaystyle{ kT }[/math] is the product of Boltzmann's constant, [math]\displaystyle{ k }[/math], and thermodynamic temperature, given by [math]\displaystyle{ T }[/math].
The probability that a molecule of a gas has a center-of-mass speed within the range [math]\displaystyle{ v }[/math] to [math]\displaystyle{ v+dv }[/math] is given by [math]\displaystyle{ f(v)dv }[/math].
As has been mentioned, the Maxwell-Boltzmann Distribution details the distribution of particle speeds in an ideal gas, and this distribution can be characterized in a few ways shown below:
A Computational Model
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