Calculate most probable energy from Maxwell-Boltzmann distribution of energy

TehDarkArchon

1. The problem statement, all variables and given/known data
If the Maxwell-Boltzmann distribution of energy is f(E)=2*pi*E^1/2*(1/pi*k*T)^3/2*e^E/kT. Can you calculate the most probable energy from this? (The answer is kt/2).


2. Relevant equations
Some kind of integration (guassian integral) is needed from 0 to infinity I believe.


3. The attempt at a solution
I tried integration by parts and some other methods with no luck. My teacher made it clear that this is not a gamma function (my initial thought since the E^1/2 is paired with an exponential function) and that the answer is relatively easy. Any help would be much appreciated!

BruceW

Wikipedia says that it is a gamma distribution...
Also, shouldn't it be e^-E/kT?

TehDarkArchon

It is supposed to be e^-E/kT, sorry about that. Unfortunately he made it clear that it's not a gamma function...and integration by parts only causes that term to be 1/2(E)^1/2 and just ends up being more complicated. We have discussed transformation of variables and guassian integrals in class, so perhaps that's the way to go about this problem (I'm still getting used to both of these as I've never done them before this class, thus probably why I'm struggling so much.)

vela

You don't need to integrate at all. You're thinking of the average value, or expectation value, of energy. The problem is asking for the most probable energy.

TehDarkArchon

Okay that makes sense...I was going off my teacher's example of how to calculate most probable velocity from the Maxwell-Boltzmann distribution. Any hint on what kind of calculation I need to do to get most probable energy? I'm probably gonna feel incredibly stupid once I figure it out...

vela

What does the phrase "most probable" translate to mathematically in terms of f(E)?

BruceW

Think about the curve of f(E) against E.

TehDarkArchon

f(E) is just a function of E, with the most probable energy being the energy that most of the molecules have in the system, so should I just compute the value to infinity?

vela

No. What does f(E) represent?

jfy4

I know we already have some people helping but, think "optimization problems"...

BruceW

Yep, jfy4 has the right idea.
Another hint: If you had the graph of f(E) against E, you could simply see by looking at it what the most probable energy was.

TehDarkArchon

Ahh I think I see how to get it now. Basically E_mp is the maximum of f(E) so you'd go about this problem how you would any graph to find the maximum, by setting the derivative to 0. By setting all of the constants equal to Q, canceling Q and the exponential function on both sides, i made [itex]\frac{1}{2\sqrt{E}}[/itex] = [itex]\frac{\sqrt{E}}{kT}[/itex]. With some rearrangement I get kT = 2E, or kt/2 = E.

jfy4

well done.

TehDarkArchon

Thanks very much everyone for your help! I appreciate it so much!