Which of the following wavelengths corresponds to the energy required to excite a hydrogen atom from the n=2 state to the n=5 state?

• A. 434 nm
• B. 4340 nm
• C. 397 nm
• D. 4340 Å

Answer: (D)

Verified step by step guidance

1. Identify the initial and final energy levels for the hydrogen atom transition: initial level \(n_i = 2\) and final level \(n_f = 5\).
2. Use the energy level formula for hydrogen: \(E_n = -13.6 \frac{1}{n^2}\) eV, where \(n\) is the principal quantum number.
3. Calculate the energy difference \(\Delta E\) between the two levels using \(\Delta E = E_{n_f} - E_{n_i} = -13.6 \left( \frac{1}{n_f^2} - \frac{1}{n_i^2} \right)\) eV.
4. Convert the energy difference \(\Delta E\) to wavelength \(\lambda\) using the relationship between energy and wavelength: \(\Delta E = \frac{hc}{\lambda}\), where \(h\) is Planck's constant and \(c\) is the speed of light. Rearranged, this is \(\lambda = \frac{hc}{\Delta E}\).
5. Calculate \(\lambda\) in meters and convert it to nanometers (1 nm = \$10^{-9}\( m) or angstroms (1 Å = \)10^{-10}$ m) to compare with the given options and identify the correct wavelength.