I was doing some exercises the other day, when I came across this question in my book:
A proton weighs about 1.66 x 10-24 g and has a diameter of
about 10-15 m. What is its density in g/cm3?
As you can see, a really simple and standard question, but… does it even make sense to say that a proton has a mass? And calculate its density?! If so, do I consider it a sphere?
It seems to me, subatomic particles are so small it doesn't really make sense to talk about mass, volume and density as if we were talking about… tennis balls!
Best Answer
A proton is a bound state of three quarks. The quarks themselves are (as far as we know) pointlike, but because you have the three of them bound together the proton has a finite size. It doesn't have a sharp edge any more than an atom has a sharp edge, but an edge is conventionally defined at a radius of 0.8768 femtometres. Protons are spherical in the same way that atoms are spherical even though they're made up of discrete electrons.
The three quarks have a mass, but actually the proton is a lot heavier than the combined mass of the three quarks. That's because the binding energy of the quarks is very high, and that energy increases the mass in line with Einstein's famous equation $E = mc^2$.
So, yes, it does make sense to calculate a density for the proton just as it makes sense to calculate a density for an atom.