Chemistry - Bohr’s model for hydrogen atom

Bohr’s model for hydrogen atom : 

Niels Bohr (1913) put forth his postulates about the atomic model for hydrogen. While doing so he used the quantum theory, wave particle duality of electromagnetic radiation and the emission line spectra of hydrogen.

Postulates of Bohr atomic theory :

Bohr’s model of hydrogen atom is based on the following postulates.

1. The electron in the hydrogen atom can move around the nucleus in one of the many possible circular paths of fixed radius and energy. These paths are called orbits, stationary states 

or allowed energy states. These orbits are arranged concentrically around the nucleus in 

an increasing order of energy.

2. The energy of an electron in the orbit does not change with time. However, the electron will move from a lower stationary state to a higher stationary state if and when the required amount of energy is absorbed by the electron. Energy is emitted when electron moves from a higher stationary state to a lower one. The energy change does not take place in a continuous manner.

3. The frequency of radiation absorbed or emitted when transition occurs between two stationary states that differ in energy by ∆E is given by the following expression

Where E1 and E2 are the energies of the lower and higher allowed energy states respectively. This expression is commonly known as Bohr’s frequency rule.

4. The angular momentum of an electron in a given stationary state can be expressed as 

Thus, an electron can move only in those orbits for which its angular momentum is integral multiple of h/2π. Thus only certain fixed orbits are allowed here.

Results of Bohr’s theory : 

Bohr’s theory is used to derive the energies of orbits, that is, the stationary states, in hydrogen atom. The results of Bohr’s theory for hydrogen atom are summarized here.

a. The stationary states for electron are numbered n = 1, 2, 3....... .These integers are known as principal quantum numbers.

b. The radii of the stationary states are

where ao= 52.9 pm (picometer). Thus, the radius of the first stationary state, called the Bohr radius is 52.9 pm.

c. The most important property associated with the electron is the energy of its stationary state. It is given by the expression.

RH is the Rydberg constant for hydrogen and
its value in joules is 2.18 × 10^-18 J
The lowest energy state is called the ground 
state. Energy of the ground 1/1^2 state is

Energy of the stationary state corresponding to
n = 2 is

E2 = -2.18 × 10^-18 (1/(2)^2) = -0.545 × 10^-18 J

d. Bohr theory can be applied to hydrogen like species. For example He⊕, Li2⊕, Be3⊕ and so on. Energies of the stationary states associated
with these species are given by :

and radii by the expression

where Z is the atomic number. From the above expression it can be seen that the energy decreases (becomes more negative) and radius
becomes smaller as the value of Z increases.

e. Velocities of electrons can also be calculated from the Bohr theory. Qualitatively it is found that the magnitude of velocity of an electron increases with increase of Z and decreases with increase in the principal quantum number n.