(a) Using Equation 6.5, calculate the energy of an electron in the hydrogen atom when n = 3 and when n = 6. Calculate the wavelength of the radiation released when an electron moves from n = 6 to n = 3. when n = 6

Verified step by step guidance

1. Identify the equation for the energy of an electron in a hydrogen atom, which is typically given by the formula: E_n = -\frac{Z^2 \times 13.6 \text{ eV}}{n^2}, where Z is the atomic number (1 for hydrogen) and n is the principal quantum number.
2. Substitute n = 3 into the energy equation to find the energy of the electron when n = 3.
3. Substitute n = 6 into the energy equation to find the energy of the electron when n = 6.
4. Calculate the energy difference (\Delta E) between the two energy levels (n = 6 and n = 3) by subtracting the energy at n = 3 from the energy at n = 6.
5. Use the energy difference calculated in the previous step to find the wavelength of the radiation released using the formula: \lambda = \frac{hc}{\Delta E}, where h is Planck's constant (6.626 x 10^{-34} J s) and c is the speed of light (3.00 x 10^8 m/s).

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 a given level can be calculated using the formula E_n = -13.6 eV/n², where E_n is the energy in electron volts and n is the principal quantum number. Higher values of n correspond to higher energy levels, and the energy becomes less negative as n increases.

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.

Calculating Energy Differences

To find the energy difference when an electron moves from a higher energy level (n=6) to a lower one (n=3), you first calculate the energies at both levels using the energy formula. The difference in energy (ΔE) will give you the energy of the photon emitted during the transition. This energy can then be used to find the wavelength of the emitted radiation, which is crucial for understanding the spectral lines of hydrogen.