For gases, the density at boiling point is used.   Size data
Copper atoms stack like cannonballs. We can calculate the atom size by assuming the atoms are shaped like either cubes or spheres. For copper atoms,
Density = D = 8900 kg/m3 Atomic mass unit= M0 = 1.661⋅10-27 kg Atomic mass = MA = 63.55 Atomic mass units Mass = M = MA⋅M0 = 9.785⋅10-26 kg Number density = N = D / M = 9.096⋅1028 atoms/m3 Cube volume = Υcube= 1 / N = 1.099⋅10-29 m3 Volume/atom if the atoms are cubes Cube length = L = Υ1/3cube = 2.22⋅10-10 m Side length of the cube Sphere fraction = f = π/(3√2) = .7405 Fraction of volume occupied by spheres in a stack o spheres Sphere volume = Υsph= Υcube f = 8.14⋅10-30 m3 = 4⁄3πR3 Volume/atom if the atoms are spheres Sphere radius = R = 1.25⋅10-10 m
The "de Broglie wavelength" of a particle is
Particle momentum = Q Planck constant = h = 6.62*10^-34 Joule seconds Particle wavelength = W = h/Q (de Broglie formula)The Bohr hypothesis states that for an electron orbiting a proton, the number of electron wavelengths is an integer. This sets the characteristic size of a hydrogen atom.
Orbit circumference = C = N W where N is a positive integer N Orbital 1 S 2 P 3 D 4 F Electron mass = m = 9.11*10-31 kg Electron velocity = V Electron momentum = Q = m V Electron charge = e = -1.60*10-19 Coulombs Coulomb constant = K = 9.0*109 Newtons meters / Coulombs2 Electric force = Fe = K e2 / R2 Centripetal force = Fc = M V2 / R Orbit radius = R = N h2 / (4 π2 K e2 m) = N * 5.29e-11 meters Electron energy = E = - .5 K e2 / (R N2) = N-2 2.18e-18 Joules = N-2 13.6 electron Volts (Ionization energy)For an electron on a circular orbit,
Fe = Fc