Bohr’s theory of the atomic spectrum of hydrogen

In 1913, Niels Bohr combined elements of quantum theory and classical physics in a treatment of the hydrogen atom. He stated two postulates for an electron in an atom: . Stationary states exist in which the energy of the electron is constant; such states are characterized by circular orbits about the nucleus in which the electron has an angular momentum mvr given by equation 1.6. The integer, n, is the principal quantum number.

where m = mass of electron; v = velocity of electron; r = radius of the orbit; h=  the Planck constant; h/2π may be written as ћ .

Energy is absorbed or emitted only when an electron moves from one stationary state to another and the energy change is given by equation 1.7 where n₁ and n₂ are the principal quantum numbers referring to the energy levels En₁ and E_n2 respectively.

If we apply the Bohr model to the H atom, the radius of each allowed circular orbit can be determined from equation 1.8. The origin of this expression lies in the centrifugal force acting on the electron as it moves in its circular orbit; for the orbit to be maintained, the centrifugal force must equal the force of attraction between the negatively charged electron and the positively charged nucleus.

From equation 1.8, substitution of n = 1 gives a radius for the first orbit of the H atom of 5.293× 10 ^-11 m, or 52.93 pm. This value is called the Bohr radius of the H atom and is given the symbol a₀. An increase in the principal quantum number from n= 1 to n=∞ has a special significance; it corresponds to the ionization of the atom (equation 1.9) and the ionization energy, IE, can be determined by combining equations 1.5 and 1.7, as shown in equation 1.10. Values of IEs are quoted per mole of atoms:

Although the SI unit of energy is the joule, ionization energies are often expressed in electron volts (eV) (1 eV = 96.4853 ≈ 96.5 kJ mol^-1). Impressive as the success of the Bohr model was when applied to the H atom, extensive modifications were required to cope with species containing more than one electron; we shall not pursue this further here.