# How to derivate Maxwell Boltzmann Distribution

• Troller
In summary, the Maxwell Boltzmann Distribution, represented by ψ(v)=(m/2*pi*kT)^(3/2) e^(-mv^2/2kT), can be derived in various ways, with some methods being more insightful than others. While a background in Statistical Mechanics or Statistics may be helpful, there are online resources available for derivation. However, it would be beneficial for the questioner to specify what aspects of previous derivations they were unsatisfied with to avoid receiving repetitive responses.
Troller
Hi PFers. I am interested in Maxwell Boltzmann Distribution. I have searched in Internet for the derivation but I am not satified with them. Can somebody show me the way to derivate it? Thanks.
ψ(v)=(m/2*pi*kT)^(3/2) e^(-mv^2/2kT)

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There are many ways to derive it - some more insightful than others.

Have you taken/taking stat mech?

I haven't taken Statistical Mechanics but Statistics. Could it be useful for this?

Troller said:
I have searched in Internet for the derivation but I am not satified with them.

It would help if you tell us why you are not satisfied. Then people won't waste time pointing you to the same things again.

Thanks. Yeap, here.

1st http://www.eecis.udel.edu/~breech/physics/physics-notes/node32.html

Sorry I haven't used LaTeX well enough to type it here again. At derivation of ψ(v)=f(vx)f(vy)f(vz) by taking derivatives with respect to vx, it is very unclear to me. I think dψ/dvx = dψ/dv * dv/dvx. And after that I don't know how to solve it.

2nd http://www.maxwellsociety.net/PhysicsCorner/Miscellaneous Topics/MaxwellBoltzmann.html

This seems better and easier. At the part argumenting for spherical shell, there are a lot of values of vx/y/z so that give the same value of v!? And the application of function g(v)=$\alpha$v^n * e^(-$\beta$v^2) is a bit evidencelessly.

## 1. What is the Maxwell Boltzmann Distribution?

The Maxwell Boltzmann Distribution is a probability distribution that describes the distribution of speeds of particles in a gas at a given temperature. It is named after James Clerk Maxwell and Ludwig Boltzmann, who independently developed the concept in the 19th century.

## 2. How is the Maxwell Boltzmann Distribution derived?

The Maxwell Boltzmann Distribution is derived from the kinetic theory of gases, which assumes that gas particles move randomly and independently. It also considers the collisions between particles and the walls of the container to be perfectly elastic. Using these assumptions, a mathematical expression for the probability of a particle having a certain speed at a given temperature can be derived.

## 3. What is the formula for the Maxwell Boltzmann Distribution?

The formula for the Maxwell Boltzmann Distribution is: f(v) = 4π (m/2πkT)^3/2 * v^2 * e^(-mv^2/2kT), where f(v) is the probability density function, m is the mass of the particle, v is the particle's speed, k is the Boltzmann constant, and T is the temperature in Kelvin.

## 4. What does the Maxwell Boltzmann Distribution tell us about gas particles?

The Maxwell Boltzmann Distribution tells us about the distribution of speeds of gas particles at a given temperature. It shows that most particles have speeds close to the average speed, and the number of particles decreases as the speed increases. It also shows that there is a maximum speed at which particles can move, known as the most probable speed, and it is directly proportional to the square root of the temperature.

## 5. How is the Maxwell Boltzmann Distribution used in real-world applications?

The Maxwell Boltzmann Distribution is used in various fields such as physics, chemistry, and engineering to understand the behavior of gas particles at different temperatures. It is also used in statistical mechanics to calculate thermodynamic properties of gases, such as pressure and internal energy. Additionally, it is used in the design and optimization of gas-based technologies, such as gas turbines and combustion engines.

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