Question

Explain how the existence of line spectra is consistent with Bohr's theory of quantized energies for the electron in the hydrogen atom.

Short answer

In Bohr's theory, the hydrogen atom consists of an electron orbiting a nucleus in fixed circular orbits with quantized energies given by \[E_n = -\dfrac{13.6\, eV}{n^2}\], where \(n\) is the principal quantum number. When an electron transitions from a higher to a lower energy level, it releases energy in the form of a photon with wavelength \(\lambda\) related to the quantized energies by \[\dfrac{1}{\lambda} = R_H\left(\dfrac{1}{n_f^2} - \dfrac{1}{n_i^2}\right)\], where \(R_H\) is the Rydberg constant for hydrogen. The fixed wavelengths of emitted photons correspond to the observed line spectra, consistent with Bohr's theory of quantized energies for the electron in the hydrogen atom.

Step-by-step solution

Understanding Bohr's theory of the hydrogen atom

According to Bohr's theory, the hydrogen atom consists of a nucleus (proton) and an electron revolving around the nucleus in fixed circular orbits. The theory assumes that the electron orbits have quantized energies and angular momenta.

Explaining the quantization of energy in Bohr's theory

According to Bohr's theory, an electron can only exist in specific energy levels, and it cannot have energy values between these levels. These energy levels are given by the expression: \[E_n = -\dfrac{13.6\, eV}{n^2}\] where \(E_n\) is the energy of the nth energy level, and \(n\) is the principal quantum number, which takes integer values \(n = 1, 2, 3, ...\)

Understanding the emission of energy and line spectra

When an electron in the hydrogen atom transitions from a higher energy level (\(n_i\)) to a lower energy level (\(n_f\)), it releases energy in the form of a photon. The energy of the emitted photon can be calculated using the energy difference between the initial and final energy levels: \[\Delta E = E_{n_f} - E_{n_i} = h\nu\] where \(\Delta E\) is the energy difference, \(h\) is Planck's constant, and \(\nu\) is the frequency of the emitted photon. This energy difference corresponds to specific wavelengths of light, leading to the formation of line spectra.

Relating quantized energies to line spectra

From Bohr's theory, the quantized energies of the electron in the hydrogen atom can be related to the line spectra as follows: \[\dfrac{1}{\lambda} = R_H\left(\dfrac{1}{n_f^2} - \dfrac{1}{n_i^2}\right)\] where \(\lambda\) is the wavelength of the emitted photon, and \(R_H\) is the Rydberg constant for hydrogen (\(R_H \approx 1.097 \times 10^7 \, m^{-1}\)). This equation demonstrates how the line spectra observed in the hydrogen atom can be explained by the quantized energies of the electron, as predicted by Bohr's theory. When an electron transitions between two energy levels, it emits a photon with a wavelength that corresponds to the energy difference between the levels. Since the energy levels are quantized, the emitted photon's wavelength will also be fixed, resulting in the line spectra observed in experiments.