Consider the three electronic transitions in a hydrogen atom shown here, labeled A, B, and C. (c) Calculate the wavelength of the photon emitted for each transition. Do any of these transitions lead to the emission of visible light? If so which one(s)?

Calculate the wavelength of the photon emitted for transition B.

Verified step by step guidance

1. Identify the initial and final energy levels for transition B. From the diagram, transition B is from n=4 to n=3.
2. Use the Rydberg formula to calculate the energy difference between these levels: \( E = -2.18 \times 10^{-18} \left( \frac{1}{n_f^2} - \frac{1}{n_i^2} \right) \) where \( n_i \) is the initial energy level and \( n_f \) is the final energy level.
3. Substitute the values for \( n_i \) and \( n_f \) into the Rydberg formula: \( E = -2.18 \times 10^{-18} \left( \frac{1}{3^2} - \frac{1}{4^2} \right) \).
4. Calculate the energy difference \( E \) in joules.
5. Convert the energy difference to wavelength using the equation \( \lambda = \frac{hc}{E} \), where \( h \) is Planck's constant (6.626 \times 10^{-34} \text{Js}) and \( c \) is the speed of light (3.00 \times 10^8 \text{m/s}).

Key Concepts

Energy Levels in Hydrogen Atom

In a hydrogen atom, electrons occupy discrete energy levels, denoted by quantum numbers (n). The energy levels are quantized, meaning electrons can only exist at specific distances from the nucleus, corresponding to specific energy values. When an electron transitions between these levels, energy is absorbed or emitted in the form of photons, with the energy difference between the levels determining the wavelength of the emitted light.

Photon Emission and Wavelength Calculation

When an electron transitions from a higher energy level to a lower one, a photon is emitted. The energy of the emitted photon can be calculated using the formula E = hν, where E is the energy difference between the two levels, h is Planck's constant, and ν is the frequency of the photon. The wavelength (λ) can then be found using the relationship c = λν, where c is the speed of light, allowing us to determine the wavelength of the emitted photon for each transition.

Visible Light Spectrum

The visible light spectrum ranges from approximately 400 nm (violet) to 700 nm (red). To determine if a transition emits visible light, the calculated wavelength of the emitted photon must fall within this range. Transitions that result in wavelengths outside this range will emit ultraviolet or infrared light, which are not visible to the human eye.