atom, hydrogen It consists of a single proton at the nucleus and one electron. On the basis of Bohr's theory the energy levels of hydrogen atom can be calculated.

Bohr's theory : The theory is based on the following two hypotheses .

(i) The electrons can revolve only in discrete orbits such that the angular momentum of the electrons are integral multiple of Planck's constant, h divided by 2p . This implies that,

mv_nr_n = nh

where v_n and r_n are velocity of the electron in the nth orbit and radius of the nth orbit respectively.

(ii) The electrons may undergo transition from one energy level to another at a lower level. The difference of energy is emitted as photon of frequency, n such that,

E_n - E_m = hn

The centripetal force of the revolving electron is provided by the Coulomb attraction between the nucleus and the electron. Hence,

(a33)

Simultaneous solution of these equations gives,

r_n = 0.053 n² [nm] (a34)

and E_n = - [eV] (a35)

Using eq.(a35) we can obtain, the expression for wavelength when the electron makes a transition from mth to nth level.

(a36)

where R = 1.0974´ 10⁷ m^-1 is called the Rydberg's constant.

Spectral lines : The spectral lines emitted by excited hydrogen atom occur in series. Lyman series is in the ultraviolet range. It arises due to transition of electron from higher levels to the ground state, n=1. The wavelengths of spectral lines can be obtained using (a36),where m = 2,3,4...Similarly `Balmar series' arises for transition to n=2 state. `Paschen series' arises for transition to n=3 state.

Quantum theory of Hydrogen atom : The hydrogen atom which was described by simple Bohr's theory could not explain all the lines obtained in the spectra of hydrogen atom.This was explained using the Quantum theory. In this theory ( details are beyond the scope of this book) an electron in an atom is described by a set of four numbers called quantum numbers.These are as follows :

1. Principal quantum number, n : It defines a principal electron orbital or shell. n takes integer values 1,2,3,4 ..etc. The larger is the value of n,the further removed is the shell from the nucleus.

2. Orbital angular momentum quantum number l : It take integer values from l =0 to l = n-1. The values of l define suborbitals (or subshells), which are denoted by spectrographic notations ;

for l = 0 1 2 3 4

notations s p d f g

3. Magnetic quantum number m_l ; Defines the spatial orientation of the sub orbitals. m can take the values, -l ,-l+1, ... -1 , 0, 1, 2, ..l-1, l that is (2l+1) values for any l.

4. Electron spin quantum number , m_l : It defines the orientation of the spin of the electron. It can have only two values , +1/2 and -1/2 .