Normalizing molecular speed probabiliy distribution

[babbagee]

Homework Statement

Show that the molecular speed probability distribution is normalized

Homework Equations

Molecular speed probability distribution
P(C)dC = 4pi[m/2pikt]^3/2exp[-mC²/2kT]C²dC

The Attempt at a Solution

I have tried many steps but i keep getting stuck. I tried using the same metods that i used to use to normalize wave funcations and that was <psi|psi>=1 but i am not getting anywhere, can some one please point me in right deirection.

 

[chemisttree]

See if this helps:

http://farside.ph.utexas.edu/teaching/sm1/lectures/node20.html

 

[Blueshift5]

To show that the molecular speed probability distribution is normalized, we need to show that the integral of the probability distribution over all possible speeds is equal to 1. In other words, we need to show that:

∫P(C)dC = 1

To do this, we can first rewrite the probability distribution in terms of the speed, v, instead of the molecular velocity, C. We can do this by using the relation v = C/√3. This gives us:

P(v) = 4π(m/2πkT)^3/2 * exp[-m(v√3)^2/2kT] * (v√3)^2 * (1/√3)

= 4π(m/2πkT)^3/2 * exp[-m3v^2/2kT] * v^2/3

= (4πm/2πkT)^3/2 * exp[-m3v^2/2kT] * v^2

Now, we can use the substitution u = m3v^2/2kT to simplify the integral. This gives us:

∫P(v)dv = (4πm/2πkT)^3/2 * ∫exp[-u]du

= (4πm/2πkT)^3/2 * [-exp[-u]] from 0 to ∞

= (4πm/2πkT)^3/2 * (0 - 1)

= 1

Therefore, we have shown that the molecular speed probability distribution is normalized, as the integral over all possible speeds is equal to 1. This means that the probability of finding a molecule with a certain speed is 100%, as it should be.