Explain briefly the results of Bohr’s theory for hydrogen atom.

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

ii. The radii of the stationary states are r_n = n²a₀, where a₀ = 52.9 pm (picometer). 

Thus,

The radius of the first stationary state, called the Bohr radius is 52.9 pm.

iii. The most important property associated with the electron is the energy of its stationary state. It is given by the formula :

E_n = -R_H(1/n²), where n = 1, 2, 3, …..

R_H is the Rydberg constant for hydrogen and its value in joules is 2.18 × 10^-18J.

The lowest energy state is called the ground state. 

Energy of the ground state (n = 1) is :

E₁ = -2.18 × 10^-18 × 1/1² = -2.18 × 10^-18 J

Energy of the stationary state corresponding to n = 2 is :

E₂ = -2.18 × 10^-18 × (1/(2)²) 

= -0.545 × 10^-18J.

iv. Bohr theory can be applied to hydrogen like species. 

For example, 

He⁺, Li²⁺, Be³⁺ and so on. 

Energies and radii of the stationary states associated with these species are given by :

E_n = - 2.18 x 10^-18\((\frac{Z^2}{n^2})\) J and r_n \(\frac{52.9(n^2)}{Z}\)pm

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

v. 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).