Estimate the energy of the highest-l state for (a) the L shell of Be+ and (b) the N shell of Ca +.

Verified step by step guidance

1. Identify the principal quantum number (n) for each shell. For the L shell, n=2, and for the N shell, n=4.
2. Determine the maximum orbital angular momentum quantum number (l) for each shell, which is n-1. For the L shell, l=1, and for the N shell, l=3.
3. Use the formula for the energy levels of a hydrogen-like atom, E = -\frac{Z^2}{n^2} \times 13.6 \text{ eV}, where Z is the atomic number of the ion minus the number of electrons lost plus one.
4. For Be+, Z=4 and it has lost one electron, so Z effective is 4. For Ca+, Z=20 and it has lost one electron, so Z effective is 20.
5. Plug in the values of Z effective and n into the energy formula to estimate the energy for each case.

Key Concepts

Quantum Energy Levels

In quantum mechanics, electrons occupy discrete energy levels around an atom's nucleus. The energy of these levels is quantized, meaning electrons can only exist in specific states defined by quantum numbers. The principal quantum number (n) indicates the shell, while other quantum numbers describe the shape and orientation of the orbital. Understanding these levels is crucial for estimating the energy of electrons in different shells.

Ionization Energy

Ionization energy is the energy required to remove an electron from an atom or ion in its gaseous state. For ions like Be+ and Ca+, the ionization energy can vary based on the electron configuration and the effective nuclear charge experienced by the outer electrons. This concept is essential for estimating the energy of electrons in various shells, as it directly relates to the stability and energy of the electron states.

Effective Nuclear Charge (Z_eff)

Effective nuclear charge is the net positive charge experienced by an electron in a multi-electron atom. It accounts for the shielding effect of inner electrons that reduce the full nuclear charge felt by outer electrons. The Z_eff influences the energy levels of electrons in different shells, as a higher effective nuclear charge typically results in lower energy levels due to increased attraction between the nucleus and the electrons.