GREPhysics.NET: GR0177 #98

GR0177 #98

Problem

GREPhysics.NET Official Solution

This problem is still being typed.

Statistical Mechanics $\Rightarrow$ }Average Energy

The average energy is given by $\sum_i E_i e^{-E_i/kT}/Z$ , where $Z=\sum_i e^{-E_i/kT}$ is the partition function.

The above is just the stat mech generalization of the baby-averaging $\bar{u}=\sum_i U_i/N$ , where $N$ is the number of energies summed over. Z is like a generalized N.

Alternate Solutions

There are no Alternate Solutions for this problem. Be the first to post one!

Comments

sparksj
2008-07-26 17:15:06

No knowledge of what this function this represents is needed to solve this problem. Use units.

The units of the given expression are the units of Ei which as given is energy.

a) Has units of energy. (yes)
b) Unitless. (no)
c) Unity is unity this one can be ruled out if you are not a moron. (n0)
d) Probability is unitless. (no)
3) Entropy has units of Q/T or energy/temperature. (no)

Reply to this comment

Blake7
2007-09-22 06:14:51

Yosun,

(For completeness sake and a deeper understanding) one might want to add that

(D) P(Ei) = exp(-Ei/kT)/Z (see how close!!!)

and

(E) S/k = Sum[(Exp(-Ei/kT))/Z)(Ei/kT+lnZ)]

(C) Unity is unity ;)

or something like that

Da Broglie
2011-04-08 17:33:44

yep pretty important, just to put the equations into LaTeX for ease of reading,

(D) $P\left(E_i\right)=\frac{e^{\frac{-E_i}{kT}}}{Z}$

(E) $\frac{S}{k}=\Sigma[\frac{e^{\frac{-E_i}{kT}}}{Z}(\frac{E_i}{kT})+ln(Z)]$

just to add here that average energy of a state is given by

$E_{avg}=\frac{1}{Z}\frac{\partial Z}{\partial\beta}$ where $\beta=\frac{1}{kT}$

which equals the given expression in the problem

Reply to this comment

Post A Comment!

You are replying to:

Yosun, (For completeness sake and a deeper understanding) one might want to add that (D) P(Ei) = exp(-Ei/kT)/Z (see how close!!!) and (E) S/k = Sum[(Exp(-Ei/kT))/Z)(Ei/kT+lnZ)] (C) Unity is unity ;) or something like that

Bare Basic LaTeX Rosetta Stone
LaTeX syntax supported through dollar sign wrappers $, ex., $\alpha^2_0$ produces $\alpha^2_0$ .
type this...	to get...
$\int_0^\infty$	$\int_0^\infty$
$\partial$	$\partial$
$\Rightarrow$	$\Rightarrow$
$\ddot{x},\dot{x}$	$\ddot{x},\dot{x}$
$\sqrt{z}$	$\sqrt{z}$
$\langle my \rangle$	$\langle my \rangle$
$\left( abacadabra \right)_{me}$	$\left( abacadabra \right)_{me}$
$\vec{E}$	$\vec{E}$
$\frac{a}{b}$	$\frac{a}{b}$

The Sidebar Chatbox...
Scroll to see it, or resize your browser to ignore it...

All-Solutions List | Latest Comments | Unanswered Questions (Cries for Help!) |::| About

This site is not affiliated with ETS, nor is it officially endorsed (yet) by ETS. This site is the work of an individual named Yosun Chang. The solutions, sans user comments and ETS questions, are © 2005 by Yosun Chang except where noted. All rights reserved. / The comments appending particular solutions are due to the respective users. / Educational Testing Service, ETS, the ETS logo, Graduate Record Examinations, and GRE are registered trademarks of Educational Testing Service. The examination questions are © 1997 and © 2001 by ETS.

Bare Basic LaTeX Rosetta Stone