# Rotational spectra - thermal population of rotational levels

1. Apr 19, 2007

### isomer

1. The problem statement, all variables and given/known data

If the thermal population of the rotational levels is given by:

Nj/No = (2J+1)*exp(-hcBJ(J+1)/kT)

Calculate which state has the highest thermal population at a given temperature T. Calculation needs to be shown, not just a result.

2. Relevant equations

Nj/No = (2J+1)*exp(-hcBJ(J+1)/kT)

where:

Nj = population in excited j state
No = population in ground state
J = rotational quantum #
B = rotational constant
k = Boltzmann constant
T = temperature

3. The attempt at a solution

I honestly don't know where to even begin..... The given equation appears to be in the form of a Boltzmann distribution. I'd assume a reasonable answer would be a function of J and T; where the function is greater than 1, obviously the Nj state would have the highest thermal population and vice versa. Other than that, i'm pretty much at a loss. Took a stab in the dark, differentiating the right side of the equation with respect to J then setting equal to zero to maximize the function and get it down to a single variable, temperature, but I seem to be getting nowhere fast.

Any help would be greatly appreciated, thanks.

Last edited: Apr 19, 2007