Using the Bohr model of the hydrogen atom, show that the...

Using the Bohr model of the hydrogen atom, show that the speed of an electron in an orbit is given by $v_{n}=2.2 \times 10^{6} / n \mathrm{m} / \mathrm{s} .$ Here $n$ is the quantum number in the Bohr model.

Key Concepts

Quantum Number

The quantum number 'n' in the Bohr model specifies the energy level or orbit in which the electron resides. It is an integer that determines not only the radius of the electron’s orbit but also contributes inversely to the electron’s velocity, explaining how the speed decreases as the orbit number increases.

Electrostatic and Centripetal Force Balance

In the Bohr model, the electron’s circular orbit is maintained by the balance between the centripetal force required for circular motion and the electrostatic (Coulomb) attraction force between the negatively charged electron and the positively charged nucleus. Setting these two forces equal provides a relationship between the electron’s speed and the orbital radius.

Quantization of Angular Momentum

A key concept in the Bohr model is that the angular momentum of an electron in orbit is quantized, meaning it can only take on specific values. This quantization is expressed by the relation that the product of the mass, velocity, and orbital radius is an integer multiple of a fundamental constant, which underpins the derivation of electron speeds in different orbits.

Bohr Model

The Bohr model describes the hydrogen atom as an electron orbiting a nucleus in fixed, circular paths without radiating energy. It introduced the idea that electrons can only occupy certain allowed orbits, and that transitions between these orbits occur with the emission or absorption of discrete amounts of energy.

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