[bmu123]

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.

[Corribus]

The forum's policy is that you have to show work to get help. If you are not sure where to start, try looking Up the definition of rotational temperature.

[bmu123]

Yeah I've got this equation: ΘR = ħ²/2k_BI and with that I get a rotational temperature of 109.55 K So this is 'the temperature at which thermal energy is comparable to the spacing between rotational energy levels'. But wouldn't this be the rotational temperature no matter what environment it was in? Seems quite high

[mjc123]

The rotational temperature is the temperature at which the thermal population of the rotational states is such as to give rise to the observed rotational spectrum, in terms of the relative intensities of the different transitions. So calculate, in terms of T, the Boltzmann distribution of the populations of the initial rotational states (taking into account degeneracy), and hence the relative intensities of the transitions, particularly the second and third lines.
I agree that your value of ΘR seems quite high; without knowing the molecular constants, I did an order of magnitude estimation and came out with something of the order of 1K, which seems more sensible. Check your maths.

[bmu123]

OK sorry I've just come back to this and just to clear it up, would I calculate the rotational energy then plug it into ni/n0 = (2J+1) exp[-BJ(J+1)/kT] to find the intensities of each line?

[bmu123]

I redid the rotational temperature calculation and got a temperature of 1.125K by the way, do I substitute this in to the other equation?

[mjc123]

(A) Yes
(B) Is this the result of the calculation in (A) or is it a recalculation of ΘR? It looks about right for ΘR, but ΘR is not the rotational temperature. What is the "other equation"?

[bmu123]

1.125K is the recalculation of ΘR
When I plug it into ni/n0 = (2J+1) exp[-BJ(J+1)/kT] I just keep getting 0 though.

[mjc123]

You don't plug ΘR into that equation. ΘR is a constant characteristic of the molecule. You are looking for the actual temperature - a variable - at which two rotational lines are of equal intensity. Express that intensity ratio as a function of T and find the value of T that makes the ratio 1. If the analytical expression is too complicated to solve, do it numerically by varying the value of T (e.g. in Excel).

[terrystrickl]

Is there a difference in acidity or basicity in aqueous and gas phase? i.e. HNO3 is more acidic than H2SO4 in aqueous solutions but what about in gas phase? My reason for wondering if there's a difference is because in aq phase, all bonds are free to rotate but in gas phase, bonds are fixed hence angles are fixed and i wonder if the "immobility" of the bonds which hence affects the steric factors will cause a change in acidity in gas phase?

[mjc123]

This is a separate topic, you should start a new thread.
The answer to your question is yes, there is a difference, but I can't say off the top of my head what it is. But where did you get the idea that bonds are fixed in the gas phase?