Maxwell Boltzmann Distribution

In summary, the conversation discusses the use of the Maxwell-Boltzmann distribution to calculate the fraction of particles between two speeds, and how to integrate this equation without approximation or help from Maple. The simplified equation and integral are mentioned, as well as the question of how to solve the problem for a certain amount of moles with hydrogen between two different velocities.
  • #1
TeslaPow
40
1
I don't know how to integrate the Maxwell-Boltzmann distribution without approximation or help from Maple.

Given the Maxwell-Boltzmann distribution:

f(v) = 4*pi*[m/(2*pi*k*T)]^(3/2)*v^2*exp[(-m*v^2)/(2*k*T)]

Observe the appearance of the Boltzmann factor exp[(-m*v^2)/(2*k*T)] with E = 1/2(mv^2)

Assuming a fixed temperature and mass, one can simplify this equation:

f(v) = a*v^2*exp[-bv^2]
a = 4*pi*[m/(2*pi*k*T)]^(3/2)
b = m/(2*k*T)

In order to calculate the fraction of particles between two speeds v1 and v2, one should evaluate the definite integral:

∫f(v)dv

Here is an link to integral-tables, http://integral-table.com/
How would I solve this problem for let's say a certain amount of moles with hydrogen between two different velocities? Tor
 
Physics news on Phys.org
  • #2
What have you tried in terms of integrating the M-B distribution?
 
  • #3
SteamKing said:
What have you tried in terms of integrating the M-B distribution?

I've just put in some values to calculate how many atoms gets affected from let's say 400-401 m/s for hydrogen. I'm not sure what solution to pick for this kind of integration, that's what I'm asking. To calculate between higher speeds like 400-500 m/s, an integration is needed.
 
Last edited:

FAQ: Maxwell Boltzmann Distribution

What is the Maxwell Boltzmann Distribution?

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

What factors affect the shape of the Maxwell Boltzmann Distribution?

The shape of the Maxwell Boltzmann Distribution is affected by three main factors: temperature, mass of the particles, and the size of the container. As temperature increases, the curve shifts to the right and becomes wider. For particles with larger mass, the curve is shifted towards lower speeds. And for a larger container, the curve becomes flatter and more spread out.

What is the significance of the Maxwell Boltzmann Distribution in statistical mechanics?

The Maxwell Boltzmann Distribution is a fundamental concept in statistical mechanics, which is the study of the behavior of large systems of particles. It helps us understand the distribution of energy among particles in a gas and how this distribution changes as temperature and other factors are varied. It also provides a basis for understanding other important concepts in thermodynamics and kinetic theory.

How is the Maxwell Boltzmann Distribution related to the ideal gas law?

The ideal gas law, which describes the relationship between pressure, volume, and temperature of a gas, is based on the average speed of particles in a gas. The Maxwell Boltzmann Distribution gives us a more detailed understanding of the distribution of particle speeds, which allows us to calculate the average speed and other important properties of the gas.

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

The Maxwell Boltzmann Distribution has many practical applications, including in the design and operation of gas turbines, refrigeration systems, and chemical reactions. It is also used in fields such as astrophysics, plasma physics, and atmospheric science to study the behavior of gases in extreme environments. Additionally, the distribution is used in medical imaging techniques to analyze the motion of particles in tissues and fluids.

Back
Top