Question

A student has written what he thinks are some ground-state electron configurations. Which ones have something wrong with them? What is wrong? (a) \(1 s^{2} 2 p^{6} 3 s^{1}\) (b) \(1 s^{2} 2 s^{6} 3 s^{2} 3 p^{6} 4 s^{2} 4 p^{6}\) (c) \(2 s^{2} 2 p^{6} 3 s^{2} 3 p^{6} 4 s^{2} 3 d^{7}\) (d) \(1 s^{2} 2 s^{2} 2 p^{7} 3 s^{3} 3 p^{6}\) (e) \(1 s^{2} 2 s^{2} 2 p^{5} 3 s^{1}\) (f) \(1 s^{2} 2 s^{2} 2 p^{6} 3 s^{2} 3 p^{6} 4 s^{2} 3 d^{10} 4 p^{6} 5 s^{1} 4 d^{4}\) [This is the transition metal niobium, which violates the four-block electron-assignment rules, so be careful.]

Short answer

The electron configurations with errors are: (b) \(1 s^{2} 2 s^{6} 3 s^{2} 3 p^{6} 4 s^{2} 4 p^{6}\): Incorrect number of electrons in 2s orbital. (c) \(2 s^{2} 2 p^{6} 3 s^{2} 3 p^{6} 4 s^{2} 3 d^{7}\): Missing 1s orbital. (d) \(1 s^{2} 2 s^{2} 2 p^{7} 3 s^{3} 3 p^{6}\): Incorrect number of electrons in 2p and 3s orbitals. (f) \(1 s^{2} 2 s^{2} 2 p^{6} 3 s^{2} 3 p^{6} 4 s^{2} 3 d^{10} 4 p^{6} 5 s^{1} 4 d^{4}\): Incorrect electron configuration for niobium (Nb).

Step-by-step solution

Identify incorrect electron configurations

For each electron configuration, we will check if the number of electrons in each subshell doesn't exceed the maximum allowed value, and we will analyze the order they are in. (a) \(1 s^{2} 2 p^{6} 3 s^{1}\): This electron configuration is correct; there's nothing wrong with it. (b) \(1 s^{2} 2 s^{6} 3 s^{2} 3 p^{6} 4 s^{2} 4 p^{6}\): The number of electrons in the 2s orbital is incorrect. The maximum number of electrons in the s orbital is 2. (c) \(2 s^{2} 2 p^{6} 3 s^{2} 3 p^{6} 4 s^{2} 3 d^{7}\): The 1s orbital is missing from this configuration. (d) \(1 s^{2} 2 s^{2} 2 p^{7} 3 s^{3} 3 p^{6}\): The number of electrons in the 2p and 3s orbitals are incorrect. The maximum number of electrons in a p orbital is six, and the maximum in an s orbital is two. (e) \(1 s^{2} 2 s^{2} 2 p^{5} 3 s^{1}\): This electron configuration is correct; there's nothing wrong with it. (f) \(1 s^{2} 2 s^{2} 2 p^{6} 3 s^{2} 3 p^{6} 4 s^{2} 3 d^{10} 4 p^{6} 5 s^{1} 4 d^{4}\): The electron configuration for niobium (Nb, atomic number 41) violates the electron-assignment rules. The correct configuration is \(1s^2\: 2s^2\: 2p^6\: 3s^2\: 3p^6\: 4s^2\: 3d^{10}\: 4p^6\: 5s^1\: 4d^{5}\).

Summary of the incorrect electron configurations and their issues

Here, we'll summarize the incorrect configurations, highlighting the error in each one. (b) \(1 s^{2} 2 s^{6} 3 s^{2} 3 p^{6} 4 s^{2} 4 p^{6}\): Incorrect number of electrons in 2s orbital. (c) \(2 s^{2} 2 p^{6} 3 s^{2} 3 p^{6} 4 s^{2} 3 d^{7}\): Missing 1s orbital. (d) \(1 s^{2} 2 s^{2} 2 p^{7} 3 s^{3} 3 p^{6}\): Incorrect number of electrons in 2p and 3s orbitals. (f) \(1 s^{2} 2 s^{2} 2 p^{6} 3 s^{2} 3 p^{6} 4 s^{2} 3 d^{10} 4 p^{6} 5 s^{1} 4 d^{4}\): Incorrect electron configuration for niobium (Nb).

Key concepts

Ground-State Electron Configuration

Understanding the ground-state electron configuration is crucial for many areas of chemistry, as it provides a way to describe the arrangement of electrons in an atom or ion at its lowest energy level.

An electron configuration lists the occupancy of an atom's orbitals by electrons. The ground state refers to the most stable, low-energy arrangement, where electrons fill the lowest available energy levels before moving to higher ones. The fundamental principle underlying this arrangement is the Aufbau principle, which means 'building up' in German. It implies that electron filling starts from the lowest energy orbital, which is the 1s orbital, and proceeds to higher energy orbitals sequentially.

An example of a correct ground-state electron configuration is that of helium: \( 1s^2 \), where both electrons of helium fill the lowest energy orbital, the 1s.

Orbital Electron Capacity

Each orbital in an atom has a maximum electron capacity dictated by quantum mechanics. It is essential to understand this capacity to determine if an electron configuration is correct.

Orbitals are divided into four types: s, p, d, and f. The s orbital can hold up to 2 electrons, as in hydrogen: \( 1s^1 \). A p orbital can hold up to 6 electrons, a d orbital can hold 10, and an f orbital can hold a maximum of 14 electrons. This capacity follows the expression \( 2n^2 \), where n represents the principal quantum number corresponding to the energy level.

One of the most common mistakes when assigning electrons to orbitals is exceeding these capacity limits, as seen in option (b) of the exercise, where the 2s orbital is incorrectly filled with six electrons instead of the maximum of two.

Niobium Electron Configuration

The electron configuration of niobium (Nb, atomic number 41) can present a bit of a challenge due to its position as a transition metal. Transition metals follow somewhat different rules because of their d orbitals. For niobium, the correct ground-state electron configuration is \( 1s^2 2s^2 2p^6 3s^2 3p^6 4s^2 3d^{10} 4p^6 5s^1 4d^{4} \), but in option (f), the 4d orbital was incorrectly filled with 4 electrons after the 5s orbital was filled. According to the electron-assignment rules, the 5s orbital is filled before the 4d orbitals start to fill, but the correct configuration for niobium would have five electrons in the 4d orbital, not four, thus violating the Aufbau principle.

Electron-Assignment Rules

When assigning electrons to an atom's orbitals, several rules must be followed: the Aufbau principle, the Pauli exclusion principle, and Hund's rule. The Aufbau principle, as mentioned, dictates that electrons fill up the lowest energy orbitals first. The Pauli exclusion principle states that no two electrons in an atom can have the same set of four quantum numbers, effectively limiting the number of electrons in each orbital based on their spin. Finally, Hund's rule explains that electrons will fill degenerate (equal energy) orbitals singly and with the same spin before doubling up.

These rules ensure the most stable, lowest energy electron configuration for an atom. Violations of these rules, such as placing too many electrons in an orbital or filling orbitals out of sequence, would result in incorrect configurations, as noted in several of the given examples from the exercise.