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

The existence of line spectra is consistent with Bohr's theory of quantized energies for the electron in the hydrogen atom because the observed discrete wavelengths in the line spectra correspond to the specific energy level transitions of the electron. Bohr's model asserts that the electron in a hydrogen atom orbits the nucleus at certain fixed energy levels and can only move between these levels by absorbing or emitting a photon. As a result, when an electron transitions from a higher energy level to a lower one, it releases a photon with an energy equal to the difference between those energy levels, which corresponds to a specific wavelength of emitted light. Since the energy levels are quantized, only certain energy differences can occur, leading to the distinctive line spectra observed.

Step-by-step solution

Understanding Line Spectra

Line spectra, also known as atomic spectra or emission spectra, are unique patterns of spectral lines produced when atoms emit light. Each element has its own characteristic line spectra, which can be used to identify the element. When an atom in an excited state (high energy) returns to a lower energy state, it releases a photon with a specific energy, which corresponds to a specific wavelength of light. The collection of these wavelengths forms the line spectra.

Introducing Bohr's Theory and Quantized Energies

In 1913, Niels Bohr developed a model for the hydrogen atom to explain the observed line spectra. According to Bohr's theory, the electron in a hydrogen atom orbits the nucleus at certain fixed energy levels, also known as orbits or shells. An electron can move between these energy levels by either absorbing or emitting a photon, but it cannot exist between the energy levels. The energy levels are quantized, meaning that the electron can only move to discrete energy levels and not to any arbitrary energy state.

The Bohr Formula for Hydrogen Atom's Energy

Bohr derived a formula for the energy levels of the hydrogen atom: \[E_n = -\frac{13.6 eV}{n^2}\] Here, \(E_n\) represents the energy of the electron in the \(n\)th energy level, and \(n\) is an integer value greater than or equal to 1 (known as the principal quantum number). The negative sign indicates that the electron is bound to the nucleus.

Connecting Line Spectra to Bohr's Quantized Energy Levels

When the electron transitions from a higher energy level (initial level, \(n_i\)) to a lower energy level (final level, \(n_f\)), it releases a photon with an energy equal to the difference between those energy levels: \[\Delta E = E_f - E_i = -\frac{13.6 eV}{n_f^2} - (-\frac{13.6 eV}{n_i^2})\] This energy difference corresponds to a specific wavelength of emitted light. Since the energy levels are quantized, only certain energy differences can occur, which leads to the discrete wavelengths observed in the line spectra. This relationship can be further described by the equation: \[\frac{1}{\lambda} = R_H (\frac{1}{n_f^2} - \frac{1}{n_i^2})\] Where \(\lambda\) is the wavelength of the emitted light, and \(R_H\) is the Rydberg constant for hydrogen. The different spectral series (Lyman, Balmer, Paschen, etc.) correspond to different values of \(n_f\). In conclusion, the existence of line spectra is consistent with Bohr's theory of quantized energies for the electron in the hydrogen atom because the observed discrete wavelengths in the line spectra correspond to the specific energy level transitions of the electron. The quantized energy levels in Bohr's model explain why only specific wavelengths of light are emitted, resulting in the line spectra observed.