# Rotational temperature problem

1. Feb 22, 2015

### burns96

1. The problem statement, all variables and given/known data
Hi there, I have a question that I'm not sure how to go about solving:
I've been given a series of transitions in the microwave spectrum of 31P14N and have assigned these Jinitial and Jfinal quantum numbers, calculated the bond length etc.
The next part says that when 31P14N is observed in the very cold environment of interstellar space by microwave spectroscopy, the second and third lines have equal intensity, and asks what the rotational temperature of the molecule in this environment would be. Any help would be greatly appreciated.

2. Relevant equations
ΘR = ħ2/2kBI
ni/n0 = (2J+1) exp [-BJ(J+1)/kT]
3. The attempt at a solution
I have 3.581 x 10-46 for I
But then plugging this in to the equation I get 1.125K
I'm a bit unsure of where to go from here, I'm told I need to work out the boltzmann distributions, is that for the two lines of equal intensity, and do I use the rotational temperature for T?

2. Feb 23, 2015

### Quantum Defect

The intensity (I) of a given rotational transition (J->J+1) is:

I(J) = N_J *S(J) where S is the line strength.

I(J) = const * (2J+1)exp [-B*J*(J+1)/kT]*S(J)

For a linear molecule, S(J) = mu^2 * (J+1) /(2J+1) -- Townes and Schalow, "Microwave Spectroscopy"

I(J=0)/I(J=1) = 1 = .....

Plug in and solve for T.