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 \(\mathrm{N}\) ? (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 \(\mathrm{Z}_{\text {eff }}\) from \(\mathrm{N}\) to \(\mathrm{O}\) is smaller than that from \(\mathrm{C}\) to \(\mathrm{N}\). Can you provide an explanation for this observation?

Short answer

In summary, the change in effective nuclear charge (Z_eff) from Carbon to Nitrogen is predicted to increase by 1 using the simple model, while Slater's rules predict a change of 0.70. When compared to the actual change in Z_eff (0.70), Slater's rules provide a more accurate estimation. The change in Z_eff from Nitrogen to Oxygen is smaller than that from Carbon to Nitrogen due to electron-electron repulsions within the 2p orbital, which are not considered in the simple model but are accounted for in Slater's rules.

Step-by-step solution

Understand effective nuclear charge and the simple model

Effective nuclear charge (Z_eff) is the net positive charge experienced by an electron in a multi-electron atom. In a simple model, we assume that core electrons completely shield valence electrons, and valence electrons do not shield each other. C: Atomic number (Z) = 6 and N: Atomic number (Z)= 7 Since we are comparing the same 2p electron for both C and N and moving from left to right in the periodic table, the atomic numbers increase, and so does the number of protons. Using the simple model, the Z_eff should increase by 1 from C to N.

Calculate Z_eff using Slater's rules

Slater's rules take into account different types of shielding for different orbitals. Here are the steps to calculate Z_eff using Slater's rules: 1. Write down the electron configuration of the given element. 2. Determine the electron for which you want to calculate Z_eff. 3. Apply the shielding coefficients depending on the type of electron. 4. Calculate the Z_eff using the equation: Z_eff = Z - S For Carbon (C): 1. Electron configuration: 1s^2 2s^2 2p^2 2. We want to calculate the Z_eff for the 2p electron. 3. Shielding coefficients: 1s electrons shield completely (2), 2s electrons shield (0.85 * 2) 4. Z_eff (for C) = 6 - (2 + 1.7) = 2.30 For Nitrogen (N): 1. Electron configuration: 1s^2 2s^2 2p^3 2. We want to calculate the Z_eff for the 2p electron. 3. Shielding coefficients: 1s electrons shield completely (2), 2s electrons shield (0.85 * 2), one 2p electron shield (0.35) 4. Z_eff (for N) = 7 - (2 + 1.7 + 0.35) = 3.00 Difference in Z_eff using Slater's rules: 3.00 - 2.30 = 0.70

Compare estimated Z_eff with actual change

The actual change in Z_eff from C to N is 0.70. Comparing the simple model with Slater's rules, the following are the changes in Z_eff: Simple Model: Change in Z_eff = 1 Slater's Rules: Change in Z_eff = 0.70 Based on these results, the Slater's rules approach to estimating Z_eff is more accurate than the simple model.

Provide an explanation for the change in Z_eff from N to O

In the case of Nitrogen (N) to Oxygen (O), the actual change in Z_eff is smaller than in the Carbon (C) to Nitrogen (N) case. This can be explained by electron-electron repulsions within the 2p orbital. As the number of electrons in the 2p orbital increases from N to O, the repulsive forces between them increase and, as a result, reduce the net effective nuclear charge. This electron-electron repulsion effect is not taken into account in the simple model but is included in the calculations using Slater's rules.

Key concepts

Slater's rules

Slater's rules help us estimate the effective nuclear charge (\(Z_{\text{eff}}\)) experienced by electrons in atoms. These rules consider the shielding effect, where inner-shell electrons block the outer electrons from the full attractive force of the nucleus.
To apply Slater's rules, follow these steps:

• Write down the electron configuration of the element.
• Identify which electron's \(Z_{\text{eff}}\) you need to calculate.
• Assign shielding coefficients based on electron types (e.g., 1s, 2s).
• Calculate \(Z_{\text{eff}} = Z - S\), where \(Z\) is the atomic number and \(S\) is the shielding constant.

These rules are more accurate than simpler models because they account for different electron interactions. When applied to elements like carbon and nitrogen, Slater’s rules give us a more realistic picture of \(Z_{\text{eff}}\).

shielding effect

The shielding effect describes how core electrons block valence electrons from feeling the nucleus's full positive charge. This is crucial for understanding variations in effective nuclear charge across different elements.
Core electrons are closer to the nucleus and repel valence electrons, reducing the attractive force these outer electrons experience. The more core electrons, the stronger the shielding effect.
As you move across a period (C to N), the number of core electrons stays the same, but the number of protons increases. This leads to a stronger pull on valence electrons, increasing \(Z_{\text{eff}}\). However, how we account for this effect changes depending on the model used.

atomic structure

Atomic structure refers to the arrangement of electrons around a nucleus. The central nucleus contains protons and neutrons, while electrons occupy orbitals in layers around it. Understanding this structure is key to predicting an element's chemical properties.
Electrons are organized in shells and subshells: for example, the configuration for carbon and nitrogen involves the 1s, 2s, and 2p orbitals. Electrons closer to the nucleus have lower energy levels and feel a higher effective nuclear charge compared to those further away.
This distribution helps explain why elements behave differently in chemical reactions, and why calculating \(Z_{\text{eff}}\) accurately is important for predicting these behaviors.

periodic trends

Periodic trends refer to predictable changes in the properties of elements as you move across the periodic table. The effective nuclear charge is one such trend.
As you move from left to right across a period, \(Z_{\text{eff}}\) generally increases. This increase is due to additional protons in the nucleus, which pull valence electrons closer, and also due to minimal changes in inner electron shielding.
However, the change in \(Z_{\text{eff}}\) can vary due to factors like electron-electron repulsions in the same orbital, reflected more accurately by Slater's rules. Recognizing these trends is essential for understanding chemical reactivity and bonding. By understanding these nuances, students can better predict how elements might interact in different scenarios.