Maxwell-Boltzmann Distribution and average speed

In summary, the conversation discusses using the Maxwell-Boltzmann distribution function to calculate the average speed of molecules in a gas. The equation for the average speed is given as v^2(average)=integral(0,infinity)(v^2*v(t)dv), where a is represented as m/(2kt). The solution to the integral is provided in the problem statement.
  • #1
Math Jeans
349
0

Homework Statement


Given the integral
integral(0,infinity)(v^3*e^(-a*v^2)dv)=a^-2/2

Calculate the average speed v(average) of molecules in a gas using the Maxwell-Boltzmann distribution function.


Homework Equations


v^2(average)=integral(0,infinity)(v^2*v(t)dv)


The Attempt at a Solution



I took the general approach.

integral(0,infinity)(f(v)dv)=(4/pi)*(m/(2*k*T))^(3/2)*integral(0,infinity)(v^2*e^(-mv^2/(2*k*t))dv)

However, I could not find out how to make a substitution for a in order to get the v^3 into the equation.
 
Physics news on Phys.org
  • #2
Maxwell-Boltzman distribution is as you wrote. So if you are looking for average speed you just multiply it with v, and than calculate the integral. It's exactly the integral you have given and of course a = m/(2kt).
 
  • #3
fikus said:
Maxwell-Boltzman distribution is as you wrote. So if you are looking for average speed you just multiply it with v, and than calculate the integral. It's exactly the integral you have given and of course a = m/(2kt).

So I should just multiply by v and do integration by parts?
 
  • #4
noup. You have already given the solution of that integral in your problem statement. You should only put in a = m/(2kt) and then multiply the result with all that constants in front (4/pi ...).

Maxwell-Boltzman distribution is: [tex]f(v)\propto v^2 e^{-av^2} [/tex] where a is as mentioned.
the average speed is [tex] \int_{0}^{\infty} v\cdot f(v)[/tex]

the solution of this integral is in your problem statement.
 

1. 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 shows the likelihood of particles having a certain speed, with the peak of the distribution representing the most probable speed.

2. How does temperature affect the Maxwell-Boltzmann distribution?

As temperature increases, the distribution shifts towards higher speeds and becomes broader. This is because at higher temperatures, particles have more kinetic energy and move faster, resulting in a wider range of speeds.

3. What is the relationship between average speed and the Maxwell-Boltzmann distribution?

The average speed of particles in a gas is directly related to the peak of the Maxwell-Boltzmann distribution. The higher the peak, the higher the average speed of the particles in the gas.

4. How does the mass of particles affect the Maxwell-Boltzmann distribution?

The mass of particles does not directly affect the shape of the Maxwell-Boltzmann distribution, but it does affect the peak. Heavier particles will have a lower peak compared to lighter particles at the same temperature, meaning they will have a lower average speed.

5. What is the significance of the Maxwell-Boltzmann distribution in understanding gas behavior?

The Maxwell-Boltzmann distribution allows us to predict the speeds of particles in a gas and understand how they are affected by temperature. This is important in various fields, such as chemistry and engineering, as it helps us understand and control gas behavior in different environments.

Similar threads

  • Introductory Physics Homework Help
Replies
5
Views
1K
  • Introductory Physics Homework Help
Replies
13
Views
1K
  • Introductory Physics Homework Help
Replies
11
Views
1K
  • Introductory Physics Homework Help
Replies
9
Views
803
  • Introductory Physics Homework Help
Replies
4
Views
907
  • Introductory Physics Homework Help
Replies
15
Views
246
  • Introductory Physics Homework Help
Replies
2
Views
590
  • Introductory Physics Homework Help
2
Replies
38
Views
1K
  • Introductory Physics Homework Help
Replies
6
Views
625
  • Introductory Physics Homework Help
Replies
9
Views
2K
Back
Top