# Probability distribution using the Maxwell Boltzmann formula

• Sonor@12
In summary, the kinetic theory of gases explains the distribution of molecules moving along the x direction using the equation Σx=(1/2)mv^2, where m is the mass and vx is the velocity in the x direction. The Boltzmann law, p(x)=e^[(-mv^2)/ekT], gives the distribution of particles over velocities ranging from -∞ to ∞. In order to normalize the distribution, a constant c is needed, which is achieved by converting the integral into a double integral using polar coordinates. The assumption of isotropy and independence between x and y allows for the simplification of the double integral to a single integral, resulting in the normalized probability distribution equation p(v)dv=[(
Sonor@12
According the the kinetic theory of gases, molecules moving along the x direction are given by Σx= (1/2) mv^2, where m = mass and vx is the velocity in the x direction
The distribution of particles over velocities is given by the Boltzmann law p(x)=e^[(-mv^2)/ekT]
where velocities range from -∞ to ∞

Here, the probability distribution p(v), needs a constant, c, that will normalize the distribution so that
c∫e^[(-mv^2)/2kT]dv=1 ( -∞ to ∞ )

The publisher used a trick converting this equation into a double integral from
I=∫e^(-ax^2)dx ( -∞ to ∞ ) where a = m/2kT
to
I^2= ∫e^(-ax^2)dx ∫e^(-ay^2)dy -∞ to ∞
and combined the exponentials from the double integral to get

I^2=∫ ∫e^-a(x+y^)2 dxdy -∞ to ∞

then converted to polar coordinates r and θ since r^2= x^2 + y^2

to simplify to
I^2= ∫rdr ∫e^-ar^2dθ from 0 to 2π

simplifying further to exchange dθ for dr

I^2= ∫dθ ∫e^-ar^2dr from 0 to 2π

and integrated the first integrand to ∫dθ from 0 to 2π = 2π

reducing the double integral to

I^2=2π∫e^-ar^2dr 0 to 2π

and finally used u substition u=-ar^2 & du = -2ardr to leave us with

I^2= (-π/a)∫e^udu from 0 to ∞

==> I^2= (-π/a)e^u integrating from 0 to ∞ gives

I^2=π/a

and I = (π/a)^-1/2

where a= m/2kT gives us

I= [(2πkT)/m]^1/2

So when used as a constant, the normalized probability distribution equation becomes
p(v)dv= [(m)/2kT)^1/2] * ∫e^[(-mv^2)/2kT]dv

My questions is

How are you able to take an Integral with respect to x and make it into a double integral with respect to x and y?

Both sides were squared in that step. So on the right you have the integral times the integral. Remember that the variable you integrate over is a "dummy variable", that means it can be changed to a different character and it doesn't matter. http://mathworld.wolfram.com/DummyVariable.html

By changing the dummy variable from "x" to "y" it is better illustrated how you are going from cartesian to polar coordinates.

Also, you should realize that the assumption was made that p(x,y)=p(x)p(y), which is not always the case. When p(x,y)=p(x)p(y), that means that the probability distribution for x has nothing to do with the value of y. In terms of velocities, it means that if I know the x-velocity is Vx, it tells me nothing about the y-velocity Vy. In probability theory, it says that the correlation between x and y is zero. It is also the assumption of isotropy, that there is no special direction in space. That means that p(x) and p(y) are the same function, different variables. If you pick an x-axis and a y-axis, you get the same result as if you choose two different axes, x' and y'. Those two assumptions are why you can build p(x,y) from p(x) by making it equal to p(x)p(y).

## 1. What is the Maxwell Boltzmann formula?

The Maxwell Boltzmann formula is a mathematical equation used to describe the probability distribution of particles in a gas at a given temperature. It takes into account the mass, velocity, and temperature of individual particles to calculate the likelihood of finding them in a certain energy state.

## 2. How is the Maxwell Boltzmann formula used in science?

The Maxwell Boltzmann formula is used in various fields of science, including physics, chemistry, and engineering. It helps to understand the behavior of gases and the distribution of particles in different systems, such as chemical reactions and thermodynamic processes.

## 3. What is the relationship between temperature and the Maxwell Boltzmann distribution?

The Maxwell Boltzmann distribution shows that as temperature increases, the number of particles with higher energy levels also increases. This means that at higher temperatures, there is a wider range of particle energies, and the distribution curve becomes broader and flatter.

## 4. Can the Maxwell Boltzmann formula be used for all types of particles?

Yes, the Maxwell Boltzmann formula can be used for any type of particle, as long as it obeys the laws of classical mechanics. This includes particles with mass, such as atoms and molecules, and even subatomic particles like protons and electrons.

## 5. How does the Maxwell Boltzmann formula relate to the ideal gas law?

The Maxwell Boltzmann formula is a key component of the ideal gas law, which describes the relationship between pressure, volume, temperature, and the number of moles of an ideal gas. The Maxwell Boltzmann distribution helps to explain the behavior of individual gas particles, while the ideal gas law describes the overall properties of a gas system.

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