Problem 32 - General Chemistry: Principles and Modern Applications

Question

Without doing detailed calculations, indicate which of the following electron transitions in the hydrogen atom results in the emission of light of the longest wavelength. (a) \(n=4\) to \(n=3 ;\) (b) \(n=1\) to \(n=2\) (c) \(n=1\) to \(n=6 ;\) (d) \(n=3\) to \(n=2\).

Step-by-Step Solution

Answer

The transition that produces the light of the longest wavelength is \(n=3\) to \(n=2\).

Step 1: Understand the Relationship between Energy, Frequency, and Wavelength

The energy of a photon is given by the equation \(E=hf\), where \(h\) is Planck's constant and \(f\) is the frequency of the light. We also know that the frequency of light is inversely proportional to the wavelength, and so this can also be written as \(E=hc/\lambda\), where \(c\) is the speed of light and \(\lambda\) is the wavelength.

Step 2: Compare the Energy Changes for Each Transition

The energy levels of the hydrogen atom are given by \(E=-13.6/n^{2} eV\), where \(n\) is the principal quantum number. The energy difference between two levels is given by \(ΔE=E_{f}-E_{i}\), where \(E_{f}\) and \(E_{i}\) represent the final and initial energy states respectively. For (a), the energy difference is \(E_3 - E_4\), for (b) it’s \(E_2 - E_1\), for (c) it’s \(E_6 - E_1\), and for (d) it’s \(E_2 - E_3\). We're looking for the smallest energy difference.

Step 3: Determine the Transition with the Smallest Energy Difference

Without detailed calculations, observe the energy levels involved in each of the transitions. (a) \(n=4\) to \(n=3\), (b) \(n=1\) to \(n=2\), (c) \(n=1\) to \(n=6\), and (d) \(n=3\) to \(n=2\). Notice that the smallest difference in the quantum numbers is for transition (b) \(n=1\) to \(n=2\), and (d) \(n=3\) to \(n=2\). Both have a change of one energy level. However, energy levels in an atom are not equally spaced; they get closer together as the energy level number increases. Therefore, the transition from \(n=3\) to \(n=2\) will have a smaller energy difference than the transition from \(n=1\) to \(n=2\).

Key Concepts

Energy Levels

Atoms, particularly hydrogen atoms, have specific locations, called energy levels, that electrons can occupy. These levels are associated with a quantum number, commonly denoted as \(n\).

In a hydrogen atom, the energy of a level is determined by the formula: \(E = -\frac{13.6}{n^2}\) eV. This means that energy levels are quantized, and the energy becomes less negative as \(n\) increases. As such, the very first level, \(n = 1\), is the most tightly bound energy state, while higher values of \(n\) represent higher energy levels that an electron can occupy.

Whenever an electron transitions between two levels, it either absorbs or emits energy. Calculating these energy differences helps us understand the nature of light produced in such processes.

Photon Emission

When an electron in a hydrogen atom transitions from a higher energy level to a lower one, it emits energy in the form of a photon. This happens because the electron has excess energy at the upper energy level, which it must release to move to a lower state.

The energy difference between the initial and final states determines the energy (and thus the frequency and wavelength) of the emitted photon. If the energy difference is large, high-energy photons like ultraviolet light are emitted. Conversely, a small energy difference results in low-energy photons like infrared light.

• Photon emission is central to how hydrogen atoms produce light in phenomena such as spectral lines.
• Lesser energy difference means larger wavelength (less energy density per photon).

The transition with the smallest energy difference is therefore associated with the emission of light of the longest wavelength.

Wavelength and Frequency Relations

Wavelength and frequency are two key characteristics of light that are inseparably linked.

The relationship between them is given by the formula: \(c = \lambda f\), where \(c\) is the speed of light, \(\lambda\) is the wavelength, and \(f\) is the frequency. This shows that wavelength is inversely proportional to frequency; when the frequency increases, the wavelength decreases and vice versa.

Consequently, in the context of electron transitions, a small energy difference leads to a low frequency and therefore a longer wavelength. This is pivotal when assessing which electron transition produces light of the longest wavelength.

• Long wavelength equates to lower energy photons.
• Visible light frequencies are mid-range, while radio waves have the longest wavelengths.

Factors like frequency and wavelength help scientists determine the type of light emitted from different electronic transitions.