Atoms — Part 2: The Bohr model and the hydrogen spectrum: visual lesson, CBSE Class 12 Physics

Lesson

The Bohr model

Part 2 — three bold postulates give the sizes and energies of hydrogen’s orbits and explain the bright lines in its spectrum.

In this lesson you will

State Bohr’s postulates
Derive the radius, speed and energy of the electron in the nth orbit of hydrogen
Explain the hydrogen line spectrum using energy levels

Levels from n = 5 upwards crowd just below 0 eV and are not drawn.

Concept

Bohr’s postulates

For the hydrogen atom:

1. Stationary orbits: the electron moves in certain circular orbits without radiating energy.
2. Quantised angular momentum: in these orbits $mvr = \dfrac{nh}{2\pi}$ , $n = 1, 2, 3, \dots$
3. Frequency condition: when the electron jumps from a higher orbit to a lower one, it emits a photon with $h\nu = E_i - E_f$ .

Concept

Radius and speed in the nth orbit

The Coulomb force provides the centripetal force, and angular momentum is quantised: $\dfrac{mv^2}{r} = \dfrac{1}{4\pi\varepsilon_0}\dfrac{e^2}{r^2},\qquad mvr = \dfrac{nh}{2\pi}$ Solving: $v_n = \dfrac{e^2}{2\varepsilon_0 nh},\qquad r_n = \dfrac{n^2 h^2\varepsilon_0}{\pi m e^2}$

With the constants, $r_1 \approx 5.3 \times 10^{-11}$ m (the Bohr radius, 0.53 Å) and $v_1 \approx 2.18 \times 10^{6}$ m s $^{-1}$ .
So $r_n = n^2 r_1$ and $v_n = v_1/n$ .

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