Problem 33 - University Physics with Modern Physics

Question

(a) The doubly charged ion \(\mathrm{N}^{2+}\) is formed by removing two electrons from a nitrogen atom. What is the ground-state electron configuration for the \(N^{2+}\) ion? (b) Estimate the energy of the least strongly bound level in the \(L\) shell of \(N^{2+} .(c)\) The doubly charged ion \mathrm{P}^{2+}\( is formed by removing two electrons from a phosphorus atom. What is the ground-state electron configuration for the \)\mathrm{P}^{2+}\( ion? (d) Estimate the energy of the least strongly bound level in the \)M\( shell of \)\mathrm{P}^{2+}$ .

Step-by-Step Solution

Answer

(a) N²⁺: 1s² 2s² 2p¹. (b) Use Z_eff to find energy for 2p electron. (c) P²⁺: 1s² 2s² 2p⁶ 3s² 3p¹. (d) Calculate 3p energy with Z_eff.

Step 1: Understand Electron Configurations

The electron configuration of an atom describes the distribution of electrons in the atomic orbitals. For the nitrogen atom (N), the atomic number is 7, so it has 7 electrons. Its neutral configuration is 1s² 2s² 2p³.

Step 2: Determine the Ground-State Configuration for N²⁺

The doubly charged ion \( N^{2+} \) is formed by removing two electrons from a neutral nitrogen atom. Removing two electrons from the 2p orbital (which is the outermost) results in: 1s² 2s² 2p¹.

Step 3: Energy of Least Strongly Bound Electron in N²⁺

The L shell corresponds to the n=2 level (2s and 2p orbitals). For \( N^{2+} \), the least strongly bound electron is the one in the 2p orbital. Calculate using the effective nuclear charge (Z_eff) and energy level formula: \( E_n \approx -Z_{eff}^2/n^2 \times 13.6 \text{ eV} \). For nitrogen (approximate): Z_eff can be about 5 for the p-electron.

Step 4: Understand Electron Configuration for Phosphorus

Phosphorus (P) has an atomic number of 15, meaning it has 15 electrons. Its neutral configuration is 1s² 2s² 2p⁶ 3s² 3p³.

Step 5: Determine the Ground-State Configuration for P²⁺

The doubly charged \( P^{2+} \) ion is formed by removing two electrons, typically from the outermost 3p orbital, resulting in: 1s² 2s² 2p⁶ 3s² 3p¹.

Step 6: Energy of Least Strongly Bound Electron in P²⁺

The M shell corresponds to the n=3 level (3s and 3p orbitals). For \( P^{2+} \), the least strongly bound electron is in the 3p orbital. Use a similar calculation as Step 3, adjusting for phosphorus with Z_eff estimated around 13.

Key Concepts

Atomic Orbitals

Atomic orbitals are regions in an atom where electrons are most likely to be found. Each electron in an atom occupies a specific atomic orbital, determined by its energy level and subshell.
The major types of atomic orbitals include:

• s orbitals: Spherical in shape and can hold up to 2 electrons.
• p orbitals: Dumbbell-shaped and can hold up to 6 electrons across three orientations (pₓ, p_y, and p_z).
• d orbitals: More complex shapes, accommodating up to 10 electrons across five orientations.
• f orbitals: Even more complex, holding up to 14 electrons across seven orientations.

Understanding atomic orbitals is crucial for determining the electron configuration of an atom. Electrons fill the lower-energy orbitals first (like s), before moving to higher-energy orbitals (like p and d). This principle follows the Aufbau Rule.

Ground State

The ground state of an atom or ion is its most stable and low-energy electron configuration. This configuration helps atom maintain the lowest possible energy.
In the ground state, electrons are distributed among orbitals in a way that they occupy the lowest available energy levels. For instance, in the nitrogen ion \(N^{2+}\), the electrons fill the 1s, 2s, and then the 2p orbitals sequentially according to their energy.
The electrons' distribution follows several key principles to establish the ground state:

• Hund’s Rule: Electrons occupy degenerate orbitals singly before pairing up.
• Pauli Exclusion Principle: No two electrons in an atom can have the same set of four quantum numbers.
• Aufbau Principle: Electrons fill orbitals of the lowest energy first.

The understanding of ground-state electron configurations provides a fundamental basis for predicting chemical reactivity and bonding.

Effective Nuclear Charge

Effective nuclear charge (Z_eff) is the net positive charge experienced by an electron in a multi-electron atom or ion, reflecting how strongly the nucleus attracts a particular electron. This concept is crucial for explaining the energy levels and behavior of electrons within an atom.
Electrons closer to the nucleus partially shield outer electrons from the full nuclear charge, a phenomenon known as electron shielding. As a result, the effective nuclear charge can be significantly less than the actual charge of the nucleus.

• Inner electron shielding: Core electrons shield outer electrons effectively, reducing the attraction from the nucleus.
• Penetration effect: Electrons in s orbitals can penetrate closer to the nucleus, experiencing greater effective nuclear charge than electrons in p, d, or f orbitals of the same energy level.

Calculating (or estimating) Z_eff is important when considering the energy calculations of least bound electrons, like in \(N^{2+}\) or \(P^{2+}\), to determine ionization energies and affinity.

Energy Levels

Energy levels refer to the distinct layers or shells containing the electrons within an atom. They help describe the arrangement and stability of electrons surrounding an atomic nucleus. Each level can house a specific number of electrons, increasing with the higher levels.
The main characteristics of energy levels are:

• Principal quantum number (): Indicates the relative size and energy of an electron shell.
• Sublevels: Each energy level is made up of sublevels or orbitals (s, p, d, f), with the number and type depending on the principal quantum number.
• Maximum electron capacity: The nth energy level can hold a maximum of \((2n^2)\) electrons. For instance, the first energy level can house up to 2 electrons, while the second can accommodate up to 8 electrons.

Energy levels determine where an electron resides around the nucleus and its potential energy. In doubly charged ions like \(N^{2+}\) and \(P^{2+}\), electrons in higher energy levels (e.g., =2 for L shell or =3 for M shell) are often the least strongly bound and most reactive. Understanding energy levels aids in predicting atomic behavior and properties.