Question

Consider the change in effective nuclear charge experienced by a \(2 p\) electron as we proceed from \(C\) to \(N\). (a) Based on a simple model in which core electrons screen the valence electrons completely and valence electrons do not screen other valence electrons, what do you predict for the change in \(Z_{\text {eff }}\) from \(C\) to \(N\) ? (b) What change do you predict using Slater's rules? (c) The actual change in \(Z_{\text {eff }}\) from \(C\) to \(N\) is \(0.70+.\) Which approach to estimating \(Z_{\text {eff }}\) is more accurate? (d) The change in \(Z_{\text {dif }}\) from \(N\) to \(O\) is smaller than that from \(C\) to \(N\). Can you provide an explanation for this observation?

Short answer

The change in effective nuclear charge (\(Z_{\text {eff }}\)) experienced by a 2p electron as we proceed from Carbon (C) to Nitrogen (N) can be calculated using a simple model and Slater's rules. The simple model predicts a change in \(Z_{\text {eff }}\) of 1, while Slater's rules predict a change of 0.65. The actual change is 0.70, showing that Slater's rules provide a more accurate estimation. The smaller change in \(Z_{\text {eff }}\) from Nitrogen (N) to Oxygen (O) can be explained by the increased electron-electron repulsion within the 2p subshell, reducing the net force experienced by the 2p electrons.

Step-by-step solution

Understand the definitions and determine the atomic numbers

The effective nuclear charge (\(Z_{\text {eff }}\)) experienced by an electron is the net positive charge experienced by that electron after accounting for the shielding effect of other electrons which reduces the attractive force of the protons in the nucleus. To proceed with the calculation, we need to know the atomic numbers of Carbon (C) and Nitrogen (N), which are 6 and 7, respectively.

Calculate the change in \(Z_{\text {eff }}\) using the simple model

According to the simple model, core electrons screen valence electrons completely, and valence electrons don't screen each other. For Carbon (C), the core electrons are 2 (from the 1s shell) and the number of protons (Z) is 6. Therefore, the effective nuclear charge of a 2p electron in Carbon is: \(Z_{\text {eff }} = Z - core \, electrons = 6 - 2 = 4\) For Nitrogen (N), the core electrons are 2 (from the 1s shell) and the number of protons (Z) is 7. Thus, the effective nuclear charge of a 2p electron in Nitrogen is: \(Z_{\text {eff }} = Z - core \, electrons = 7 - 2 = 5\) Using this simple model method, the change in effective nuclear charge from Carbon (C) to Nitrogen (N) is: \(ΔZ_{\text {eff }} = Z_{\text {eff }}(N) - Z_{\text {eff }}(C) = 5 - 4 = 1\)

Calculate the change in \(Z_{\text {eff }}\) using Slater’s rules

According to Slater's rules, the screening constant for a 2p electron can be calculated using weighted contributions from electrons in other shells. For both Carbon and Nitrogen, these contributions are as follows: - All other electrons in the 2p shell: 0.35 each - All electrons in the 2s shell: 0.85 each - All electrons in the n=1 shell: 1.00 each Starting with Carbon, we have the following: \(Core \, shielding \, (1s) = 2(1.00) = 2.00\) \(Valence \, shielding \, (2s) = 2(0.85) = 1.70\) \(Z_{\text {eff }}(C) = Z - Total \, shielding = 6 - (2.00 + 1.70) = 2.30\) Now for Nitrogen: \(Core \, shielding \, (1s) = 2(1.00) = 2.00\) \(Valence \, shielding \, (2s) = 2(0.85) = 1.70\) \(Valence \, shielding \, (2p) = 1(0.35) = 0.35\) \(Z_{\text {eff }}(N) = Z - Total \, shielding = 7 - (2.00 + 1.70 + 0.35) = 2.95\) Using Slater's rules, the change in effective nuclear charge from Carbon (C) to Nitrogen (N) is: \(ΔZ_{\text {eff }} = Z_{\text {eff }}(N) - Z_{\text {eff }}(C) = 2.95 - 2.30 = 0.65\)

Compare the predictions and the actual change

We now have the following predictions: Simple model: \(ΔZ_{\text {eff }} = 1\) Slater's rules: \(ΔZ_{\text {eff }} = 0.65\) The actual change in effective nuclear charge from Carbon (C) to Nitrogen (N) is \(0.70\). The approach based on Slater's rules (\(0.65\)) is closer to the actual change, making it the more accurate method between the two.

Explain the smaller change from Nitrogen (N) to Oxygen (O)

The smaller change in effective nuclear charge from Nitrogen (N) to Oxygen (O) can be explained by considering electron-electron repulsion. When moving from Nitrogen to Oxygen, there is an additional electron within the same 2p subshell. The increased electron-electron repulsion within the subshell leads to a reduction in the net force experienced by the 2p electrons, resulting in a smaller change in the effective nuclear charge observed.

Key concepts

Slater's rules

Understanding Slater's rules is crucial for anyone delving into the world of atomic structure and chemistry. These rules were formulated by John C. Slater in the 1930s and provide a simple and pragmatic method to estimate the effective nuclear charge (\( Z_{\text{eff}} \) that an electron experiences within an atom. Slater's rules take into account the screening effect or 'shielding' by other electrons in the atom, significantly altering the net charge felt by any given electron.

The rules assign electrons a screening constant based on their position in the atomic orbitals. Electrons closer to the nucleus, according to Slater, shield outer electrons more effectively than those further away. His rules present us with a set of priorities:

• Electrons in the same group as the one in question shield partially.
• Electrons in groups closer to the nucleus shield more effectively than those further out.

Application of these rules allows us, as seen in the exercise with Carbon and Nitrogen, to calculate a more realistic effective nuclear charge experienced by the 2p electrons, refining our understanding beyond the simple assumption that only core electrons shield valence electrons. Slater's rules introduce the concept that valence electrons also shield each other, albeit to a lesser extent. This method, able to predict more accurate differences in effective nuclear charge with changes in atomic number, lends itself to an enhanced comprehension of atomic behavior.

Electron shielding

Electron shielding, also known as screening, is a fundamental concept that helps explain the behavior of atoms and their electrons in various chemical environments. It addresses the way in which inner shell electrons can diminish the pull of the nucleus on outer shell electrons.

Let's illustrate this with a common kitchen analogy: Picture the nucleus of an atom as a magnet and the electrons as layers of fabric wrapped around it. The innermost fabric layer feels the magnet's pull most strongly, while the subsequent layers experience a progressively diminished force due to the buffering effect of the material between them and the magnet. Similar to this, electrons in the inner shells create a buffer that weakens the nuclear attraction for electrons in the outer shells, essentially 'shielding' them.

As a result, when calculating the effective nuclear charge (\( Z_{\text{eff}} \)), we must account for this shielding to comprehend the actual strength of the nucleus's attraction on a given electron. The calculations become even more complex when considering electrons in the same outer shell because, as noted in the exercise, the introduction of additional electrons increases repulsion, thus affecting the effective nuclear charge. This dynamic interplay between attraction and repulsion due to electron shielding is critical for understanding atomic properties and their variations across the periodic table.

Atomic structure

The atomic structure pertains to how an atom is built, comprising protons, neutrons, and electrons. The protons and neutrons form the atomic nucleus while electrons whirl around this core within regions known as orbitals. Each atomic element features a unique number of protons, known as the atomic number, which defines its position on the periodic table.

The structure is not chaotic; it follows well-defined rules. Electrons fill orbitals starting from the lowest energy level progressing to higher ones—a principle known as the Aufbau principle. This orderly filling of orbitals leads to the establishment of distinct electron 'shells' and 'subshells', which determine the chemical properties of an element.

An electron in a 2p orbital of Nitrogen experiences a different effective nuclear charge compared to one in Carbon due to these electrons being in different atomic environments. Understanding atomic structure, including how many protons there are in the nucleus (atomic number) and the distribution of electrons (electron configuration), is vital for any discussion on effective nuclear charge, as it lays the foundation for concepts such as Slater's rules and electron shielding, ultimately explaining how an electron experiences the nucleus' pull.