How many different sets of equivalent protons are there for para-xylene (1,4-dimethylbenzene)? only 1 two three four?

Two  

Step-by-step explanation:

There are two ways to determine the number sets of equivalent protons in a compound.  

(a) By substitution

Replace each H with a different group and see if you get a different compound

For example, the six CH₃ H atoms are an equivalent set, because, if I replace any one of them, I will always get the same compound: 1-chloromethyl-4-methylbenzene (Fig. 1).

Similarly, the four aromatic H atoms are an equivalent set. If I replace any one of them, I will always get the same compound: 2-chloro-1,4-dimethylbenzene (Fig. 2).

Thus, there are two sets of equivalent protons in p-xylene.

(b) By symmetry

Two atoms are equivalent if one can be converted into the other by a symmetry operation on the molecule.

The methyl hydrogens can be interconverted by a combination of reflections about the AB and CD mirror planes and by rotations about the C-C bonds to the ring (Fig. 3).

Similarly, the four aromatic H atoms can be interconverted by a combination of reflections about the AB and CD mirror planes.

Again, we find two sets of equivalent protons.