The energy of different orbitals in an atom or ion having only one electron, depends on — Atomic Structure Chemistry Question

💡 Solution & Explanation

### Step 1 - Energy Expression for Single-Electron Systems For hydrogen and hydrogen-like single-electron systems (such as $\ce{H}$, $\ce{He^+}$, $\ce{Li^{2+}}$, and $\ce{Be^{3+}}$), the energy of an electron in a given orbital is given by: $$E_n = -R_H \cdot \frac{Z^2}{n^2}$$ where: * $R_H \approx 2.18 \times 10^{-18}\text{ J}$ (Rydberg constant) * $Z$ = atomic number (constant for a given atom or ion) * $n$ = principal quantum number For any specific atom or ion, $Z$ is fixed, so the energy depends **strictly on $n$**. --- ### Step 2 - Absence of Shielding and Subshell Degeneracy In **single-electron systems**: 1. There are no other electrons → inter-electronic repulsion = 0 2. There is no shielding or screening effect (screening constant $\sigma = 0$) Because there is no shielding, all orbitals within a given shell (same $n$) experience the exact same nuclear charge $Z$. Therefore, all subshells within a given shell are **degenerate** (have the same energy): $$E_{2s} = E_{2p}$$ $$E_{3s} = E_{3p} = E_{3d}$$ In contrast, in multi-electron systems, different subshells within a shell have different energies due to shielding, and the energy depends on both $n$ and $l$ (via the $n+l$ rule). --- ### Step 3 - Systematic Option Analysis * **Option (A) $n$ only:** Correct. For a single-electron system, energy is a function of $n$ only, due to the complete absence of inter-electronic repulsion. * **Option (B) $n$ and $l$ only:** Incorrect. The energy depends on both $n$ and $l$ only in multi-electron systems. For single-electron systems, the energy is independent of $l$. * **Option (C) $n$, $l$ and $m$ only:** Incorrect. The magnetic quantum number $m$ determines spatial orientation and does not affect energy (unless an external magnetic field is applied). Additionally, energy is independent of $l$ in single-electron systems. * **Option (D) $n$, $l$, $m$ and $s$:** Incorrect. The spin quantum number $s$ describes the spin state of the electron and does not change the energy eigenvalues of the spatial orbitals. $$\text{Correct Option: } \boxed{\text{A}}$$