a. Using Equation 6.5, calculate the energy of an electron in the hydrogen atom when n = 2 and when n = 6. Calculate the wavelength of the radiation released when an electron moves from n = 6 to n = 2.

Verified step by step guidance

1. Identify the formula for the energy of an electron in a hydrogen atom: \(E_n = -\frac{R_H}{n^2}\), where \(R_H\) is the Rydberg constant (2.18 \times 10^{-18} \text{ J}) and \(n\) is the principal quantum number.
2. Calculate the energy of the electron when \(n = 2\) using the formula: \(E_2 = -\frac{R_H}{2^2}\).
3. Calculate the energy of the electron when \(n = 6\) using the formula: \(E_6 = -\frac{R_H}{6^2}\).
4. Determine the energy difference between the two states: \(\Delta E = E_2 - E_6\). This energy difference corresponds to the energy of the photon released.
5. Use the relationship between energy and wavelength: \(\Delta E = \frac{hc}{\lambda}\), where \(h\) is Planck's constant (6.626 \times 10^{-34} \text{ J s}) and \(c\) is the speed of light (3.00 \times 10^8 \text{ m/s}), to solve for the wavelength \(\lambda\).

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 an electron in these levels can be calculated using the formula E_n = -13.6 eV/n², where E_n is the energy at level n. This means that as n increases, the energy becomes less negative, indicating that the electron is less tightly bound to the nucleus.

Photon Emission and Wavelength

When an electron transitions between energy levels, it can emit or absorb a photon, which corresponds to the energy difference between the two levels. The energy of the emitted photon can be calculated using the equation ΔE = E_initial - E_final. The wavelength of the emitted radiation can then be determined using the equation λ = hc/ΔE, where h is Planck's constant and c is the speed of light.

Planck's Constant and the Speed of Light

Planck's constant (h) is a fundamental constant that relates the energy of a photon to its frequency, given by E = hf, where f is the frequency. The speed of light (c) is the speed at which light travels in a vacuum, approximately 3.00 x 10^8 m/s. These constants are essential for converting energy differences into wavelengths, allowing for the calculation of the wavelength of emitted radiation during electron transitions.