What Is the Rotational Temperature of 31P14N in Interstellar Space?

[burns96]

Homework Statement

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 J_initial and J_final 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.

Homework Equations

ΘR = ħ²/2k_BI
n_i/n₀ = (2J+1) exp [-BJ(J+1)/kT]

The Attempt at a Solution

I have 3.581 x 10^-46 for I
But then plugging this into 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?

 

[Quantum Defect]

Homework Statement

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 J_initial and J_final 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.

Homework Equations

ΘR = ħ²/2k_BI
n_i/n₀ = (2J+1) exp [-BJ(J+1)/kT]

The Attempt at a Solution

I have 3.581 x 10^-46 for I
But then plugging this into 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?

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.