Problem 15 - Chemistry The Central Science In Si Units

Question

Detailed calculations show that the value of \(Z_{\text {eff }}\) for the outermost electrons in Na and \(\mathrm{K}\) atoms is \(2.51+\) and \(3.49+\), respectively. (a) What value do you estimate for \(Z_{\text {eff }}\) experienced by the outermost electron in both \(\mathrm{Na}\) and \(\mathrm{K}\) by assuming core electrons contribute 1.00 and valence electrons contribute 0.00 to the screening constant? (b) What values do you estimate for \(Z_{\text {eff }}\) using Slater's rules? (c) Which approach gives a more accurate estimate of \(Z_{\text {eff }}\) ? (d) Does either method of approximation account for the gradual increase in \(Z_{\text {eff }}\) that occurs upon moving down a group? (e) Predict \(Z_{\text {eff }}\) for the outermost electrons in the \(\mathrm{Rb}\) atom based on the calculations for \(\mathrm{Na}\) and \(\mathrm{K}\).

Step-by-Step Solution

Answer

(a) Na: 1.00, K: 1.00; (b) Na: 1.15, K: 2.20; (c) Slater's rules are more accurate; (d) Neither fully accounts for gradual increase; (e) Estimate Rb's \( Z_{\text{eff}} \approx 4.5 \).

Step 1: Estimate Z_eff assuming core electron contribution

For sodium (Na), we consider the core electrons as the ones in the 1s, 2s, and 2p shells. There are 10 core electrons contributing 1.00 each to the screening constant. So, the effective nuclear charge, \( Z_{\text{eff,Na}} = Z - S \), where \( Z = 11 \) and \( S = 10 \). Thus, \( Z_{\text{eff,Na}} = 11 - 10 = 1.00 \).For potassium (K), core electrons include 1s, 2s, 2p, 3s, and 3p shells, totaling 18 electrons. The screening constant \( S \) is thus 18. Therefore, \( Z_{\text{eff,K}} = 19 - 18 = 1.00 \).

Step 2: Estimate Z_eff using Slater's rules for Na

Slater's rules say: 1. Electrons in the same group (nl set) contribute 0.35.2. Electrons in groups with one lower principal quantum number contribute 0.85.3. Electrons in groups with two or more lower quantum numbers contribute 1.00.For Na, the configuration is \([\text{Ne}]3s^1\). Using Slater's rules, the electron in the \(3s\) orbital does not shield itself, so \( S = 9 \times 1.00 + 1 \times 0.85 = 9.85 \), giving \( Z_{\text{eff,Na}} = 11 - 9.85 = 1.15 \).

Step 3: Estimate Z_eff using Slater's rules for K

The electronic configuration of K is \([\text{Ar}]4s^1\) and Slater's rules are applied.1. \(3d\), \(3p\), and \(3s\) electrons contribute 0.85 each.2. The \(1s\), \(2s\), and \(2p\) electrons each contribute 1.00.So, \( S = 8 \times 1.00 + 8 \times 0.85 = 16.80 \), giving \( Z_{\text{eff,K}} = 19 - 16.80 = 2.20 \).

Step 4: Compare methods and evaluate accuracy

The core electron contribution method gives \( Z_{\text{eff}} \) values too low compared to the given data (Na: 2.51, K: 3.49). Slater's rules, although still underestimating it, are much closer to the given values. Therefore, Slater's rules provide a more accurate estimate of the effective nuclear charge (\( Z_{\text{eff}} \)).

Step 5: Consider comparison with periodic trends

Effective nuclear charge generally increases down a group. This trend is somewhat reflected in the results using Slater's rules, but not accurately represented by the core electron contribution method, which gives the same \( Z_{\text{eff}} \) value for different atoms.

Step 6: Predict Z_eff for Rb based on trend

Based on the trend observed (Na: 2.51, K: 3.49), and anticipating that the effective nuclear charge will continue increasing as we move down the group, for \( \text{Rb} \), we estimate a \( Z_{\text{eff,Rb}} \approx 4.5 \). This assumes continued increase in effective nuclear charge.

Key Concepts

Slater's Rules

Slater's Rules are a set of guidelines used to estimate the screening constant, which helps calculate the effective nuclear charge \(Z_{\text{eff}}\) experienced by an electron in an atom. These rules account for the fact that electrons in an atom do not equally shield each other from the nuclear charge. Instead, their shielding effect varies depending on their position in the atom's electron configuration.
To apply Slater's Rules, follow these steps:

• Group the electrons in the same principal quantum number \(n\), such as \(1s\), \(2s\), \(2p\), etc.
• Electrons within the same group contribute 0.35 to the screening constant, except for \(1s\) which contribute 0.30.
• Electrons in groups with a principal quantum number one less (\(n-1\)) contribute 0.85.
• Electrons in shells with even lower quantum numbers contribute a full 1.00.

The primary advantage of these rules is that they give a more nuanced estimate of \(Z_{\text{eff}}\), recognizing the varying degrees of electron shielding within the atom. When applied to sodium (Na) and potassium (K), Slater's Rules provide a closer estimate of the actual effective nuclear charge compared to simpler methods.

Screening Constant

The screening constant \(S\) is a crucial component for calculating the effective nuclear charge \(Z_{\text{eff}}\), which represents the positive charge from the nucleus that an electron "feels". The screening constant accounts for the repulsive effects of other electrons, which lessen the pull of the nucleus on any given electron.
In simpler terms, each electron in an atom acts like a shield, partially blocking the force of the nucleus from reaching other electrons. The screening constant is the sum of these blocking effects, calculated based on Slater's Rules.
For example:

• Core electrons contribute significantly (either 0.85 or 1.00 for lower \(n\) shells) to the screening constant because they efficiently shield outer electrons.
• Electrons in the same shell as the target electron contribute a lesser amount (0.35), asserting that they have a weaker shielding effect.

Using the screening constant, you can derive the effective nuclear charge using the formula:\[Z_{\text{eff}} = Z - S\]where \(Z\) is the atomic number of the element. This concept helps explain why \(Z_{\text{eff}}\) increases across a period, as electrons added to the same shell do not screen each other effectively.

Periodic Table Trends

The effective nuclear charge \(Z_{\text{eff}}\) increases across a period from left to right on the periodic table and generally also increases when moving down a group. This trend is due to both the effective nuclear charge increasing, and the electron shielding varying as the atomic numbers increase.
When we move across a period:

• The nuclear charge \(Z\) increases with each additional proton in the nucleus, which tends to increase \(Z_{\text{eff}}\) for electrons in that period.
• Electrons are added to the same energy level, meaning they do not effectively shield one another, leading to a minimal increase in the screening constant.

This results in a stronger effective nuclear charge as you move across a period, making electrons more tightly bound to the nucleus.
When we move down a group:

• Electrons are added to new shells, farther from the nucleus, and while \(Z\) increases significantly, the increased \(S\) from additional inner-shell electrons offers stronger shielding.
• The overall effect still often results in an increased \(Z_{\text{eff}}\), yet the increase is less dramatic than moving across a period due to better shielding.

Understanding these trends helps predict and explain the chemistry and reactivity of elements, such as why elements in the same group react similarly and why atomic sizes vary across the table.