Question

the rest of the atom is apparently \empty space\
this was proposed as a result of what data?
when most of the alpha particles passed through the gold foil

the central, dense core of the atom is positively charged, with the nuclear charge equal
to about one - half the atomic mass.
this was proposed as a result of what data?

Explanation:

In Rutherford's gold - foil experiment, when alpha particles were fired at a thin gold foil:

• For the statement "The rest of the atom is apparently 'empty space'": Most alpha particles passed through the gold foil with little or no deflection. This implies that there is a lot of "empty space" in the atom as the alpha particles (which are relatively massive for sub - atomic particles at the time of the experiment) could pass through without significant obstruction.
• For the statement "The central, dense core of the atom is positively charged, with the nuclear charge equal to about one - half the atomic mass": A small fraction of alpha particles were deflected at large angles (some even bounced back). This indicates that there is a small, dense, positively charged region (the nucleus) in the atom. The fact that the nuclear charge was estimated to be about one - half the atomic mass (later refined with the discovery of neutrons) was based on the analysis of the scattering angles and the known charge and mass of alpha particles.

Answer:

For "The rest of the atom is apparently 'empty space'": Most alpha particles passed through the gold foil.
For "The central, dense core of the atom is positively charged, with the nuclear charge equal to about one - half the atomic mass": A small fraction of alpha particles were deflected at large angles (some bounced back).