Question

How did scientists calculate the no. of protons/electrons in the atom of different elements?

Answer 1

In order to determine the number of protons, neutrons and electrons that are in an atom, the first thing you need to do is find the atomic number and the atomic mass of the element you are interested in. You can use the Periodic Table of Elements to find this information. Let's use mercury as an example:

From the Table of Elements, we can see that mercury's atomic number is 80 and its atomic mass is 200.59. We'll need these numbers to get the information you want. The atomic number is the number of protons in an atom's nucleus, so we can tell right away that an atom of mercury contains 80 protons. Atoms, by definition, are electrically neutral. Protons carry a positive charge, so the nucleus of an atom of mercury carries a charge of +80. This positive charge is balanced by electrons, which carry a negative charge. 80 electrons are needed to balance the 80 protons. The atomic mass is sort of an average of all of the different isotopes of an element. Mercury's atomic mass is 200.59, but we can round that to 201 (this is known as the mass number). This is a tally of the total number of particles in an atom's nucleus, so an 'average' atom of mercury contains 201 particles in its nucleus. We know that 80 of those particles are protons. The rest must be neutrons. So, an atom of mercury contains 201 - 80 = 121 neutrons. In summary:

protons = atomic number
electrons = atomic number
neutrons = rounded atomic mass - atomic number

Answer 2

^this is good, but it's assuming that we know that protons + neutrons = mass number, which was not the case when people first began characterizing elements.

from http://en.wikipedia.org/wiki/Proton,
"Following the discovery of the atomic nucleus by Ernest Rutherford in 1911, Antonius van den Broek proposed that the place of each element in the periodic table (its atomic number) is equal to its nuclear charge. This was confirmed experimentally by Henry Moseley in 1913 using X-ray spectra."

X-ray spectroscopy is the way the number of protons were determined.