Problem 50 - Chemistry The Science in Context

Question

Without calculating any frequency values, rank the following transitions in the hydrogen atom in order of increasing frequency of the electromagnetic radiation that could produce them. a. from \(n=4\) to \(n=6\) b. from \(n=6\) to \(n=8\) c. from \(n=9\) to \(n=11\) d. from \(n=11\) to \(n=13\)

Step-by-Step Solution

Answer

Question: Rank the following transitions in the hydrogen atom from highest to lowest frequency: a) from n=4 to n=6, b) from n=6 to n=8, c) from n=9 to n=11, d) from n=11 to n=13. Answer: a, b, c, d.

Step 1: Deriving energy differences for each transition

First, let's find the energy difference for each transition: a. from \(n=4\) to \(n=6\): \(\Delta E_a = -13.6 eV \left(\frac{1}{6^2} - \frac{1}{4^2}\right)\) b. from \(n=6\) to \(n=8\): \(\Delta E_b = -13.6 eV \left(\frac{1}{8^2} - \frac{1}{6^2}\right)\) c. from \(n=9\) to \(n=11\): \(\Delta E_c = -13.6 eV \left(\frac{1}{11^2} - \frac{1}{9^2}\right)\) d. from \(n=11\) to \(n=13\): \(\Delta E_d = -13.6 eV \left(\frac{1}{13^2} - \frac{1}{11^2}\right)\)

Step 2: Comparing energy differences

Now, let's compare the energy differences: \(\Delta E_a > \Delta E_b > \Delta E_c > \Delta E_d\). The energy difference between the energy levels decreases as the initial and final levels increase.

Step 3: Relate the energy difference to frequency

As mentioned in the analysis, we know that the higher the energy difference, the higher the frequency. Therefore, the order of the frequency for the given transitions is as follows: \(\nu_a > \nu_b > \nu_c > \nu_d\) So, from highest to lowest frequency, the transitions are ranked: a, b, c, d.

Key Concepts

Energy Levels

Energy levels in a hydrogen atom represent different states in which an electron can exist. These levels are quantified by principal quantum numbers denoted as \( n \). The lower the value of \( n \), the closer the electron is to the nucleus and the lower the potential energy of the electron. Each energy level has a distinct, fixed amount of energy, and the electron can only inhabit these specified levels. When an electron moves between these energy levels, it does so through transitions.

• Higher energy levels (larger \( n \)) have energy levels closer together compared to lower levels.
• The energy difference between levels decreases as \( n \) increases.

Understanding how these energy levels work is crucial since they dictate the properties and behavior of the electrons in an atom.

Electromagnetic Radiation

Electromagnetic radiation refers to the waves of the electromagnetic field, propagating through space. When electrons in a hydrogen atom transition between energy levels, they absorb or emit electromagnetic radiation. This radiation is what we perceive as light or other forms of energy.

• In the context of atoms, this radiation is most commonly in the form of visible light, ultraviolet light, or other forms on the electromagnetic spectrum.
• Each transition corresponds to a specific wavelength or frequency of radiation.

The type and amount of electromagnetic radiation depend on the energy difference between the initial and final states of the electron. More substantial energy changes result in higher-frequency radiation.

Frequency

Frequency is a fundamental property of electromagnetic waves, referring to the number of wave cycles that pass a given point per second. It is measured in Hertz (Hz). In the realm of atomic transitions, the frequency \( u \) of the electromagnetic radiation is directly linked to the energy difference \( \Delta E \) between two energy levels through Planck's equation: \( E = h u \), where \( h \) is Planck's constant.

• The greater the energy difference during an electron transition, the higher the frequency of the electromagnetic radiation emitted.
• This means that transitions involving higher energy differences will produce radiation with higher frequency.

For students learning about atomic structures, understanding frequency helps make connections between energy changes and observable radiation.

Rydberg Formula

The Rydberg formula is a mathematical equation used to predict the wavelength (or frequency) of light resulting from an electron moving between different energy levels in a hydrogen atom. It is expressed as:\[\frac{1}{\lambda} = R_H \left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right)\]where \( \lambda \) is the wavelength, \( R_H \) is the Rydberg constant for hydrogen, \( n_1 \) and \( n_2 \) are the principal quantum numbers of the initial and final energy levels.

• The formula helps in determining the specific spectral lines of hydrogen.
• It shows that shorter wavelengths (or higher frequencies) correspond to larger energy level differences, supporting the relationship seen in frequency.

The Rydberg formula is a crucial tool in understanding spectral lines and the quantized nature of energy in atomic physics.