So the two Hydrogen atoms on the same carbon causing two different signals, are different in R/S not cis/trans? I remember having learned about this some years ago... And I believe that this was a part that I really understood. Thanks!
Yes, the two hydrogen atoms on the same carbon cause two different signals b/c of the chiral center. It is similar to the case with cis/trans alkenes, where two hydrogens on the same carbon give different signals. The reasons are slightly different, though.
For alkenes, the lack of rotation around the C=C double bond causes the hydrogens to be in different chemical environments all the time. With a chiral molecule, although you may have free rotation around single bonds, the two hydrogens of the CH2
still never experience the same chemical environment.
The added complexity is that diastereotopic hydrogens on the same carbon (same carbon = geminal) not only cause different signals, they also split each other. They can also still be split by hydrogens one carbon away (one carbon away = vicinal).
Geminal splitting frequently gives different J-values than vicinal, and the N+1 rule only works if all the coupling constants are the same. If the J-values are different, you start getting doublets of doublets (dd), doublets of triplets (dt), doublets of quartets (dq) or other combinations (ddd, dddd, ddq, etc.).
I'll try to post something more visual later, when I get to my laptop.