GR 8677927796770177 | # Login | Register

GR0177 #77
Problem
 GREPhysics.NET Official Solution Alternate Solutions
This problem is still being typed.
Statistical Mechanics$\Rightarrow$}Maxwell-Boltzmann Distributions

The Maxwell-Boltzmann distribution is $N\propto g e^{-\epsilon/(kT)}$, where g is the degeneracy.

Given $\epsilon_a = 0.1+\epsilon_b$, one finds the ratio of distributions (thus ratio of numbers) to be $e^{-(0.1+\epsilon_b)/kT}/e^{-\epsilon_b/kT}= e^{-0.1/kT}$.

The problem gives $kT = 0.025$ eV, and thus the above ratio becomes $e^{-.1/.025}=e^{-4}$, as in choice (E).

Alternate Solutions
 Ning Bao2008-02-01 07:39:44 Quick elimination: higher states are less likely->D or E. Ratio of given energy to kT mist be important: as Energy of A increases, likelihood in state A decreases ->E.Reply to this comment
 gbenga2014-10-18 22:35:18 semi-fast soln: rn$\frac{N_{A}}{N_{B}}=\frac{e^{\frac{-E_A}{kT}}}{e^{\frac{-E_B}{kT}}}=e^{\frac{-\triangle{E}}{kT}}=e^{\frac{-0.1}{0.025}}$ rnThis power is negative so A,B, & C are eliminated. The denominator is small so D is unlikely. Remaining is E Reply to this comment gbenga2014-10-18 22:32:03 semi-fast soln: $\frac{N_{A}}{N_{B}}=\frac{e^{\frac{-E_A}{kT}}}{e^{\frac{-E_B}{kT}}}=e^{\frac{-\triangle{E}}{kT}}=e^{\frac{-0.1}{0.025}}$ This power is negative so A,B, & C are eliminated. The denominator is small so D is unlikely. Remaining is EReply to this comment QuantumCat2014-09-01 10:39:10 A quick way to solve this problem (knowing that the occupation number depends on the energy) is to say that state B is at zero energy so that the exponential for state B just becomes 1, which is infinitely easier to deal with. Reply to this comment Ning Bao2008-02-01 07:39:44 Quick elimination: higher states are less likely->D or E. Ratio of given energy to kT mist be important: as Energy of A increases, likelihood in state A decreases ->E.Reply to this comment

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}\$ $\frac{a}{b}$