Question

Give the postulates of Bohr's atomic model and draw the energy level diagram for the first three lines of the Balmer series in the hydrogen atomic spectrum.

Hint:

State Bohr's three postulates (stable orbits, quantised angular momentum \( mvr = nh/2\pi \), photon on transition); Balmer lines are jumps to \( n=2 \) from \( n = 3, 4, 5 \).

Answer 1

Step 1: Postulates of Bohr's atomic model.
(1) Stable orbits: An electron revolves around the nucleus only in certain fixed circular orbits without radiating energy. These are called stationary (stable) orbits.
(2) Quantisation of angular momentum: The allowed orbits are those in which the electron's angular momentum is an integral multiple of \( \dfrac{h}{2\pi} \):
\[ mvr = \frac{nh}{2\pi}, \quad n = 1, 2, 3, \dots \]
(3) Emission/absorption of energy: Energy is emitted or absorbed only when an electron jumps from one orbit to another. If it falls from a higher level \( E_2 \) to a lower level \( E_1 \), the frequency of the emitted photon is
\[ h\nu = E_2 - E_1 \]

Step 2: Energy of the levels.
For hydrogen the energy of the \( n \)-th level is
\[ E_n = -\frac{13.6}{n^2}\ \text{eV} \]
giving \( E_2 = -3.40\ \text{eV},\ E_3 = -1.51\ \text{eV},\ E_4 = -0.85\ \text{eV},\ E_5 = -0.54\ \text{eV} \).

Step 3: Balmer series.
The Balmer series (visible region) is produced when electrons jump from higher orbits down to \( n = 2 \). The first three lines are:
• Hα : \( n=3 \to 2 \), \( \Delta E = 1.89\ \text{eV} \), \( \lambda \approx 6563\ \text{Å} \) (red).
• Hβ : \( n=4 \to 2 \), \( \Delta E = 2.55\ \text{eV} \), \( \lambda \approx 4861\ \text{Å} \) (blue-green).
• Hγ : \( n=5 \to 2 \), \( \Delta E = 2.86\ \text{eV} \), \( \lambda \approx 4340\ \text{Å} \) (violet).

Step 4: Energy level diagram (described).
Draw a set of horizontal lines, one above the other, marking the energy levels. The lowest drawn line is \( n=2 \) at \( -3.40\ \text{eV} \); above it \( n=3 \) at \( -1.51\ \text{eV} \), then \( n=4 \) at \( -0.85\ \text{eV} \), and \( n=5 \) at \( -0.54\ \text{eV} \); the spacing between the lines gets smaller as \( n \) increases. Now draw three downward vertical arrows, all ending on the \( n=2 \) line: one starting from \( n=3 \) (Hα), one from \( n=4 \) (Hβ) and one from \( n=5 \) (Hγ). Each arrow represents an emitted photon of the Balmer series, the longer jumps giving photons of higher energy (shorter wavelength).

\[\boxed{\text{Balmer lines: } n = 3, 4, 5 \to n = 2}\]