Question

Estimate the energy of the highest-ll state for (a) the LL shell of Be+ and (b) the NN shell of Ca+.

Accepted Answer

Understand the problem: The goal is to estimate the energy of the highest-l state for the given shells of the ions Be+ and Ca+. The energy levels in an atom or ion are determined by the principal quantum number (n), the azimuthal quantum number (l), and the effective nuclear charge (Z_eff). The highest-l state corresponds to l = n - 1 for a given shell. Recall the formula for the energy of an electron in a hydrogen-like ion: $$ E_n = - \frac{Z_{	ext{eff}}^2 \cdot 13.6 	ext{ eV}}{n^2} $$, where $$ Z_{	ext{eff}}  is $$the effective nuclear charge and $$ n  is $$the principal quantum number. For multi-electron atoms, $$ Z_{	ext{eff}} $$ accounts for the shielding effect of inner electrons. For part (a), the L shell corresponds to $$ n = 2 . $$The highest-l state is $$ l = n - 1 = 1 . $$Estimate $$ Z_{	ext{eff}} $$ for Be+ (atomic number 4) by considering the shielding effect of the inner electrons. Use $$ Z_{	ext{eff}} \approx Z - 	ext{shielding} . $$Substitute $$ n = 2 $$ and the estimated $$ Z_{	ext{eff}} $$ into the energy formula. For part (b), the N shell corresponds to $$ n = 4 . $$The highest-l state is $$ l = n - 1 = 3 . $$Estimate $$ Z_{	ext{eff}} $$ for Ca+ (atomic number 20) by considering the shielding effect of the inner electrons. Use $$ Z_{	ext{eff}} \approx Z - 	ext{shielding} . $$Substitute $$ n = 4 $$ and the estimated $$ Z_{	ext{eff}} $$ into the energy formula. Compare the results for the two ions. Note that the energy of the highest-l state depends on both $$ Z_{	ext{eff}} $$ and $$ n . $$The higher the $$ Z_{	ext{eff}} $$, the more tightly bound the electron is, resulting in a lower (more negative) energy.

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

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Energy Levels in Atoms

Quantum Numbers

Ionization Energy