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Boltzmann equation problem

by Benzoate
Tags: boltzmann, equation
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Benzoate
#1
Sep17-07, 10:03 AM
P: 569
1. The problem statement, all variables and given/known data
Consider a pure hydrogen hydrogen gas ata tempeature of 10080 K. What is the ratio of the populations of the ground state(n=1) to the first excited state(n=2). Note that the energy difference is 1.634e-18 joules between these two states. At what temperature would both levels have equal populations?


2. Relevant equations

Boltzmann equation: N(B)/N(A)= g(b)/g(a) exp[(E(A)-E(B))/kT
g=2*n^2
3. The attempt at a solution
finding N(B)/N(A) was easy. N(2)/N(1)=2(2)^2/2(1)^2*exp(1.634e-18 joules)/(10080 K)(1.380 *10^-23))= .000031665

I had a difficult time finding the Temperature when both populations are equal. When both populations are equal, does that imply N(a)/N(b) = 1 because N(a)=N(B)?

Now finding the temperature is simple : 1=4 exp(1.634e-18 joules/(T(1.380e-23)) => T= -85411.7 Kelvins.

The only problem is I don't know if I should assume N(B)/N(A) = 1 , just because the temperatures are equal
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CompuChip
#2
Sep17-07, 11:07 AM
Sci Advisor
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P: 4,300
No, you shouldn't assume N(B)/N(A) = 1 because the temperatures are equal.
You should assume it because you assume that the populations are equal; then you fill in the formula and solve for the temperature where this occurs.

So you have obtained the right answer with the right method but with a wrong way of thought
Benzoate
#3
Sep17-07, 01:24 PM
P: 569
Quote Quote by CompuChip View Post
No, you shouldn't assume N(B)/N(A) = 1 because the temperatures are equal.
You should assume it because you assume that the populations are equal; then you fill in the formula and solve for the temperature where this occurs.

So you have obtained the right answer with the right method but with a wrong way of thought
sorry, I meant to say that the populations are equal, so should I assume that N(a)/N(b)=1 since N(a)=N(b)

CompuChip
#4
Sep17-07, 01:47 PM
Sci Advisor
HW Helper
P: 4,300
Boltzmann equation problem

Then you got the right answer by the right method and a typo :)


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