Question

Which of the following energy level changes for an electron is most energetic: \(5 \rightarrow 2,4 \rightarrow 2,\) or \(3 \rightarrow 2 ?\)

Short answer

The most energetic transition is from level 5 to level 2.

Step-by-step solution

Understand the Concept

The energy emitted or absorbed by an electron when it changes levels is calculated using the formula \[ E = -13.6 imes rac{1}{n^2} \]for each level, where \( n \) is the principal quantum number. More negative values correspond to lower energy levels, and transitioning down in levels emits energy.

Calculate Initial and Final Energies

For each transition, calculate the energy for both initial \((n_i)\) and final \((n_f)\) levels:1. **For transition \(5 \rightarrow 2\):** - Initial energy: \( E_i = -13.6 \times \frac{1}{5^2} \) - Final energy: \( E_f = -13.6 \times \frac{1}{2^2} \)2. **For transition \(4 \rightarrow 2\):** - Initial energy: \( E_i = -13.6 \times \frac{1}{4^2} \) - Final energy: \( E_f = -13.6 \times \frac{1}{2^2} \)3. **For transition \(3 \rightarrow 2\):** - Initial energy: \( E_i = -13.6 \times \frac{1}{3^2} \) - Final energy: \( E_f = -13.6 \times \frac{1}{2^2} \)

Calculate Energy Changes

Energy change for each transition \( \Delta E \) is given by:- **\(5 \rightarrow 2\):** - \( E_i = -13.6 \times \frac{1}{25} = -0.544 \) - \( E_f = -13.6 \times \frac{1}{4} = -3.4 \) - \( \Delta E = |E_f - E_i| = |-3.4 + 0.544| = 2.856 \, \text{eV} \)- **\(4 \rightarrow 2\):** - \( E_i = -13.6 \times \frac{1}{16} = -0.85 \) - \( E_f = -13.6 \times \frac{1}{4} = -3.4 \) - \( \Delta E = |E_f - E_i| = |-3.4 + 0.85| = 2.55 \, \text{eV} \)- **\(3 \rightarrow 2\):** - \( E_i = -13.6 \times \frac{1}{9} = -1.51 \) - \( E_f = -13.6 \times \frac{1}{4} = -3.4 \) - \( \Delta E = |E_f - E_i| = |-3.4 + 1.51| = 1.89 \, \text{eV} \)

Determine the Most Energetic Transition

Compare the calculated energy changes to determine which transition is most energetic:- Transition \(5 \rightarrow 2\) has \( \Delta E = 2.856 \, \text{eV} \)- Transition \(4 \rightarrow 2\) has \( \Delta E = 2.55 \, \text{eV} \)- Transition \(3 \rightarrow 2\) has \( \Delta E = 1.89 \, \text{eV} \)The most energetic transition is \(5 \rightarrow 2\) with \(2.856 \, \text{eV}\).

Key concepts

Energy Transition

Energy transitions refer to the movement of an electron between different energy levels within an atom. When an electron moves from a higher energy level to a lower one, it releases energy, typically in the form of light. Conversely, moving to a higher energy level requires absorption of energy. These transitions are fundamental to understanding atomic behavior since they play a crucial role in the emission and absorption spectra of elements. Every element emits or absorbs specific wavelengths of light based on the transitions of its electrons. This is why each element has a unique spectral fingerprint, allowing scientists to identify them by the light they emit or absorb.

Principal Quantum Number

The principal quantum number, denoted by \( n \), is an important quantum number assigned to each electron in an atom. It defines the overall size of the electron cloud and its average distance from the nucleus. The larger the quantum number, the higher the energy level and the farther the electron is from the nucleus.

• The principal quantum number has integral values: \( n = 1, 2, 3, \ldots \).
• Electrons in higher \( n \) values are at higher energy states.
• Transitions involving larger changes in \( n \) involve higher energy differences.

Energy levels in an atom are quantized, meaning electrons can only occupy specific levels, each represented by a distinct \( n \). Understanding the principal quantum number helps in computing energy changes during transitions.

Energy Change Calculation

To calculate the energy change during an electron's transition between different energy levels in a hydrogen atom, we use the following formula:\[ E = -13.6 \times \frac{1}{n^2} \] where each \( n \) represents an integer principal quantum number. The negative sign indicates that higher energy values are less negative, meaning they are closer to zero, and thus at higher energy levels.

The process involves the following steps:

• Determine the initial and final principal quantum numbers \( n_i \) and \( n_f \).
• Calculate the initial energy \( E_i \) and final energy \( E_f \) using the formula above.
• Evaluate the energy change \( \Delta E \) as the absolute value of the difference between \( E_f \) and \( E_i \):\[ \Delta E = |E_f - E_i| \]

Understanding this calculation allows us to identify which transitions release more energy, such as the \( 5 \rightarrow 2 \) transition, which is the most energetic in this exercise.