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Chapter 6: Problem 79

What is wrong with the following electron configurations for atoms in their ground states? (a) \(1 s^{2} 2 s^{2} 3 s^{1}\), (b) \([\mathrm{Ne}] 2 s^{2} 2 p^{3}\), (c) \([\mathrm{Ne}] 3 s^{2} 3 d^{5}\).

Short Answer

Expert verified
The electron configuration (a) \(1s^2 2s^2 3s^1\) is correct as it follows the Aufbau Principle, Pauli Exclusion Principle, and Hund's Rule. However, configurations (b) \([\text{Ne}]2s^2 2p^3\) and (c) \([\text{Ne}]3s^2 3d^5\) are incorrect. Configuration (b) incorrectly fills already occupied 2s and 2p orbitals in the Neon configuration, while configuration (c) incorrectly fills the 3d orbitals before the 3p orbitals.

Step by step solution

01

Validate configuration (a)

To validate the configuration \(1s^2 2s^2 3s^1\), let's check if the electrons are filling up the lowest energy levels available based on the Aufbau Principle: 1. \(1s^2\) - This is correct, as the 1s orbital is the lowest energy level and holds a maximum of 2 electrons. 2. \(2s^2\) - This is also correct, as the 2s orbital is the next lowest available energy level and holds a maximum of 2 electrons. 3. \(3s^1\) - This is also correct, as the 3s orbital is the next available energy level. The configuration \(1s^2 2s^2 3s^1\) follows the Aufbau Principle, the Pauli Exclusion Principle, and Hund's Rule. There is nothing wrong with this electron configuration.
02

Validate configuration (b)

To validate the configuration \([\text{Ne}]2s^2 2p^3\), let's first expand the \([\text{Ne}]\) notation, which represents the electron configuration of a Neon atom: \(1s^2 2s^2 2p^6\). The full electron configuration is: \(1s^2 2s^2 2p^6 2s^2 2p^3\) The third and fourth terms are incorrect, as the 2s and 2p orbitals are already fully occupied in the Neon configuration (\(2s^2 2p^6\)). Instead of using \([\text{Ne}]\), this configuration should have started with a higher energy level (\([\text{Xe}]\) for example). Therefore, the configuration \([\text{Ne}]2s^2 2p^3\) is incorrect.
03

Validate configuration (c)

To validate the configuration \([\text{Ne}]3s^2 3d^5\), let's first expand the \([\text{Ne}]\) notation, which represents the electron configuration of a Neon atom: \(1s^2 2s^2 2p^6\). The full electron configuration is: \(1s^2 2s^2 2p^6 3s^2 3d^5\) The \(3s^2\) part is correct, as the 3s orbital is the next available energy level after Neon's configuration. However, the \(3d^5\) part is not correct because before filling the 3d orbitals, electrons should fill the 3p orbitals. The correct electron configuration will have 3p orbitals filled first before filling any 3d orbitals. Therefore, the configuration \([\text{Ne}]3s^2 3d^5\) is incorrect. In conclusion: - The electron configuration \(1s^2 2s^2 3s^1\) is correct. - The electron configurations \([\text{Ne}]2s^2 2p^3\) and \([\text{Ne}]3s^2 3d^5\) are incorrect because they do not comply with the rules for electron configuration.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Aufbau Principle
At the heart of electron configurations is the Aufbau Principle, which dictates that electrons fill atomic orbitals in order of increasing energy levels, from lowest to highest.

Imagine a set of stairs, with each step representing a different energy level for the electrons. Just as you would climb the stairs from the bottom to the top, electrons fill the lower-energy orbitals before moving to higher-energy ones. This principle explains why the first electron enters the 1s orbital鈥攊t's the lowest energy level available.

However, it's not always a straight climb. The energy levels of atomic orbitals increase generally from 1s to 2s to 2p to 3s, and so on, but there are exceptions. To remember the correct order, many students use a diagram known as the Aufbau diagram or memorize the specific sequence of orbital filling.

For instance, in the exercise we're looking at, part (c) suggests filling the 3d orbitals before the 3p orbitals. This is a common mistake because the 3d orbitals are actually of higher energy than the 3p orbitals, meaning 3p must be filled first according to the Aufbau Principle.
Pauli Exclusion Principle
Complementing the Aufbau Principle is the Pauli Exclusion Principle, which is rooted in quantum mechanics. This principle states that no two electrons in an atom can have the same set of four quantum numbers.

Adhering to this rule is like ensuring no two people in a room are wearing an identical combination of hat, shirt, pants, and shoes鈥攊t's necessary for maintaining uniqueness. In the context of electron configurations, it means that each orbital can hold a maximum of two electrons, and they must have opposite spins. That's why you see configurations like '1s虏', indicating two electrons with opposite spins in the 1s orbital.

If we circle back to the problematic electron configuration in part (b) of the exercise, the repetition of '2s虏' after the neon core notation '[Ne]' violates the Pauli Exclusion Principle since these two electrons with identical quantum numbers have already been accounted for in neon's electron configuration.
Hund's Rule
Hund's Rule addresses the distribution of electrons among orbitals of equal energy, known as degenerate orbitals. It is often referred to as the 'bus seat' rule鈥攋ust as passengers prefer to sit alone rather than next to someone else on the bus, electrons fill empty orbitals before pairing up.

This rule dictates that electrons must occupy all the orbitals in a subshell singly, with parallel spins, before any orbital is doubly occupied. Hund's Rule is crucial for understanding why the 'p', 'd', and 'f' orbitals are each filled with one electron first before any of them gets a second electron.

In the context of our exercise, while all configurations properly account for Hund's Rule, it is key for students to recognize that this principle applies only after both the Aufbau Principle and Pauli Exclusion Principle have been considered, ensuring electrons are in the proper orbitals and arranged correctly within them.

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Most popular questions from this chapter

(a) Account for formation of the following series of oxides in terms of the electron configurations of the elements and the discussion of ionic compounds in Section \(2.7 :\) $\mathrm{K}_{2} \mathrm{O}, \mathrm{CaO}, \mathrm{Sc}_{2} \mathrm{O}_{3}, \mathrm{Ti} \mathrm{O}_{2}, \mathrm{V}_{2} \mathrm{O}_{5}, \mathrm{CrO}_{3} .$ (b) Name these oxides. (c) Consider the metal oxides whose enthalpies of formation (in kJ mol \(^{-1}\) ) are listed here. Calculate the enthalpy changes in the following general reaction for each case: $$ \mathrm{M}_{n} \mathrm{O}_{m}(s)+\mathrm{H}_{2}(g) \longrightarrow n \mathrm{M}(s)+m \mathrm{H}_{2} \mathrm{O}(g) $$ (You will need to write the balanced equation for each case and then compute \(\Delta H^{\circ} . )(\mathbf{d})\) Based on the data given, estimate a value of \(\Delta H_{f}^{\circ}\) for \(S c_{2} \mathrm{O}_{3}(s) .\)

(a) Consider the following three statements: (i) A hydrogen atom in the \(n=3\) state can emit light at only two specific wavelengths, (ii) a hydrogen atom in the \(n=2\) state is at a lower energy than the \(n=1\) state, and (iii) the energy of an emitted photon equals the energy difference of the two states involved in the emission. Which of these statements is or are true? (b) Does a hydrogen atom "expand" or "contract" as it moves from its ground state to an excited state?

(a) What is the frequency of radiation whose wavelength is \(0.86 \mathrm{~nm}\) ? (b) What is the wavelength of radiation that has a frequency of \(6.4 \times 10^{11} \mathrm{~s}^{-1}\) ? (c) Would the radiations in part (a) or part (b) be detected by an X-ray detector? (d) What distance does electromagnetic radiation travel in \(0.38 \mathrm{ps}\) ?

What is the maximum number of electrons in an atom that can have the following quantum numbers? (a) \(n=3, m_{l}=-2 ;\) (b) \(n=4\), \(l=3 ;(\mathrm{c}) n=5, l=3, m_{l}=2,(\mathrm{~d}) n=4, l=1, m_{l}=0\).

Bohr's model can be used for hydrogen-like ions-ions that have only one electron, such as \(\mathrm{He}^{+}\)and \(\mathrm{Li}^{2+}\). (a) Why is the Bohr model applicable to \(\mathrm{He}^{+}\)ions but not to neutral He atoms? (b) The ground-state energies of \(\mathrm{H}, \mathrm{He}^{+}\), and \(\mathrm{Li}^{2+}\) are tabulated as follows:
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