[CHM 331] Re: question

[Andrew J. Pounds]

On 12/14/2011 03:19 PM, Vanessa.Marie.Breslin wrote:
> Dr. Pounds,
>
> On HW set 8 your problem #2, I had two questions. The problem is: 
> Calculate the fractional occupations (Ni/N) of the energy levels of a 
> particle in a 1-D box at a temperature at which the population of the 
> n=2 level is 0.421 of the population of the n=1 level. The energy 
> levels of the particle in the 1-D box are nondegenerate. The relative 
> energies of the levels can simply be taken as 1, 4, 9, 16, 25,...
>
> My first question is about a calculation. When trying to find the 
> temperature, I got ln(N2/N1)=-3/kT, where (N2/N1)=0.421. So I 
> calculated my temperature to be 2.511*10^23 Kelvin. Is that correct? 
> Although that temperature checks out with the other calculation steps 
> later in the problem, I was just curious because this does not seem 
> like a reasonable temperature to me.
>
You are really only looking for the "relative" temperature here -- so it 
would be relative to some other state that is not specified.  That being 
said, there is no need to calculate the actual temperature in this 
problem -- as can keep "kT" together.

ln(N_2 /N_1 ) = -3 kT

N_2 /N_1 = 0.421 (from the problem definition)

kT = -3 / ln(N_2 /N_1 )

kT = -3 / ln(.421)

kT = 3.47



> A second question I have about the same homework set problem is when 
> you calculated the molecular partition function (q), how do you know 
> when to stop adding energy terms? Why did you stop at n=8?
>
> Those are all the questions I have.
>
> Thanks,
> Vanessa
Lets calculate terms 1 and 8

exp(-1/3.47) = 0.7497

exp(-64/3.47) = 9.77e-9

My calculator has a 10 digit display.  If I added any other terms the 
value in the display didn't change...



The fraction in the first level is 0.651

The fraction in the second level is 0.274

The fraction in the third level is 0.0649

The fraction in the fourth level is 0.0086

The fraction in the fifth level is 0.0006


-- 
Andrew J. Pounds, Ph.D.  (pounds at theochem.mercer.edu)
Associate Professor of Chemistry and Computer Science
Mercer University,  Macon, GA 31207   (478) 301-5627

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