[CHM 331] Textbook Problem & Your Problem

[Andrew J. Pounds]

On 11/09/13 21:33, Jeffrey Mimbs wrote:
> Dr. Pounds,
>
> Is problem 57 in Chapter 13 referring to Chapter 16? The equation 
> makes no sense to me, and none of us know what to do with it.
>
> Also, for part b of your problems, the first 5 energy  levels include 
> the zeroth energy level, right? So it would be v = 0,1,2,3,4 for the 
> first 5.
>
> Thanks
>
> Jeffrey Mimbs

Yes -- I think they deleted something from the text when they changed 
editions.  The equations in 16 have an implication that you already know 
something about all of the populations.  Fortunately you can use 13.73 
to answer this question.

$f_v = \left(1-e^{-h\nu/kT}\right) e^{-vh\nu/kT}$

Now, in working with spectroscopic problems I prefer to switch to 
spectroscopic units (the centemeter-gram-second, or CGS system).

In this unit system the unit of energy is the erg, the unit of force is 
the dyne.

$h = \mathrm{6.626}\times\mathrm{10^{-27}\ \ erg\cdot s}$

$k = \mathrm{1.381}\times\mathrm{10^{-16}\ \ erg\cdot K^{-1}}$

$c = \mathrm{2.998}\times\mathrm{10^{10}\ \ cm\cdot s^{-1}}$


AND if we switch to using wavenumbers (in cm) then the term $h\nu/kT$ 
becomes $hc\tilde{\nu}/kT$ and we can use data straight from table 13.4.

For HCI I get the value of $hc\tilde{\nu}/kT$ to be 14.3364.  If I then 
plug this into equation 13.73 for the ground state (v=0) I get 0.9999994 
for the fraction of molecules in the ground state at 300K.  If I repeat 
this process for the first vibrational excited state (v=1) I 
$5.94\times10^{-7}$.  The book problem asks for the ratio of these two 
quantities.  Since the population of the ground stat is so close to one, 
the ratio $f_1/f_0$ is simply $5.94\times10^{-7}$.

For iodine you will get something quite different.

And yes, vibrational state numbering starts at zero.

-- 
Andrew J. Pounds, Ph.D.  (pounds_aj at mercer.edu)
Professor of Chemistry and Computer Science
Mercer University,  Macon, GA 31207   (478) 301-5627
http://faculty.mercer.edu/pounds_aj

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