# How is the Maxwell-Boltzmann Distribution Derived?

• Heisenberg2001
In summary, the OP is trying to find the contribution of one particle to the partition function. There should be a factor of ##V## and the exponential should have a minus sign.
Heisenberg2001
Homework Statement
Consider a monoatomic gas, whose momentum relation is ##\vec{p}=m\vec{v}##, where ##\vec{v}## is the three-dimensional velocity of the gas particles.

1. Show that the classical partition function can be written as $$z=\frac{4\pi m^3}{h^3}\int dv v^2e^{\beta \frac{mv^2}{2}}.$$

2. Let ##f(v)=Av^2e^{\beta \frac{mv^2}{2}}.## Find ##A## such that ##\int_{0}^{\infty}dv f(v)=1.##
Relevant Equations
##\vec{p}=m\vec{v}\, , \,z=\frac{4\pi m^3}{h^3}\int dv v^2e^{\beta \frac{mv^2}{2}}##

##f(v)=Av^2e^{\beta \frac{mv^2}{2}}\, , \,\int_{0}^{\infty}dv f(v)=1##
1.

##\vec{p}=m\vec{v}##

##H=\frac{\vec{p}^2}{2m}+V=\frac{1}{2}m\vec{v}^2##

##z=\frac{1}{(2\pi \hbar)^3}\int d^3\vec{q}d^3\vec{p}e^{-\beta H(\vec{p},\vec{q})}##

##z=\frac{Vm^3}{(2\pi \hbar)^3}\int d^3 \vec{v}e^{-\beta \frac{mv^2}{2}}##

##z=\frac{Vm^3}{(2\pi \frac{h}{2\pi})^3}\int d^3 \vec{v}e^{-\beta \frac{mv^2}{2}}##

##z=\frac{Vm^3}{h^3}\int d^3 \vec{v}e^{-\beta \frac{mv^2}{2}}##

I am lost at this point of the solution. Can someone assist me with this?

2.

##\int_{0}^{\infty}dv f(v)=1##

##\int_{0}^{\infty}Av^2e^{\beta \frac{mv^2}{2}}dv=1##

##A\int_{0}^{\infty}v^2e^{\beta \frac{mv^2}{2}}dv=1##

##A\frac{1}{4\left(\frac{\beta m}{2} \right)}\sqrt{\frac{\pi}{\frac{\beta m}{2}}}=1##

##A=\sqrt{\frac{4\beta ^3m^3}{2\pi}}##

##A=4\pi \sqrt {\left(\frac{\beta m}{2\pi} \right)^3}##

##A=4\pi \left( \frac{m}{2\pi k_BT} \right)^\frac{3}{2}##

Last edited by a moderator:
Heisenberg2001 said:
I am lost at this point of the solution. Can someone assist me with this?
Change to spherical coordinates and integrate the angles out.

vela said:
Change to spherical coordinates and integrate the angles out.
There would still be a minus sign left in the exponential and a volume term V outside.

gurbir_s said:
There would still be a minus sign left in the exponential and a volume term V outside.
It appears ##z## is the contribution from just one particle to the partition function, and there should be a factor of ##V##. Also, the exponential should have a minus sign. It could simply be typos by the OP.

vela said:
It appears ##z## is the contribution from just one particle to the partition function, and there should be a factor of ##V##. Also, the exponential should have a minus sign. It could simply be typos by the OP.
Yes, ##z## should be the contribution from one particle only as the integral is over a 6d phase space.

Also note that a - is missing in the exponential ;-).

vanhees71 said:
Also note that a - is missing in the exponential ;-).
How do I edit the equations that I have typed? I can't see an edit button.

## What is the Maxwell-Boltzmann distribution?

The Maxwell-Boltzmann distribution is a statistical distribution that describes the distribution of speeds (or energies) of particles in a gas. It is derived from the principles of classical mechanics and statistical mechanics, and it applies to ideal gases where particles do not interact with each other except through elastic collisions.

## How is the Maxwell-Boltzmann distribution derived?

The Maxwell-Boltzmann distribution is derived by considering the number of ways particles can be distributed among various energy states, subject to the constraints of energy conservation and the indistinguishability of particles. This involves using the principles of statistical mechanics, particularly the Boltzmann factor, which relates the probability of a particle being in a particular energy state to the temperature of the system and the energy of that state.

## What is the significance of the Maxwell-Boltzmann distribution in physics?

The Maxwell-Boltzmann distribution is significant because it provides a fundamental description of the behavior of particles in an ideal gas. It helps in understanding various properties of gases, such as pressure, temperature, and viscosity. The distribution also plays a crucial role in explaining phenomena like the spread of molecular speeds and the rates of chemical reactions.

## How does temperature affect the Maxwell-Boltzmann distribution?

Temperature has a direct impact on the Maxwell-Boltzmann distribution. As the temperature increases, the peak of the distribution curve shifts to higher speeds, and the curve becomes broader. This indicates that at higher temperatures, particles have higher average speeds and a wider range of speeds. Conversely, at lower temperatures, the peak shifts to lower speeds, and the distribution becomes narrower.

## What are the key characteristics of the Maxwell-Boltzmann distribution curve?

The Maxwell-Boltzmann distribution curve has several key characteristics: it is asymmetric, with a peak that represents the most probable speed of particles; it has a long tail extending towards higher speeds, indicating that there are particles with very high speeds, though fewer in number; and the area under the curve is normalized to unity, reflecting the total probability of all possible speeds. The shape and position of the curve depend on the temperature and the mass of the particles.

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