Modifications of the Bohr Model

Despite the success of the Bohr model, there were some serious shortcomings present. On the experimental side, more detailed analysis of the emission spectra for hydrogen found a single emission line was actually at times composed of two or more closely spaced lines, a feature not present in the Bohr model. Theoretically, the Bohr model mixes up a particle and wave picture of electrons, which was considered by many to be unsatisfactory. For these reasons a better treatment of the hydrogen atom was sought where the electron is considered as a wave from the outset. This theory, developed by Heisenberg, Pauli, Schrödinger, Sommerfeld, and others, is fairly detailed mathematically. The main result coming from this theory is that there are four quantum numbers describing a state of an electron, compared to the one quantum number n present in the Bohr model. These quantum numbers are as follows:

n

, the principal quantum number. This corresponds to n of the Bohr model, and can assume values 1,2,3,....

l

, the orbital quantum number. This is a label characterizing the magnitude of the angular momentum of the electron. For a given n , l can assume values 0,1,2,...,n - 1 .

m_l

, the spin orbital quantum number. This is a label characterizing a component of the angular momentum vector of the electron. For a given l , m_l can assume values - l, - l + 1,..., - 1,0,1,...,l - 1,l .

m_s

, the spin quantum number. This is a label which, in a very limited sense, can be considered as characterizing the direction that the electron is spinning on its axis. m_s can assume one of two values, $\pm$ ${\frac{1}{2}}$ .

Thus, for a given n , there can be 2n ² states with different values of l , m_l and m_s .

Historically, the principal quantum number n labels what is called the shell, and the n = 1,2,3,... shell is sometimes referred to as the K,L,M,... shell. The orbital quantum number l labels the subshell, and the l = 0,1,2,3,4,... subshell is also referred to as the s,p,d,f,... subshell.

If we imagine starting to add electrons in order to form various atoms, then one would expect all the electrons to go into the lowest energy state possible, which is n = 1 and l = 0 = m_l . This does not happen in nature, however. Pauli explained this by postulating that electrons obey what is now called the Pauli exclusion principle:

No two electrons in a system may have the same sets of quantum numbers.

Table 28.1 indicates what happens as we begin to add one additional electron in turn, taking into account this principle.

 

# of e ^-'s n l m_l m_s

$${\frac{1}{2}}$$ 1 1

$${\frac{1}{2}}$$ 2 1

$${\frac{1}{2}}$$ 3 2

$${\frac{1}{2}}$$ 4 2

$${\frac{1}{2}}$$ 5 2 1 - 1

$${\frac{1}{2}}$$ 6 2 1 - 1

$${\frac{1}{2}}$$ 7 2 1 0

$${\frac{1}{2}}$$ 8 2 1 0

$${\frac{1}{2}}$$ 9 2 1 + 1

$${\frac{1}{2}}$$ 10 2 1 + 1

$$\vdots \vdots \vdots \vdots \vdots$$

We thus see that we have to fill up more and more shells as we add more and more electrons. As we shall see later in some examples, this behaviour explains qualitatively the structure of the periodic table, and provided another success of the early quantum theory.