(Original post by aster100)
Bit more lol, except the 2nI + 1... no idea how that comes about. I also got a bit confused only on the last paragraph
But very good.
Ok, so, the total energy of a system in a magnetic field, measured relative to that without a magnetic field, is given by
m is the magnetic quantum number (m = -I, -I=1, ..., I-1, I -- just like the quantum numbers l and m in atomic orbitals you're familiar with)
is the gyromagnetic ratio (a number, essentially a constant of proportionality)
is the reduced Planck constant (i.e.
) (to keep the units in order)
B = magnetic field strength in the z-direction
The energy DIFFERENCE between two states differing by
(that's the selection rule for NMR transitions) then is
(this is the Larmor frequency I mentioned earlier)
When spin-spin coupling is included, the energy of two coupled nuclei, A and X, is
is the spin-spin coupling constant. You can rationalise this by considering the scalar product of the vector operators of nuclear spin magnetic moments - but that's quite complicated and you'd need some knowledge of quantum mechanics for it. So let's just assume it's true.
Then the Larmor frequency of nucleus A is simply going to be
So, consider the coupling of a proton with FOUR equivalent protons (for protons, m=+1/2 or m=-1/2, since I=1/2):
In other words, you have 5 lines, with intensities 1
1 (as shown by the number of times you get a particular frequency out). This is EXACTLY analogous to the stick diagram from my first post, except that you do it with numbers. But the principle is the same.
More generally, for an AXn
system, where all X are equivalent spin I=1/2 nuclei, you have (n+1) lines in the spectrum, and the amplitude of the i-th line, for i=0,1,2,...,n, is given simply by the number of ways you can put i spins up and the remainder down (or 1/2 and -1/2, respectively), so
The 2nI+1 formula follows directly when you consider a spin with I>1/2.
As for the last paragraph of the previous post, consider the coupling of proton A to TWO protons, X1 and X2, which are inequivalent. Let's construct the same table as above:
Since J_1 and J_2 are DIFFERENT, each line will give us a different frequency, and you get FOUR lines. If the two coupling constants were the same, then the second and third line would be the same, and you'd get a 1
1 triplet (as expected).
So, having inequivalent protons means that lines will SPLIT. If you considered each proton X1 and X2 in turn, you'd see that X1 gives us a doublet (n+1 = 2), and so does X2, but overall we get a 1
1:1 quartet - i.e. 2*2. That's the formula I gave in the last paragraph.
Hope that clarifies things somewhat. I think if you're only doing A-levels, this is a bit of an overkill; but if you find it interesting, you can find a decent and relatively straightforward explanation of simple NMR in the Oxford Chemistry Primer 'Nuclear Magnetic Resonance' by Peter J Hore (OCP #32).