Determine bond length via rotational spectroscopy Estimate the bond length of 12C16O (pure rotational spectrum) given J''=3 (15.356 cm-1) and J'=9 (38.356)


I understand that the first step is the calculate the rotational constant. So to do this I assume I would use one frequency value and one J value? And the equation to use would be 

$$\nu = BJ(J+1)$$ 

If I do this with the numbers for $J=3$, then I get $1.28$ for $B$, which is apparently incorrect. I have seen someone else's attempt and they use 

$$12B = (38.356 - 15.365)=1.916$$ 

And solved for B this way and they get the correct answer. Why is the way I thought not correct? Why is the second way correct and where did B come from?
The method they are using seems to be looking at the separation between two different transitions to determine $B$. 

$J''$ denotes the lower energy state in a transition and $J'$ denotes the higher energy state. Here, you are looking at two different transitions: $$J_1'=4\to J_1''=3 \text{ with } \nu_{4\to 3}=\pu{15.365cm^-1}$$ $$J_2'=9\to J_2''=8 \text{ with } \nu_{9\to 8}=\pu{38.356cm^-1}$$

We can see in the figure below that $\Delta\nu=2B|J_2'-J_1''|$ or that the separation between transitions depends on difference between the $J$ of the higher state of the higher energy transition and the lower state of the lower energy transition. In your case, this gives that $\Delta\nu=2B|9-3|=12B$.

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You could also get $B$ from the individual transitions, but you have used the wrong formula. $E_J=BJ(J+1)$ is the energy of the rotational state $J$, but you are given the energy of the transitions, which can be computed as $\nu=2B(J''+1)$. Checking with either transition should give the same result as using the spacing between the transitions.

One reason you might use the spacing to compute $B$, rather than individual transitions, is that if you have any systematic error in measuring transition frequencies, taking the difference will cancel out this error.