Question

Consider the transition from the energy levels \(\mathbf{n}=2\) to \(\mathbf{n}=5\) (a) What is the wavelength associated with this transition? (b) In what spectral region does the transition occur? (c) Is energy absorbed?

Short answer

The wavelength associated with this transition is \(4.56 \times 10^{-7}\ \text{m}\). (b) In which spectral region does this transition occur? The transition occurs in the violet color region of the visible light spectrum. (c) Is energy absorbed during this transition? Yes, energy is absorbed during this transition.

Step-by-step solution

(1) Calculate the change in the energy levels

For the transition between the energy levels, first, we need to find the change in the energy levels. In this case, the electron is moving from n = 2 to n = 5, which means: Δn = n_final - n_initial Δn = 5 - 2 Δn = 3

(2) Apply the Rydberg formula

To find the wavelength associated with this transition, we will use the Rydberg formula: \(\frac{1}{\lambda} = R_H \cdot (\frac{1}{n_{1}^2} - \frac{1}{n_{2}^2})\) Where λ is the wavelength, R_H is the Rydberg constant for hydrogen (approximately \(1.097 \times 10^7 \ m^{-1}\)), and n_1 and n_2 are the initial and final energy levels. Plug in the values into the formula, we get: \(\frac{1}{\lambda} = (1.097 \times 10^7) \cdot (\frac{1}{2^2} - \frac{1}{5^2})\)

(3) Calculate the wavelength

Solve for λ: λ = \(\frac{1}{(1.097 \times 10^7) \cdot (\frac{1}{2^2} - \frac{1}{5^2})}\) = \(4.56 \times 10^{-7}\ \text{m}\)

(4) Determine the spectral region

To determine the spectral region of this wavelength, we can compare it to the known ranges of the electromagnetic spectrum. We find that this transition occurs within the visible light range, specifically in the violet color region.

(5) Determine whether energy is absorbed

As the electron is moving from a lower energy level (n = 2) to a higher energy level (n = 5), energy must be absorbed. Therefore, energy is absorbed during this transition. (a) The wavelength associated with this transition is \(4.56 \times 10^{-7}\ \text{m}\). (b) The transition occurs in the violet color region of the visible light spectrum. (c) Yes, energy is absorbed during this transition.