Spin Spin Splitting

Consider 1-bromor-2,2-dichloroethane shown below.  There are two distinct nonequivalent types of protons (blue and red), therefore there should be two signals in the HNMR.  The blue protons are more deshielded since it's attached to a carbon with two highly electronegative Cl atoms and is at about 6 ppm on the spectra below.  The ratio of the red to the blue protons is 2:1.

1-bromo-2,2-dichloroethane

Imagine you shrink down to the size of the red protons.  Lets think about the magnetic field you will experience or feel.  Well, you will feel the B_o field, but it will be split into two fields created by the neighboring blue proton. In a given molecule the blue proton can only have one of two possible spin states, up or down.  Don't forget we many molecules of 1-bromo-2,2-dichloroethane in the NMR tube.  So if we had 100 molecules in our NMR tube half would be up and half would be down.  Thus the red signal is slit into a doublet.  This is  known as the N+1 rule where N is the number of neighboring protons.  The red protons have one blue neighboring proton therefore N=1 and N+1=2 a doublet. 

No image your the blue proton which experiences the two red protons.  Below are the three possible spin states; 1) both soins up, 2) one up and one down (these two have the same resonance energy) and 3) both spins down.  If you had 100 molecules you'd have a 25:50:25 ratio of molecues in teh trhee states and the NMR signal would appear as a triplet with a 1:2:1 relative intensity (see spectra above).  In this case N=2 and N+1=3, a triplet is observed.