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Since the bonds can't rotate, would there be 2 peaks? Likewise, how many peaks would there be in trans-but-2-ene and cis-but-2-ene?

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You need to consider whether the protons in ethene are chemically equivalent, magnetically equivalent, or both. The lewis structure of ethene:

Lewis structure of ethene

Because of the symmetry of the planar ethene molecule, all H atoms are chemically and magnetically equivalent. The spin system is therefore A$_4$. The signal is not split because the environment is the same for all H atoms. The signal would therefore be a single peak. Compared to an "average" methyl group, the ethylene protons are less shielded, and would appear at a higher chemical shift than methyl groups.

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