Question

A photon with a wavelength in the visible region (between 400 and \(700 \mathrm{nm}\) ) causes a transition from the \(n\) to the \((n+1)\) state in doubly ionized lithium (Li \(^{2+}\) ). What is the lowest value of \(n\) for which this could occur?

Short answer

The lowest value of \(n\) for which this transition could occur is \(n = 3\).

Step-by-step solution

Understanding the Photon Transition

We know that the transition occurs due to the absorption of a photon by the doubly ionized lithium, Li\(^{2+}\). This indicates a transition from energy level \(n\) to \((n+1)\). The energy of the photon must match the energy difference between these two levels.

Determine the Formula for Energy Transition

For a hydrogen-like atom, the energy levels are given by the formula \(E_n = -Z^2 \dfrac{13.6 \text{ eV}}{n^2}\), where \(Z\) is the atomic number. For Li\(^{2+}\), \(Z = 3\), so the energy levels are \(E_n = -9 \dfrac{13.6 \text{ eV}}{n^2} \). We need the energy for states \(n\) and \(n+1\).

Calculate the Energy Difference

The energy difference between two levels is \( \Delta E = E_{n+1} - E_n = -9 \cdot 13.6 \left(\dfrac{1}{(n+1)^2} - \dfrac{1}{n^2}\right) \text{ eV} \). The wavelength of the photon \( \lambda \) is related to this energy by \( E = \dfrac{hc}{\lambda} \), where \( h \) is Planck's constant and \( c \) the speed of light.

Relate Wavelength to Energy

Using \( E = \dfrac{hc}{\lambda} \) and given \( \lambda \) between \(400 \text{ nm}\) and \(700 \text{ nm}\), calculate the corresponding photon energy. Planck's constant \( h = 4.14 \times 10^{-15} \text{ eV}\cdot \text{s} \), speed of light \( c = 3 \times 10^8 \text{ m/s} \), convert \( \lambda \) to meters for each bound of \( \lambda \).

Solve for n with Energy Constraints

Calculate energy transitions \( \Delta E \) for increasing values of \(n\), and equate to energy from \( \lambda \). Use the energy values from step 4 to iteratively solve for \(n\) such that the photon energy lies within this range. Begin with the lowest possible \(n\) and increase until you find \(n\) that satisfies the range constraints.

Conclusion: Determine the Lowest n

The calculation will show that the smallest energy difference corresponding to \(400 \text{ nm}\) is achieved when \(n = 3\). Thus, the lowest value of \(n\) for which the transition can occur is \(n = 3\).

Key concepts

Wavelength

The concept of wavelength is critical when discussing photon transitions. Wavelength refers to the distance between consecutive peaks of a wave, such as light. In the context of photons, which are particles of light, wavelength directly relates to the color we perceive in the visible spectrum.
For this exercise, the focus is on photons with wavelengths between 400 nm and 700 nm, which are within the visible light range. Shorter wavelengths correspond to higher-energy light (like violet and blue), while longer wavelengths represent lower-energy light (such as red).
Understanding wavelength is essential because the energy of a photon changes inversely with wavelength as described by the formula:

1. The shorter the wavelength, the higher the energy.
2. Conversely, the longer the wavelength, the lower the energy.

This relationship is given by the formula: \[ E = \dfrac{hc}{\lambda} \]where:- \( E \) is the photon energy,- \( h \) is Planck's constant,- \( c \) is the speed of light, and- \( \lambda \) is the wavelength.

Energy Levels

Energy levels in atoms represent the distinct amounts of energy that an electron within an atom can have. These levels are quantized, meaning electrons exist only in specific energy states rather than in a continuous range. The concept plays a significant role when understanding photon transitions.
In hydrogen-like atoms, such as doubly ionized lithium (Li^{2+}), the energy of each level is given by the formula:\[ E_n = -Z^2 \dfrac{13.6 \text{ eV}}{n^2} \]where:- \( E_n \) is the energy of the nth level,- \( Z \) is the atomic number (for Li^{2+}, \( Z = 3 \)),- \( n \) is the principal quantum number (indicating the energy level).
For a photon transition to occur from one level to another, the photon must possess energy equal to the difference in the energy levels involved. This difference, or the energy required for the transition, is calculated using:\[ \Delta E = E_{n+1} - E_n \]This concept is crucial as the energy stored in electron transitions corresponds directly to the energy of emitted or absorbed photons.

Hydrogen-like Atom

A hydrogen-like atom is an atom that, despite having more than one electron originally, behaves as if it has only one remaining due to ionization. Ionization means the atom has lost electrons, often causing it to mimic the behavior of a hydrogen atom. This simplification makes calculations easier while retaining accuracy for high atomic ions.
In this exercise, Li^{2+} is considered "hydrogen-like" because it has been doubly ionized, leaving it with one electron in a similar configuration to hydrogen:

• These atoms follow the same basic energy level formula as hydrogen, adjusting for their unique nuclear charge \((Z)\).
• Li^{2+} specifically has \(Z = 3\), which increases the energy of its states compared to hydrogen.

This concept helps calculate energy levels using the formula:\[ E_n = -Z^2 \dfrac{13.6 \text{ eV}}{n^2} \] Understanding this concept simplifies the process of calculating photon transitions in more complex atoms once they're ionized sufficiently to exhibit hydrogen-like properties.

Photon Energy

Photon energy is the energy carried by a photon. It is an essential aspect of understanding photon-induced transitions between energy levels in atoms. Photon energy can be described by how it relates to wavelength:\[ E = \dfrac{hc}{\lambda} \]The formula highlights two important relationships:

• The inversely proportional relationship between energy and wavelength. When the wavelength is smaller, the energy is greater.
• Direct proportionality to Planck’s constant \((h)\) and the speed of light \((c)\).

Photon energy is central to determining if a transition between atomic energy levels is possible. This is because an electron will only change energy levels when absorbing or releasing a photon of energy precisely matching the energy difference between its original and final states. In Li^{2+}, and similar transitions:

• For a transition to occur, the photon's energy must precisely match the energy difference given by \( \Delta E = E_{n+1} - E_n \).
• This relationship ensures photon absorption or emission will have a distinct role in transitions.

By understanding photon energy, we can predict or explain why certain transitions occur in atoms when interacting with specific light wavelengths.