Chapter 4, Problem 1Q

To determine

The assumptions of Bohr’s model of atom, its significance and its contradiction with classical physics.

Answer to Problem 1Q

The significance of Bohr’s model is that it predicts the allowed energy levels of the hydrogen atom. However, it contradicts with the fact that orbiting electron doesn’t radiate energy.

Explanation of Solution

The assumptions of Bohr’s model are given as follows:

• Electrons in an atom orbit around the nucleus in circular orbits.
• Not all but only certain orbits are stable. Electrons in stable orbits do not radiate energy. The allowed orbits are also called stationary orbits.
• When an electron jumps from a higher to lower orbit, photon of energy same as the energy difference between the two levels is emitted.

The energy of the emitted photon is,

    $hf=E_{\text{i}}-E_{\text{f}}$

Here, $h$ is the Planck’s constant, $f$ is the frequency, $E_{\text{i}}$ is the energy of the initial orbit and $E_{\text{f}}$ is the energy of the final orbit.

• The angular momentum of the electron in a stationary orbit is an integral multiple of $\hslash$.

The electron’s orbital angular momentum is,

    $mvr=n\hslash$

Here, $\hslash$ is the reduced Planck’s constant, $m$ is the mass of the electron, $v$ is the velocity of the electron, $r$ is the radius of its orbit and $n$ is the principal quantum number.

Using the above, assumptions, the allowed energy levels and hence the emission and absorption wavelengths of the hydrogen atom can be calculated.

Conclusion:

Even though the energy levels can be predicted, one of the limitations of the Bohr’s model is that it assumes that orbiting electrons don’t radiate energy. According to classical electromagnetism, any accelerated charge must radiate energy.

Therefore, Bohr’s atomic model contradicts with the classical physics.