Calculating the average speed of molecules in a gas

[matt_crouch]

Homework Statement

Given that the integral [between infinity and 0] v³e^-av2dv = 1/2a²

calculate the average speed v_av of molecules in the gas using the maxwell-boltzmann distribution function

Homework Equations

The Attempt at a Solution

i don't really know what to do some guidance please =]

 

[ideasrule]

Let's say I have N particles in a box and I want to find out how many particles have speeds between v and v+dv. How do you use the Maxwell-Boltzmann distribution to do this? How do I find the sum of the speeds of every particle?

Average speed is calculated by adding every particle's speed together and dividing the result by the number of particles.

 

[kerimek]

I would recommend breaking down the problem into smaller steps and using the appropriate equations and principles to solve it. First, let's define the variables in the problem: v represents the speed of the molecules, a is a constant, and e is the mathematical constant known as Euler's number. The integral given represents the probability distribution function for the speed of molecules in a gas, known as the Maxwell-Boltzmann distribution.

To calculate the average speed (vav), we can use the following formula:

vav = ∫vP(v)dv

Where P(v) is the probability distribution function. In this case, we can substitute the given integral for P(v) and solve for vav.

vav = ∫v(v3e-av2)dv

Next, we can use integration by parts to solve this integral. This involves breaking down the integral into two parts and using a specific formula to solve it. Once we have solved the integral, we can substitute the limits of infinity and 0 to get our final answer for vav.

It's important to note that the given integral represents the probability of finding a molecule with a speed between v and v+dv. Therefore, the average speed is the sum of all the possible speeds multiplied by their respective probabilities. This calculation is known as the first moment of the distribution.

I hope this provides some guidance in solving the problem. It's important to have a good understanding of probability and integration techniques to successfully solve this type of problem. Additionally, it's always helpful to consult with your peers or a professor if you need further clarification.