Problem 46 - Chemistry: The Central Science in SI Units, Global Edition

Question

Place the following transitions of the hydrogen atom in order from shortest to longest wavelength of the photon emitted: \(n=5\) to \(n=2, n=4\) to \(n=3, n=8\) to \(n=4,\) and \(n=4\) to \(n=2\).

Step-by-Step Solution

Answer

The transitions in order from shortest to longest wavelength of the photon emitted are: n=4 to n=3, n=4 to n=2, n=8 to n=4, and n=5 to n=2.

Step 1: Calculate the wavelength of each transition

Using the Rydberg formula for the hydrogen atom, we can calculate the wavelength of the emitted photon for each transition. For the transition n=5 to n=2: \[ \frac{1}{\lambda_{5 \to 2}} = R_H \left( \frac{1}{2^2} - \frac{1}{5^2} \right) \] For the transition n=4 to n=3: \[ \frac{1}{\lambda_{4 \to 3}} = R_H \left( \frac{1}{3^2} - \frac{1}{4^2} \right) \] For the transition n=8 to n=4: \[ \frac{1}{\lambda_{8 \to 4}} = R_H \left( \frac{1}{4^2} - \frac{1}{8^2} \right) \] For the transition n=4 to n=2: \[ \frac{1}{\lambda_{4 \to 2}} = R_H \left( \frac{1}{2^2} - \frac{1}{4^2} \right) \]

Step 2: Arrange the transitions in order of wavelength

Once we have calculated the wavelengths of the emitted photons for each transition, we can easily arrange them in order from shortest to longest wavelength: 1. \( \lambda_{4 \to 3} \), n=4 to n=3. 2. \( \lambda_{4 \to 2} \), n=4 to n=2. 3. \( \lambda_{8 \to 4} \), n=8 to n=4. 4. \( \lambda_{5 \to 2} \), n=5 to n=2. Thus, the transitions in order from shortest to longest wavelength of the photon emitted are n=4 to n=3, n=4 to n=2, n=8 to n=4, and n=5 to n=2.

Key Concepts

Rydberg Formula

The Rydberg Formula is a fundamental equation used in quantum physics to determine the wavelengths of light emitted from electron transitions in a hydrogen atom. It is given by:\[ \frac{1}{\lambda} = R_H \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \]where:

• \(\lambda\) is the wavelength of the emitted light.
• \(R_H\) is the Rydberg constant, approximately valued at \(1.097 \times 10^7\, \text{m}^{-1}\).
• \(n_1\) and \(n_2\) are the principal quantum numbers of the electron's initial and final energy levels, respectively.

You use this formula to calculate individual wavelengths for specific transitions. For instance, to find the wavelength for a transition from \(n = 5\) to \(n = 2\), insert the values into the formula, which involves some straightforward arithmetic using fractions of integers squared. The Rydberg Formula is an essential tool in calculating emission spectra and in various practical applications such as spectroscopy.

Photon Wavelengths

Photon wavelengths refer to the specific distances over which a light wave's shape repeats, measured in nanometers for visible light. When transitions occur between different energy levels in an atom, photons are emitted or absorbed, each corresponding to a specific wavelength of light. Understanding how to calculate proper photon wavelengths in electron transitions allows us to identify elements and compounds from their emission spectra. Longer wavelengths are associated with lower energy photons, while shorter wavelengths correspond to higher energies. In the hydrogen atom scenario, transitions to lower energy states emit photons with wavelengths that can be calculated by the Rydberg Formula, revealing information such as color or type of light emitted, which can range from visible to ultraviolet and beyond.

Electron Transitions

Electron transitions are changes in an electron's position from one energy level to another within an atom. This movement is quantized, meaning electrons jump between discrete energy levels according to specific rules governed by quantum mechanics, rather than moving continuously. When an electron moves from a higher energy level to a lower one, energy is released in the form of a photon. The energy difference between the starting and ending levels determines the photon's wavelength,which can be calculated using the Rydberg Formula. For example, transitions such as \(n=4\) to \(n=3\) or \(n=8\) to \(n=4\) in hydrogen will emit photons of different wavelengths.It's important to remember that the principal quantum numbers \(n\) define each level, with higher values indicating levels further from the nucleus. These transitions are responsible for the diverse lines in the spectrum of the hydrogen atom, leading to the formation of patterns called spectral lines.

Emission Spectra

Emission spectra refer to the spectrum of wavelengths emitted by an atom as electrons move from higher to lower energy levels, resulting in photons' release. Each element has a unique emission spectrum, which acts like a fingerprint, allowing scientists to identify the element by analyzing the light it emits. Emission spectra can be observed via spectral lines, which represent different transitions between energy levels. These spectra are categorized into two main types:

• Line spectra: Narrow lines at specific wavelengths emitted by atoms.
• Continuous spectra: Emitted by solids, liquids, or dense gases, covering a broad range of wavelengths.

In analyzing emission spectra, each line corresponds to a specific electron transition, such as \(n=5\) to \(n=2\) or \(n=4\) to \(n=3\). Thus, emission spectra not only help in identifying elements but also provide insight into the atom's structure and electron energy levels.