Atomic Structure History I was reading a book about atomic structure history (Chemistry in Context by Graham Jill and John Holman), and got interested in few points which is not explained in the book. 


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*"Calculations based on the results obtained by Geiger and Marsden showed that number of positive charges was about half of its relative atomic mass."


How did they calculate it and came to that conclusion? 
2."Moseley bombarded various elements with high speed electrons and found that the wavelength and frequency depended on the element he used."
I think that the X-ray produced was characteristic X-rays. Am I right?
If so, then if the electron knocked was from a different shell (even though if its from the same element), then the X-ray produced will be of different wavelength and frequency. Then why did the book state that "Whatever the  conditions was, a particular element always formed X-rays of same frequency."
(I was reading about Lyman and Balmer series, where emitted light from hydrogen has different wavelength depending on in which shell the electron was located, and came to this question.)


*Chadwick's Experiment
"Alpha particles knocked the neutrons from Beryllium sheet and the emitted neutrons knocked of the protons from paraffin wax."


Why did the neutrons of Beryllium got separated from its proton and electron?


*(a bit off topic) 
Mass spectrometer calculate the charge to mass ratio. But what if there are 2 reactions which happen in the chamber;


Atoms of Ar=20 gets ionised into +1 ions
Atoms of Ar=40 gets ionised into +2 ions
Will their ratio of charge to mass be equal, i.e. +1 to 20? If so, how can we distinguish them? 
That is all I am curious about for now, Thank you very much. :) 
 A: I'll be updating this as I find answers to your other questions. 
First question
Geiger and Marsden actually did several experiments all very related (and all explained at this website). The one related to your question was done in 1913, to prove relationships that Rutherford calculated and published in a 1911 paper (The Scattering of α and β Particles by Matter and the Structure of the Atom). 
Geiger and Marsden didn't know what the positive charge of the nucleus of their metals was (they had only just discovered the nucleus after all) but they assumed it was proportional to the atomic weight. They specifically tested whether it was proportional to the atomic weight squared. So what they did was use the apparatus pictured below.

They covered the holes in the disc with foils of gold, tin, copper, and aluminum and measured each foil's stopping power by equating it with an equivalent thickness of air. They counted the number of scintillations per minute that the foil produced on the screen. They then divided the scintillations per minute by the foil's air equivalent, and then divided again by the square root of the atomic weight. Therefore, Geiger and Marsden obtained the fixed number of scintillations a fixed number of atoms produces for each metal. Then, for each metal, they divided this number by the square of the atomic weight and found that the ratios were more or less the same. They therefore proved that  $s ∝ Q_n^2$ (where $Q_n$ is the positive charge of the atomic nucleus).
Second question
For your second question, about Moseley's experiment, yes, they were characteristic x-rays. This website gives more information about Moseley's law and this website gives more information about Moseley himself and his other achievements (fun fact: he predicted the existence of element 61, which ended up being named promethium).
In answer to the second part of your second question, the spectra of light emitted by atoms is proportional to the square of $Z$, the charge on their nucleus (in the Bohr model of the atom). Moseley was able to confirm that the spectra of light emitted was indeed proportional to $Z$, and he formulated Moseley's law:
$${\sqrt  f}=k_{1}\cdot \left(Z-k_{2}\right)$$
Where $f$ is the of the main, or $K$ x-ray emission line, and $k_1$ and $k_2$ are constants depending on the type of line.
Third question
For your third question, he used the beryllium to create radiation. He then aimed the radiation at paraffin wax, and since paraffin wax has a high hydrogen content and therefore offers a target dense with protons, and neutrons have almost equal mass, the protons easily scattered when the radiation hit them. Chadwick looked at the distance the protons scattered and how the radiation impacted atoms of various gases and concluded that the radiation was made up of uncharged particles with around the same mass as the proton - aka the neutron.
Now, for why the beryllium was emitting neutrons. (See this website for more information; a summary is given here.) $^9Be$ releases more neutrons than it absorbs - this particular isotope under goes an (n, 2n) reaction, which can be described below as 
$$\frac {9}{4}Be + n → 2(\frac {4}{2}He) + 2n$$
Neutrons can also be liberated when beryllium nuclei are struck by energetic alpha particles and when beryllium is under bombardment by gamma rays.
Fourth question
I'm not quite sure I understand this question. Yes, mass spectrometers calculate the charge to mass ratio, but they are also detected by a mechanism such as an electron multiplier and the atoms/ions in the sample can be correlated with known fragmentation patterns. This website and this website might be helpful to you; especially the second link.
Hope this helps!
