Why do the Hs in (E)-1-bromo-2-chloroethene split into doublets instead of staying as a singlet. Wouldn't they be isolated from each other since the double bond does not rotate?
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The hydrogens are not 'isolated' - they are still vicinal protons (3-bonds away) and so the J-coupling between them is strong. The strength of a splitting interaction is primarily determined by the number of bonds in between the hydrogens involved.
The angle between them also plays a role but to a lesser extent. The Karplus equation describes the three-bond J-coupling between two protons as a function of their dihedral angle.