Question

Detailed calculations show that the value of \(Z_{\text {eff }}\) for the outermost electrons in \(\mathrm{Na}\) 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 \(\mathrm{K}\).

Short answer

Using the core and valence electron assumption, we estimate Zeff to be 1 for both Na and K. However, using Slater's rules, we find better estimations with Zeff(Na) = 2 and Zeff(K) = 3.7. Comparing these estimations to the detailed calculations, Slater's rules provide a more accurate estimate. Neither method precisely accounts for the gradual increase in Zeff upon moving down a group. For Rb, based on Slater's rules, and taking the average increase in Zeff between Na and K, we predict Zeff(Rb) to be approximately 4.7.

Step-by-step solution

Estimating Zeff using core and valence electron assumption

For Na and K, let's use the given assumption that core electrons contribute 1.00 and valence electrons contribute 0.00 to the screening constant S. Na (sodium) has the electron configuration [Ne]3s1, the outermost shell is the 3s orbital, so it has one valence electron and 10 core electrons. K (potassium) has the electron configuration [Ar]4s1, the outermost shell being the 4s orbital, so it has one valence electron and 18 core electrons. For Na: Zeff = Z - S, Where Z is the atomic number and S is the screening constant. Zeff(Na) = 11 - (10 × 1) = 1 For K: Zeff(K) = 19 - (18 × 1) = 1

Estimating Zeff using Slater's rules

Next, we will use Slater's rules to estimate Zeff. Slater's rules are a set of empirical formulas that help us to approximate the screening of charges on a given electron in an atom. Using Slater's rules: For Na: 3s: S(Na) = 0.35 × (0) + 0.85 × (10) Zeff(Na) = 11 - S(Na) = 11 - (0.85 × 10) = 2 For K: 4s: S(K) = 0.35 × (0) + 0.85 × (18) Zeff(K) = 19 - S(K) = 19 - (0.85 × 18) = 3.7

Comparing estimated values with detailed calculations

Now, let's compare the Zeff values from the core and valence electron assumption (Step 1) and Slater's rules (Step 2) with the given detailed calculation values (2.51 for Na and 3.49 for K). a) Core and valence electron assumption: Zeff(Na) = 1 (vs. 2.51 from detailed calculations) Zeff(K) = 1 (vs. 3.49 from detailed calculations) b) Slater's rules: Zeff(Na) = 2 (vs. 2.51 from detailed calculations) Zeff(K) = 3.7 (vs. 3.49 from detailed calculations)

Evaluating accuracy and accounting for the gradual increase in Zeff

From our comparison in Step 3, we can conclude that: c) Slater's rules give more accurate estimates of Zeff than the core and valence electron assumption. d) Slater's rules seem to provide a better estimation, but neither method precisely accounts for the gradual increase in Zeff upon moving down a group as they only provide rough estimations. The given detailed calculation values, however, show the gradual increase in Zeff.

Predicting Zeff for the outermost electron in Rb

We'll predict Zeff for the outermost electron in Rb based on the calculations for Na and K. Rb (rubidium) has the electron configuration [Kr]5s1, meaning its outermost shell is the 5s orbital with one valence electron and 36 core electrons. Because Slater's rules provided more accurate estimations for Na and K, we'll use it for our prediction. However, the increase in Zeff from Na to K is not constant, so we can't easily determine the value for Rb. A linear approximation could be used, based on the increase from Na to K, but this is a rough estimation. Let's take the average increase in Zeff between Na and K from the detailed calculations and apply it to the Slater's rules estimation: Increase in Zeff from Na (2.51) to K (3.49) is 3.49 - 2.51 = 0.98 Estimate for Rb: Zeff(Rb) = Zeff(K) + 0.98 Zeff(Rb) = 3.7 + 0.98 ≈ 4.7 So, our rough prediction for Zeff for the outermost electron in Rb is approximately 4.7.

Key concepts

Slater's Rules

Understanding Slater's Rules is crucial for grasping how effective nuclear charge (\( Z_{eff} \) is estimated. These rules, devised by John C. Slater, quantify the screening effect that electrons have on each other. Particularly, they give us a method to calculate the screening constant (\( S \)), which in turn helps to determine the net positive charge felt by an individual electron in a multi-electron atom.

Each electron in an atom is screened from the full nuclear charge by the other electrons. Slater's Rules provide a set of guidelines to calculate this screening effect for electrons in different orbitals. According to the rules, electrons are grouped based on their principal quantum number and their orbital type (such as s, p, d, or f). Electrons in the same group shield one another differently than electrons from other groups.

For instances, in sodium (\( \text{Na} \)), Slater's Rules help us to determine the screening constant for the valence electron in the 3s orbital. Here, we take into account the screening by the core electrons (configured as [Ne]) using certain factors as defined by the rules. By doing this, we get an approximate value of the effective nuclear charge felt by the outermost electron. The approximate nature of this charge is why Slater's Rules are considered empirical; they're based on observation and experiment rather than purely theoretical calculation.

Screening Constant

The screening constant (\( S \)) plays a pivotal role in determining the effective nuclear charge experienced by an electron in an atom. Essentially, it represents the degree to which other electrons in the atom block the positive charge of the nucleus from an electron of interest.

The concept of a screening constant is paramount when applying both Slater's Rules and more simplistic core and valence electron assumptions in estimating Zeff. The value of S depends on the distribution of all the other electrons in the atom and how effectively they 'screen' or shield the nuclear charge. The core electrons, being closer to the nucleus, tend to shield the valence electrons more effectively, which is why they are assigned higher values in the calculation of S. In contrast, the valence electrons, themselves being the outermost electrons, do not shield each other to the same degree and are often assigned a screening value of zero as seen in the given exercise.

Atomic Number

The atomic number (\( Z \)) of an element is the number of protons in the nucleus of its atoms and is also equivalent to the number of electrons in a neutral atom. For instance, sodium (\( \text{Na} \)) has an atomic number of 11, indicating it has 11 protons and, in its neutral state, 11 electrons.

The atomic number is central to the concept of effective nuclear charge, as it initially represents the full positive charge the electrons would experience if there were no inter-electronic repulsions or screening effects. When estimating the effective nuclear charge, we subtract the screening constant from the atomic number, effectively adjusting the nuclear charge to reflect the repulsive forces of the other electrons. For example, in the exercise, to find the Zeff for sodium, we subtract the screening constant calculated based on the number of core and valence electrons from sodium's atomic number.

Valence Electrons

Valence electrons are the outermost electrons of an atom and are fundamental in determining an element's chemical properties, such as its reactivity and bonding behavior. For the purpose of calculating Zeff, valence electrons are considered to contribute very little to the screening of nuclear charge, as reflected in the example where they were assigned a contribution of 0.00 to the screening constant.

The simplification used in this approach is not fully accurate since it does not account for the varying degrees of repulsion among valence electrons themselves. However, for many practical purposes, especially within the realm of chemistry, this simplification is often sufficient to provide a basic understanding of an atom's chemical nature. Moreover, knowing the configuration of valence electrons assists in applying Slater’s Rules, as these rules take into account the specific types and arrangements of electrons in an atom when estimating screening effects.

Electron Configuration

Electron configuration describes the distribution of electrons in an atom's orbitals. It provides a roadmap of where the electrons are located around the nucleus, detailing the specific subshells they occupy. This information is vital when applying Slater's Rules or estimating effective nuclear charge because the configuration determines the sequence and grouping of electrons that contribute to the screening effect.

In the example regarding sodium (\( \text{Na} \)) and potassium (\( \text{K} \)), the electron configurations are [Ne]3s1 and [Ar]4s1, respectively. This tells us that in each case, there is a single valence electron in the outermost s orbital, with the remaining electrons serving as the inner-shell or core electrons. The configuration directly informs the calculation of the screening constant, and, therefore, the effective nuclear charge (\( Z_{eff} \)) felt by the valence electron. Understanding electron configurations is also crucial for predicting the chemical properties of elements and the nature of the bonds they form.