Question

Explain postulates of Bohr's model for Hydrogen atom. Show the number of lines in the (A) emission and (B) absorption spectra of Hydrogen atom corresponding to transition between energy states \(n = 1\) and \(n = 4\).

Hint:

The number of spectral lines increases with the number of possible energy level transitions.

Answer 1

Bohr’s Postulates for Hydrogen Atom:

(A) The electron in the atom revolves in specific orbits without radiating energy.
(B) The electron can only occupy discrete energy levels or orbits. The energy of the electron in the n-th orbit is given by:

\[ E_n = - \frac{k e^2}{2r_n} \]

where k is Coulomb's constant, e is the electron charge, and r_n is the radius of the n-th orbit.

(C) The transition of the electron between orbits results in the emission or absorption of energy corresponding to the difference in energy levels.

Emission and Absorption Spectra:

For transition between n = 1 and n = 4, the emission and absorption spectra can be derived using the following formulas:

1. Emission:

When the electron transitions from a higher to a lower energy state, the frequency of the emitted radiation is:

\[ f = \frac{E_{\text{higher}} - E_{\text{lower}}}{h} \]

2. Absorption:

The electron absorbs energy when it moves to a higher energy state.

(A) Emission Spectrum Lines:

The number of lines in the emission spectrum is given by the number of possible transitions from n=4 to n=1, including intermediate states. The transitions are as follows:

\[ 4 \to 3, \quad 4 \to 2, \quad 4 \to 1, \quad 3 \to 2, \quad 3 \to 1, \quad 2 \to 1. \]

There are a total of 6 emission lines.

(B) Absorption Spectrum Lines:

Similarly, for absorption, the number of lines will be the same, as absorption also happens when the electron moves from a lower to a higher energy state. The number of lines is 6.