Calculate the wavelength in nm of the light emitted when an electron makes a transition from an orbital in n = 5 to an orbital in n = 2 in the hydrogen atom. (LO 5.8)(a) 2.31 * 10-3 nm (b) 4.34 * 10-2 nm (c) 231 nm(d) 434 nm

Verified step by step guidance

1. Identify the initial and final energy levels of the electron. In this problem, the electron transitions from n = 5 to n = 2.
2. Use the Rydberg formula to calculate the wavelength of the emitted photon: \( \frac{1}{\lambda} = R \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \), where \( R \) is the Rydberg constant (approximately 1.097 x 10^7 m^-1), \( n_1 \) is the lower energy level, and \( n_2 \) is the higher energy level.
3. Substitute the values into the Rydberg formula: \( n_1 = 2 \) and \( n_2 = 5 \).
4. Calculate \( \frac{1}{\lambda} \) using the values substituted into the Rydberg formula.
5. Convert the result from meters to nanometers by multiplying by 10^9, as 1 meter equals 10^9 nanometers.

Key Concepts

Energy Levels in Hydrogen Atom

In a hydrogen atom, electrons occupy discrete energy levels, denoted by the principal quantum number 'n'. The energy of these levels is quantized, meaning electrons can only exist in specific states. When an electron transitions between these levels, energy is either absorbed or emitted in the form of light, with the wavelength of this light being determined by the difference in energy between the two levels.

Rydberg Formula

The Rydberg formula is used to calculate the wavelengths of spectral lines in hydrogen. It relates the wavelength of emitted light to the principal quantum numbers of the initial and final energy levels. The formula is given by 1/λ = R_H (1/n_f^2 - 1/n_i^2), where R_H is the Rydberg constant, n_f is the final level, and n_i is the initial level. This formula is essential for determining the wavelength of light emitted during electron transitions.

Wavelength and Energy Relationship

The wavelength of light is inversely related to its energy, as described by the equation E = hc/λ, where E is energy, h is Planck's constant, c is the speed of light, and λ is the wavelength. This relationship indicates that shorter wavelengths correspond to higher energy transitions. Understanding this concept is crucial for calculating the wavelength of light emitted during electron transitions in the hydrogen atom.