Calculate most probable energy from Maxwell-Boltzmann distribution of energy


by TehDarkArchon
Tags: distribution, energy, maxwellboltzmann, probable
TehDarkArchon
TehDarkArchon is offline
#1
Sep10-11, 02:56 PM
P: 18
1. The problem statement, all variables and given/known data
If the Maxwell-Boltzmann distribution of energy is f(E)=2*pi*E1/2*(1/pi*k*T)3/2*eE/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 E1/2 is paired with an exponential function) and that the answer is relatively easy. Any help would be much appreciated!
Phys.Org News Partner Science news on Phys.org
Review: With Galaxy S5, Samsung proves less can be more
Making graphene in your kitchen
Study casts doubt on climate benefit of biofuels from corn residue
BruceW
BruceW is offline
#2
Sep11-11, 08:54 AM
HW Helper
BruceW's Avatar
P: 3,337
Wikipedia says that it is a gamma distribution...
Also, shouldn't it be e-E/kT?
TehDarkArchon
TehDarkArchon is offline
#3
Sep11-11, 09:33 AM
P: 18
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
vela is offline
#4
Sep11-11, 12:35 PM
Emeritus
Sci Advisor
HW Helper
Thanks
PF Gold
P: 11,527

Calculate most probable energy from Maxwell-Boltzmann distribution of energy


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
TehDarkArchon is offline
#5
Sep11-11, 02:19 PM
P: 18
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
vela is offline
#6
Sep11-11, 02:25 PM
Emeritus
Sci Advisor
HW Helper
Thanks
PF Gold
P: 11,527
What does the phrase "most probable" translate to mathematically in terms of f(E)?
BruceW
BruceW is offline
#7
Sep11-11, 03:33 PM
HW Helper
BruceW's Avatar
P: 3,337
Think about the curve of f(E) against E.
TehDarkArchon
TehDarkArchon is offline
#8
Sep11-11, 04:12 PM
P: 18
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
vela is offline
#9
Sep11-11, 04:19 PM
Emeritus
Sci Advisor
HW Helper
Thanks
PF Gold
P: 11,527
No. What does f(E) represent?
jfy4
jfy4 is offline
#10
Sep11-11, 04:53 PM
jfy4's Avatar
P: 647
I know we already have some people helping but, think "optimization problems"...
BruceW
BruceW is offline
#11
Sep11-11, 05:41 PM
HW Helper
BruceW's Avatar
P: 3,337
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
TehDarkArchon is offline
#12
Sep11-11, 10:57 PM
P: 18
Ahh I think I see how to get it now. Basically Emp 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
jfy4 is offline
#13
Sep11-11, 11:04 PM
jfy4's Avatar
P: 647
well done.
TehDarkArchon
TehDarkArchon is offline
#14
Sep11-11, 11:51 PM
P: 18
Thanks very much everyone for your help! I appreciate it so much!


Register to reply

Related Discussions
Average Energy of Boltzmann Distribution Advanced Physics Homework 7
Help with Maxwell-Boltzmann Energy Distribution stuff Advanced Physics Homework 1
The Boltzmann distribution of uniformly spaced energy levels General Engineering 2
Boltzmann distribution vs. Distribution of energy General Physics 4
Maxwell-Boltzmann distribution General Physics 1