# Maxwell-boltzman distrubution

• E92M3

#### E92M3

If I have two particles that follows the maxwell velocity distribution:
$$\phi(v_i)dv_i=4 \pi v_i^2 \left ( \frac{m_iv_i}{2\pi kT} \right ) ^{3/2}e^{\frac{-m_iv_i^2}{2kT}}dv_i$$
Why is their combined distribution:
$$\phi(v)dv=4 \pi v^2 \left ( \frac{\mu v}{2\pi kT} \right ) ^{3/2}e^{\frac{-\mu v^2}{2kT}}dv$$
where mu is the reduced mass and v=v2-v1
I have these questions because I don't quite follow these derivations.
http://dissertations.ub.rug.nl/FILES/faculties/science/2007/a.matic/c2.pdf [Broken]
http://www.astro.psu.edu/users/rbc/a534/lec11.pdf
Namely, I not sure why the following holds:
$$\int_0^\infty \int_0^\infty \phi(v_1) \phi(v_2) v_1 v_2 \sigma dv_1 dv_2 = 4\pi \left ( \frac{\mu v}{2\pi kT} \right ) ^{3/2} \int_0^\infty v^3 \sigma e^{\frac{-\mu v^2}{2kT}}dv$$

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You have v=v1-v2. Let u=v1+v2. The new integration is straightforward as long as the integral limits are -∞ to ∞. You then need |v1v2| instead of v1v2 in the integrand. The u integration should leave you with what you want.

## What is the Maxwell-Boltzmann distribution?

The Maxwell-Boltzmann distribution is a probability distribution that describes the speed 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 late 19th century.

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

The Maxwell-Boltzmann distribution is significant because it provides a mathematical relationship between the temperature of a gas and the average speed of its particles. This relationship has wide-ranging applications in fields such as thermodynamics, statistical mechanics, and atmospheric science.

## What factors affect the shape of the Maxwell-Boltzmann distribution curve?

The shape of the Maxwell-Boltzmann distribution curve is affected by two main factors: temperature and molecular mass. As temperature increases, the curve shifts to the right, indicating a higher average speed of particles. As molecular mass increases, the curve becomes narrower and taller, indicating a smaller range of speeds and a higher peak at the most probable speed.

## How is the Maxwell-Boltzmann distribution related to the ideal gas law?

The Maxwell-Boltzmann distribution is derived from the ideal gas law, which states that the pressure of a gas is directly proportional to its temperature and the number of molecules present. The distribution describes the probability of finding a particle at a given speed in an ideal gas, and is used to calculate various thermodynamic properties of gases.

## What are some real-world applications of the Maxwell-Boltzmann distribution?

The Maxwell-Boltzmann distribution has numerous applications in various fields of science and engineering. Some examples include predicting the speed distribution of molecules in a gas, analyzing the behavior of particles in a plasma, and understanding the distribution of kinetic energy in a chemical reaction. It is also used in the design of gas turbines, combustion engines, and other heat engines.