The Bohr Model of Hydrogen-like Atoms

An outer electron in an atom is subject to a force due to a net positive central charge of Z of

F = −K/r²

The angular momentum the outer electron is quantized; i.e.,

mωr² = nh

where m is the mass of the electron, ω is the angular rotation rate, r is the orbit radius, n is an integrer (called the principal quantum number) and h is Planck's constant divided by 2π.

A balance of the electrostatic force with the centrifugal force due to revolution about the nucleus results in the orbit radius of

r_n = n²h²/(mKZ)

The quantized total energy of the electron is then

E_n = mZ²K²/(2²hn²)

The principal quantum number n is not the only quantum number for an electron state. There are four quantum numbers including n. The other three are called the orbital quantum number l, the magnetic quantum number m and the spin number s. For a given principal quantum number n the magnetic quantum number m can take on values from 0 to (n-1). The orbital quantum number l can take on any number from −m to +m. Thus there are 2m+1 possible values of the orbital quantum number l. The spin quantum number can take on the values of ±½. The total number of possible states for a given value of n is then twice the sum 1+3+5+…(n-1). The sum of the first (n-1) odd numbers is n² and hence the total possible number of states for a principal quantum number n is 2n².

The Empirical Levels of the Filled Electron Shells

The noble gases are the ones that do not enter into any chemical reaction. This includes helium, neon,argon, krypton, xenon and radon. The atomic numbers of these gases are {2, 10, 18, 36, 54, 86}. These can be called the magic numbers for electron shells. The successive differences of these numbers are the capacities of the various electron shells; i.e., {2, 8, 8, 18, 18, 32}, including 2 as the capacity of the first filled shell.

There are major puzzles in these numbers. It appears that the principal quantum number for the third shell is the same as that of the second and likewise for the fifth and fourth shells. Such however is not the case.

Resolution of the Anomalies

Electron states are filled upwards from the lowest (most negative) energy values. In the Bohr model the energy of an electron state depens only upon the prinicipal quantum number. More generally the energy depends also upon the magnetic quantum number m. The computation of the energies of possible states as a function of n and m is complicated and must be done numerically. Douglas Hartree developed an iterative method for making that calculation that gave results of sufficient accurary to demonstrate that the energy states for quantum numbers n=3 and m=2 is higher (less negative) than the energy for the states with n=4 and m=0. Thus the energy states for the third shell (n=3) are filled only for m=0 and m=1. There are two states for m=0 and six for m=1. Thus the capacity of the third shell is eight rather than eighteen. That is to say, the capacity of the third shell is twice the sum of the first two odd numbers rather than the first three. Likewise for n=4 (fourth shell) there are not 32=2·4² states but only 18=2·3² because for n=4 and m=3 the energy is higher (less negative) than for n=5 and m=0. Hence the capacity of the fourth shell is twice the sum of the first three odd numbers; i.e., 2(1+3+5).