Question

(a) Explain various series of spectral lines of spectrum of hydrogen atom and draw energy level diagram.

(b) What are limitations of Bohr’s theory?

Answer 1

(a) According to the equation

\(\overrightarrow v = R_H(\frac{1}{n_1^2} - \frac{1}{n_2^2})\)

where n₂ = 2, 3, 4, ..........

1. Lyman series: When an electron jumps from any higher orbit to the first orbit, i.e. if n1 = 1, and n2 = 2, 3, 4, ........., the series obtained is called Lyman series. It is given by

\(\overrightarrow v = R_H(\frac{1}{1^2} - \frac{1}{n_2^2})\)

It lies in the ultraviolet region.

2. Balmer series: When an electron jumps from any higher orbit to the second orbit, i.e. if n₁ = 2, and n₂ = 3, 4, 5 ......, the series obtained is called Balmer series. It is given by

\(\overrightarrow v = R_H(\frac{1}{2^2} - \frac{1}{n_2^2})\)

It lies in the visible region.

3. Paschen series: When an electron jumps from any higher orbit to the 3rd orbit, i.e. if n₁ = 3, and n₂ = 4, 5, 6, ........., the series obtained is called Paschen series. It is given by

\(\overrightarrow v = R_H(\frac{1}{3^2} - \frac{1}{n_2^2})\)

It lies in the infrared region.

4. Brackett series: When an electron jumps from any higher orbit to the 4th orbit, i.e. if n₁ = 4, and n₂ = 5, 6, 7, ..........., the series obtained is called Brackett series. It is given by

\(\overrightarrow v = R_H(\frac{1}{4^2} - \frac{1}{n_2^2})\)

It lies in the infrared region.

5. Pfund series: When an electron jumps from any higher orbit to the 5th orbit, i.e. if n₁ = 5, and n₂ = 6, 7, 8, .........., the series obtained is called Pfund series. It is given by

\(\overrightarrow v = R_H(\frac{1}{5^2} - \frac{1}{n_2^2})\)

It lies in the infrared region.

Energy Level Diagram of H-atom

The energy level diagram of various permitted stationary orbit is as shown in Fig. It is called Gossel's diagram.

The total energy of an electron in the nth orbit of hydrogen atom is

The energy of H-atom is the lowest when principal quantum number n = 1.

E₁ = \(-\frac{13.6}{(1)^2}\) eV

It is called ground state energy of H-atom. The state of the hydrogen when n = 1 is called normal or ground state.

The higher energy values of H-atom corresponding to n = 2, 3, ........ are called excited state energies. The state corresponding to n = 2 is called first excited state.

The energy of 1st excited state

E₂ = \(\frac{-13.6}{(2)^2}\) eV = -3.4 eV

The state corresponding to n = 3 is called second excited state.

E₃ = \(\frac{-13.6}{32}\) eV = -1.51 eV

The energies of 3rd, 4th, .......... excited states of H-atom are -85 eV, -54 eV ..........

(b) Limitations of Bohr theory

(a) When Bohr's theory is applied to atom with more than one electron, the predictions do not agree with experiments entirely. Certain perdicted lines are not observed at all.

(b) When the spectral lines are studied with a high resolution spectrograph, each line is found to consist of a group of several lines very close together. This is called fine structure of hydrogen lines. Bohr's theory does not explain this structure.

(c) It is observed experimentally that some spectral lines are stronger and others are weaker in intensity. Bohr's theory does not predict anything about the relative intensity of spectral lines.

(d) Bohr's postulates look arbitrary. There is no reason for assuming the orbit to be circular and having an angular momentum equal to integral multiple of h/2π.