Solution to the symmetrical molecule NMR challenge

The ¹H-NMR spectrum of a mixture of trans and cis isomers of an aliphatic amine hydrochloride shows three peaks at the vicinity of 10 ppm. The smaller signals observed at 9.7 ppm and 10.5 ppm correspond to the cis isomer. This is because the cis configuration of the amine makes protons of the –NH₂ ⁺ – group to become chemically non-equivalent: one of the –NH₂ ⁺ – protons is spatially closer to the –CH₂– group of cyclopropane whereas the other is closer to the protons of the CH groups. The protons of the –NH₂ ⁺ – group of the trans isomer, on the other hand, are in the same spatial environment, so they appear as one signal at 10.1 ppm. From the peak integrals given in the NMR spectrum, we can therefore estimate the trans/cis ratio to be approximately 2.11/(0.54 + 0.42) ≈ 2.2.