Question

Consider the isoelectronic ions \(\mathrm{F}^{-}\) and \(\mathrm{Na}^{+}\). (a) Which ion is smaller? (b) Using Equation \(7.1\) and assuming that core electrons contribute \(1.00\) and valence electrons contribute \(0.00\) to the screening constant, \(S\), calculate \(Z_{\text {eff }}\) for the \(2 \mathrm{p}\) electrons in both ions. (c) Repeat this calculation using Slater's rules to estimate the screening constant, \(S\). (d) For isoelectronic ions, how are effective nuclear charge and ionic radius related?

Short answer

For isoelectronic ions F- and Na+, Na+ is smaller in size due to its higher atomic number. The effective nuclear charge (Zeff) for 2p electrons in F- is 7 using Equation 7.1, and 5.95 using Slater's rules. For Na+, the Zeff for 2p electrons is 1 using Equation 7.1, and 8.30 using Slater's rules. In isoelectronic ions, the effective nuclear charge and ionic radius are inversely related: as Zeff increases, the ionic radius decreases, and vice versa.

Step-by-step solution

(a) Comparing the size of F- and Na+

The ions F- and Na+ are isoelectronic, which means they have the same number of electrons. F- has gained an extra electron, resulting in a total of 10 electrons. Na+ has lost an electron, also resulting in 10 electrons. To determine which ion is smaller, we can compare the number of protons in the nucleus (i.e., the atomic number). Fluorine has an atomic number of 9, while sodium has an atomic number of 11. Therefore, Na+ has a stronger positive charge in its nucleus compared to F-, which causes the electrons to be pulled in more closely, making Na+ smaller than F-.

(b) Calculating Zeff using Equation 7.1

Equation 7.1 states that the effective nuclear charge (Zeff) can be calculated by subtracting the screening constant (S) from the atomic number (Z): \[Z_{\text{eff}} = Z - S\] In this case, we are given that core electrons contribute 1.00 and valence electrons contribute 0.00 to the screening constant. For F-, there are 2 core electrons and 8 valence electrons. Therefore, the screening constant S = (2 × 1.00) + (8 × 0.00) = 2. The atomic number of F is 9, so the Zeff for the 2p electrons in F- is: \(Z_{\text{eff}} = 9 - 2 = 7\) For Na+, there are 10 core electrons and 0 valence electrons. The screening constant S = (10 × 1.00) + (0 × 0.00) = 10. The atomic number of Na is 11, so the Zeff for the 2p electrons in Na+ is: \(Z_{\text{eff}} = 11 - 10 = 1\)

(c) Calculating Zeff using Slater's rules

Slater's rules are a set of empirical rules to estimate the screening constant (S) in a more accurate way. Using Slater's rules, the screening constant for F- and Na+ can be determined by considering the contributions from different electron groups: For F-: 1s electrons: 2 × 0.30 = 0.60 2s and 2p electrons: 7 × 0.35 = 2.45 Total S: 0.60 + 2.45 = 3.05 The Zeff for F- in the 2p orbital is: \(Z_{\text{eff}} = 9 - 3.05 = 5.95\) For Na+: 1s electrons: 2 × 0.30 = 0.60 2s and 2p electrons: 6 × 0.35 = 2.10 3s and 3p electrons: 0 (no valence electrons) Total S: 0.60 + 2.10 = 2.70 The Zeff for Na+ in the 2p orbital is: \(Z_{\text{eff}} = 11 - 2.70 = 8.30\)

(d) Effective nuclear charge and ionic radius relationship

For isoelectronic ions, the effective nuclear charge (Zeff) and the ionic radius are inversely related. As the effective nuclear charge increases, the electrons are more strongly attracted to the nucleus, causing the ions to become smaller in radius. Conversely, as the effective nuclear charge decreases, the positive charge of the nucleus is less capable of attracting the electrons, resulting in a larger ionic radius.

Key concepts

Ionic Radius

The concept of ionic radius is fundamental when discussing isoelectronic ions, which are species with the same number of electrons.

In the context of isoelectronic ions, ionic radius refers to the distance between the nucleus of the ion and its outermost shell of electrons. As ions possess different nuclear charges, the ionic radius varies—ions with higher nuclear charges will have a smaller radius, as the increased positive charge pulls the electrons closer to the nucleus.

Specifically, in isoelectronic ions like F- and Na+, both have the same number of electrons, but Na+ has more protons, resulting in a higher effective nuclear charge and thus a smaller ionic radius compared to F-. This illustrates the general principle that, within a set of isoelectronic species, the more positively charged ion will be smaller.

Effective Nuclear Charge (Zeff)

The effective nuclear charge (Zeff) is the net positive charge experienced by an electron in a multi-electron atom.

This value becomes significant in comparing atoms and ions because it helps explain the strength of the attraction between the nucleus and the electrons. The concept ties closely with the size of the ions; a higher Zeff means the electrons are held more tightly by the nucleus, leading to a smaller ionic radius.

When applying Equation 7.1 to calculate the Zeff for F- and Na+, it's apparent that the difference in proton number directly influences the Zeff. Despite both ions having identical electron configurations, Na+ has a higher Zeff due to its additional protons, justifying its smaller size compared to F-.

Slater's Rules

Slater's rules provide a more nuanced method to calculate the screening constant, which in turn refines our estimation of the Zeff. Developed by John C. Slater, these rules take into account the repulsion effects of different electron groups on each other.

According to Slater's rules, electrons are divided into groups based on their principal quantum number and subshells. Electrons in the same group shield each other less effectively than electrons in lower groups. In the case of F- and Na+, applying Slater's rules yields different screening constants because the rules differentiate between the repulsion from core electrons (1s, for example) and those at the same energy level (2s and 2p). The calculations demonstrate a more accurate Zeff by factoring in the varying shielding effects of the different electron groups.

Screening Constant

The screening constant, denoted by S, defines the extent to which electrons in an atom shield or screen the nuclear charge from other electrons.

In essence, this constant is a measure of how much the repulsion from other electrons reduces the attractive force that each electron feels from the nucleus. The higher the screening constant, the less net positive charge the electrons will experience from the nucleus, which in turn affects the effective nuclear charge (Zeff).

When using the simplified model given in the problem, where core electrons contribute a constant value to the screening constant and valence electrons contribute none, it becomes clear how core electrons play a critical role in determining the size of the ion. When applying Slater's rules for a more accurate measurement, the calculated Zeff is typically lower than with the simplified approach, revealing the importance of considering electron-electron interactions to accurately assess ionic sizes.