Three ways to find proton mass include calculation from theory, from atomic molar mass, and charge/mass comparisons with electrons. Using theory to find what proton mass “should be” is realistic only for experts in the field. Charge/mass and molar mass calculations can be done at undergraduate and secondary-school levels.
Derive proton mass from quantum and relativity theories. Protons have internal structure—three particles (quarks) held together by attractive forces (gluons). Naive assumptions would give each quark 1/3 proton mass. About 95 percent to 98 percent of proton mass is not contributed by quark mass. In truth, most proton mass is derived from interaction energy between the quarks. Recall “E = mc^2” in case there’s confusion about the phrase “mass is derived from energy.”
Calculate knowing molar hydrogen mass. One mole equals 6.022e23, just as one dozen equals 12 or one pair equals two. We can take it as a given that one mole of hydrogen atoms (not “H2” molecules) weighs 1.0079 g. Each hydrogen atom contains one proton, so one mole of protons weighs 1.0079 g. Since each mole equals 6.022e23 units, we know that 6.022e23 protons weigh 1.0079 grams. Dividing 1.0079 g by mole number (1.0079 / 6.022e23) gives proton mass: 1.6737e-24 g.
Note that hydrogen atoms have an electron to balance proton charge. Protons without electrons, either as dissolved in solution or as plasma, are very different from hydrogen atoms. Since the calculation does not stop there, we can ignore the physical absurdity in pretending electrons don’t exist.
Keep in mind that the “molar mass” calculation method can be done with any element. However, three sources of error creep up. First, protons in hydrogen atoms are not bonded. In other elements, protons are bonded to neutrons. Protons bonded in a nucleus have less energy—therefore slightly less mass—than isolated protons. Second, electron number, and the error if ignoring them, begins to add up. Accounting for electrons makes the whole endeavor more complicated. Lastly, heavier elements are radioactive. Consider decay paths, isotope presence, half-lives, etc. Again, the calculation is still possible, but it becomes more difficult than it needs to be.
Use charge/mass ratios. This method measures particle curvature upon entering calibrated electric and magnetic fields. Curvature magnitude would indicate proton mass compared with electron mass. The experimental idea is similar to influencing the motion of a rolling ball. A constant mechanical force will deflect a heavy watermelon (proton) from straight-line travel to a small extent. The same force will deflect a light golf ball (electron) much more.
The molar mass method ignores electron-mass. Electrons are about 1/1837 as massive as protons, and there’s only one electron per hydrogen atom. Scientific notation such as “1.6737e-24” is convenient for very large or very small numbers. In decimal notation, one proton weighs 0.0000000000000000000000016737 g.