A triply ionized beryllium ion, Be3+ (a beryllium atom with three electrons removed), behaves very much like a hydrogen atom except that the nuclear charge is four times as great. (c) For the hydrogen atom, the wavelength of the photon emitted in the n = 2 to n = 1 transition is 122 nm (see Example 39.6). What is the wavelength of the photon emitted when a Be3+ ion undergoes this transition?

Verified step by step guidance

1. Identify the formula for the energy levels of a hydrogen-like atom, which is given by E_n = -\frac{Z^2 \times 13.6 \text{ eV}}{n^2}, where Z is the atomic number and n is the principal quantum number.
2. Calculate the energy difference between the n = 2 and n = 1 levels for the Be3+ ion. Since Be3+ has a nuclear charge Z = 4, substitute Z = 4 into the formula for both energy levels and find the difference: \Delta E = E_1 - E_2.
3. Use the energy difference to find the frequency of the emitted photon using the relation E = h \nu, where E is the energy difference and h is Planck's constant (6.626 x 10^{-34} J s). Solve for \nu (frequency).
4. Convert the frequency to wavelength using the speed of light equation c = \lambda \nu, where c is the speed of light (3.00 x 10^8 m/s), \lambda is the wavelength, and \nu is the frequency calculated in the previous step.
5. Solve for \lambda to find the wavelength of the photon emitted during the transition from n = 2 to n = 1 in the Be3+ ion.

Key Concepts

Bohr Model of the Atom

The Bohr model describes the behavior of electrons in atoms, particularly hydrogen-like ions. It posits that electrons occupy specific energy levels and can transition between these levels by absorbing or emitting photons. The energy of the emitted or absorbed photon corresponds to the difference in energy between the two levels, which can be calculated using the formula E = -13.6 eV/n² for hydrogen, where n is the principal quantum number.

Energy Levels and Transitions

In a hydrogen-like atom, energy levels are quantized and depend on the nuclear charge (Z). For a triply ionized beryllium ion (Be3+), the energy levels are given by E_n = -Z² * 13.6 eV/n², where Z = 4. The transition from a higher energy level (n=2) to a lower one (n=1) results in the emission of a photon, and the wavelength of this photon can be determined using the energy-wavelength relationship, E = hc/λ.

Wavelength Calculation

The wavelength of a photon emitted during an electronic transition can be calculated using the formula λ = hc/E, where h is Planck's constant and c is the speed of light. By first calculating the energy difference between the two levels for Be3+, we can then find the wavelength of the emitted photon. This process involves substituting the energy values derived from the energy level equations into the wavelength formula.