Problem 35 - Chemistry The Molecular Science

Question

The Bohr equation for hydrogen can be modified to apply to one-electron species other than uncharged hydrogen atoms, for example \(\mathrm{Li}^{2+},\) to calculate the energy of electron transitions in the ion. The modified equation is \(E_{n}=-Z^{2} / n^{2}\left(2.179 \times 10^{-18} \mathrm{~J}\right) . Z\) is the pos- itive charge of the nucleus and \(n\) is the principal quantum number. Calculate the energy of the photon emitted for the transition from the \(n=4\) to the \(n=1\) state in this ion. In what region of the electromagnetic spectrum does it lie?

Step-by-Step Solution

Answer

The energy of the photon is \(1.8388 \times 10^{-17} \mathrm{~J}\), in the ultraviolet region.

Step 1: Understand the Problem

We need to find the energy of a photon emitted when an electron in the \(\mathrm{Li}^{2+}\) ion transitions from the \(n=4\) level to the \(n=1\) level using the modified Bohr equation. \(\mathrm{Li}^{2+}\) means the lithium nucleus has a charge \(Z=3\).

Step 2: Write the Relevant Equation

The modified Bohr equation for the energy of a level \(n\) is given by: \[ E_{n} = -\frac{Z^2}{n^2} \left(2.179 \times 10^{-18} \mathrm{~J}\right) \] where \(Z\) is the nuclear charge and \(n\) is the principal quantum number. We need to calculate the energies \(E_4\) and \(E_1\).

Step 3: Calculate Energy at n=4

Substitute \(n=4\) and \(Z=3\) into the equation: \[ E_{4} = -\frac{3^2}{4^2} \left(2.179 \times 10^{-18} \mathrm{~J}\right) = -\frac{9}{16} \times 2.179 \times 10^{-18} \mathrm{~J} = -1.2234 \times 10^{-18} \mathrm{~J} \]

Step 4: Calculate Energy at n=1

Substitute \(n=1\) and \(Z=3\) into the equation: \[ E_{1} = -\frac{3^2}{1^2} \left(2.179 \times 10^{-18} \mathrm{~J}\right) = -9 \times 2.179 \times 10^{-18} \mathrm{~J} = -1.9611 \times 10^{-17} \mathrm{~J} \]

Step 5: Calculate the Energy of the Photon

The energy emitted as a photon when the electron transitions from \(n=4\) to \(n=1\) is given by the difference: \[ \Delta E = E_{1} - E_{4} = -1.9611 \times 10^{-17} \mathrm{~J} - (-1.2234 \times 10^{-18} \mathrm{~J}) \] \[ \Delta E = -1.9611 \times 10^{-17} \mathrm{~J} + 1.2234 \times 10^{-18} \mathrm{~J} = -1.8388 \times 10^{-17} \mathrm{~J} \] Since energy is released, we take the absolute value: \[ E_{\text{photon}} = 1.8388 \times 10^{-17} \mathrm{~J} \]

Step 6: Determine the Electromagnetic Spectrum Region

Use the relationship \(E = \frac{hc}{\lambda}\) to find the wavelength \(\lambda\), where \(h = 6.626 \times 10^{-34} \mathrm{~J}\cdot\mathrm{s}\) and \(c = 3 \times 10^8 \mathrm{~m/s}\). Solve for \(\lambda\): \[ \lambda = \frac{6.626 \times 10^{-34} \times 3 \times 10^8}{1.8388 \times 10^{-17} } = 1.08 \times 10^{-8} \mathrm{~m} \] This wavelength corresponds to the ultraviolet region of the electromagnetic spectrum.

Key Concepts

One-electron species

A one-electron species refers to an ion or atom that has only a single electron orbiting its nucleus. These include ions like

• Hydrogen (H): The most well-known one-electron species.
• Lithium ion (\( \text{Li}^{2+}\)): A lithium atom that has lost two electrons, leaving just one electron.
• Helium ion (\( \text{He}^{+}\)): A helium atom with a single electron, formed by losing one electron.

One-electron species are crucial when applying the Bohr model since calculations become simple and predictable. Their simplicity allows scientists to study electron transitions without complex electron-electron interactions. By considering single-electron systems, the Bohr model provides clear insights into the energy levels and spectral lines observed in atomic spectra.

Photon emission

Photon emission occurs when an electron jumps from a higher energy level to a lower one, releasing energy in the form of a photon. This phenomenon is best understood using the Bohr model:

• Energy Levels: Electrons orbit the nucleus at specific energy levels, defined by the principal quantum number, \(n\).
• Transition: When an electron falls to a lower energy level, it emits a photon, the energy of which corresponds to the difference between the levels.
• Conservation of Energy: The energy lost by the electron becomes the energy of the emitted photon.

In our case with \( \text{Li}^{2+}\), when the electron transitions from \(n=4\) to \(n=1\), it emits a photon with energy calculated using the Bohr model. The emitted photon's energy helps determine its position in the electromagnetic spectrum.

Electromagnetic spectrum

The electromagnetic spectrum describes the range of all possible frequencies of electromagnetic radiation, from radio waves to gamma rays. When discussing the energy of a photon, we consider where it fits on this spectrum:

• Visible Light: The small part of the spectrum visible to the human eye, ranging from about 400 to 700 nm.
• Ultraviolet and Infrared: Ultraviolet has shorter wavelengths and higher energy, while infrared has longer wavelengths and lower energy than visible light.
• Other Regions: X-rays, microwaves, and gamma rays represent other parts of the spectrum, each with unique properties and applications.

In our example, the energy of the photon emitted by the \( \text{Li}^{2+}\) ion transition was calculated to correspond to the ultraviolet region of the spectrum. This means the photon's wavelength falls in a range invisible to humans but detectable with special equipment.

Principal quantum number

The principal quantum number, represented by \(n\), is a key quantum number that describes the electron's energy level in an atom. It plays a fundamental role in the Bohr model:

• Energy Levels: \(n\) determines the size and energy of the electron's orbit.
• Values of \(n\): They are positive integers, like \(1, 2, 3, \) etc., each representing a successive energy level further from the nucleus.
• Energy Relation: As \(n\) increases, the energy level also increases, and the electron is less tightly bound to the nucleus.

In our problem, the electron moved between energy levels four (\(n=4\)) and one (\(n=1\)). The difference in these levels corresponds directly to the energy of the photon emitted during the electron's transition. This concept allows us to understand how atoms absorb and emit energy, providing a simplified view of atomic structure and spectral lines.