Propose structures for compounds with the following formulas that show only one peak in their ${}^1\mathrm{H}$NMR spectra:

a.${\mathrm{C}}_5{\mathrm{H}}_{12}$

b.${\mathrm{C}}_5{\mathrm{H}}_{10}$

c.${\mathrm{C}}_4{\mathrm{H}}_8{\mathrm{O}}_2$

The proton which are having the different environment and all proton give different peaks are the non-equivalent proton where if the proton sharing the same environment and all protons give the same singles or peaks are the equivalent protons.

All equivalent proton produces a single signal and all the non-equivalent proton produces a separate signal for each proton.

As only one peak is produced by the compound that means all hydrogens are equivalent protons that are surrounded by the same environment so, the structure must be like the center carbon surrounded by 4 methyl groups, all methyl groups produces 1 signal as shown:

Compound A

As only one peak is produced by the compound that means all hydrogens are equivalent protons that are surrounded by the same environment so, the structure must be like the cyclic pentane so all methylene groups produces 1 signal as shown:

Compound b

As only one peak is produces by the compound that means all hydrogens are equivalent protons that are surrounded by the same environment so, the structure must be like the cyclic hetero hexane with 2 oxygen as a heteroatom so all methylene groups produces 1 signal as shown:

Compound C