Question

(a) If the core electrons were totally effective at screening the valence electrons and the valence electrons provided no screening for each other, what would be the effective nuclear charge acting on the $3s$ and $3p$ valence electrons in P? (b) Repeat these calculations using Slater's rules. (c) Detailed calculations indicate that the effective nuclear charge is $5.6+$ for the $3s$ electrons and $4.9+$ for the $3p$ electrons. Why are the values for the $3s$ and $3p$ electrons different? (d) If you remove a single electron from a Patom, which orbital will it come from?

Short answer

In summary, for Phosphorus (P) with the electron configuration 1s² 2s² 2p⁶ 3s² 3p³, the effective nuclear charge (Z_eff) for both 3s and 3p valence electrons without valence electron screening would be +5. Using Slater's rules, Z_eff is +12.05 for 3s valence electrons and +12.9 for 3p valence electrons. The difference between these values and the detailed calculation result (5.6+ for 3s and 4.9+ for 3p electrons) is due to Slater's rules being an approximation and not accounting for certain electron interactions accurately. The 3s and 3p electrons experience different shielding levels due to their different spatial distributions. When a P atom loses an electron, the electron is removed from the 3s orbital since it has a lower energy and shielding effect than the 3p orbital.

Step-by-step solution

1. Calculate Core and Valence Electrons for Phosphorus

Phosphorus has an atomic number of 15, meaning it has 15 electrons. Its electron configuration is 1s² 2s² 2p⁶ 3s² 3p³. The 1s² 2s² 2p⁶ electrons (10 electrons) are the core electrons, while the 3s² 3p³ electrons (5 electrons) are the valence electrons.

2. Calculate Effective Nuclear Charge (a)

Assuming total core electron screening and no valence electron screening, we can calculate the effective nuclear charge (Z_eff) as follows: Z_eff = Z - core_electrons Z_eff = 15 - 10 (atomic number of P - number of core electrons) Z_eff = 5 Here, the effective nuclear charge for both 3s and 3p valence electrons would be +5.

3. Calculate Effective Nuclear Charge using Slater's Rules (b)

To find the Z_eff using Slater's rules, we first need to sum up the screening constants (S) for each of the orbitals: For 3s electron: S = (0.35 * 2) + (0.85 * 3) = 2.95 For 3p electron: S = (0.35 * 2) + (0.85 * 2) = 2.1 Now, we can calculate the Z_eff using Slater's rules: Z_eff(3s) = Z - S = 15 - 2.95 = 12.05 Z_eff(3p) = Z - S = 15 - 2.1 = 12.9 Using Slater's rules, the effective nuclear charge for 3s valence electrons is +12.05 and for 3p valence electrons, it's +12.9.

4. Discuss discrepancy between calculated values and actual values (c)

The calculated effective nuclear charges using Slater's rules are higher than the detailed calculation results (5.6+ for 3s and 4.9+ for 3p electrons). The discrepancy occurs because Slater's rules are an approximation and do not account for certain electron interactions accurately. The values obtained are different for 3s and 3p because they experience different levels of shielding due to their different spatial distributions around the nucleus.

5. Determine which orbital loses an electron (d)

To determine which orbital loses an electron, we should look at the ionization energy for each orbital. Electrons are removed from the orbital with the lowest ionization energy. In general, the ionization energy increases as we move from lower to higher energy orbitals. Since the 3s orbital has a lower energy and shielding effect than the 3p orbital, the 3s electron will be removed first.

Key concepts

Slater's rules

Slater's rules are a set of guidelines used to estimate the effective nuclear charge ($Z_{eff}$) experienced by an electron in an atom. These rules provide a way to calculate how much positive charge from the nucleus effectively acts on a given electron, after accounting for the electron-electron repulsion or shielding. Slater's rules help predict atomic properties by giving insights into how tightly an electron is held by the nucleus.

Use Slater's rules by assigning electrons into groups based on quantum numbers and then applying different shielding constants:

• Electrons within the same group (n, l) but to the left of the electron under consideration each contribute 0.35 to the shielding constant.
• Those that are one principal quantum number below ($n-1$) each contribute 0.85.
• Electrons further below ($n-2$) contribute a full 1.00.

Unlike detailed quantum mechanical calculations, Slater's rules provide an approximation but are intuitive and quicker for understanding trends. For example, in the phosphorus atom where the effective nuclear charge for 3s and 3p electrons is calculated, Slater’s rules help to deduce how these electrons perceive the nuclear charge as opposed to other electrons in different arrangements.

Electron Shielding

Electron shielding, or screening, is a concept where inner electrons in an atom reduce the effective nuclear charge felt by the outer electrons. This is similar to casting a shadow on some of the nuclear strength, preventing the valence electrons from feeling the full force of the nucleus. When calculating effective nuclear charge, shielding is a crucial factor as it affects the stability and chemical properties of an atom.

In elements with multiple electron shells, inner electrons act as shields:

• Core electrons, being closest to the nucleus, are quite efficient in shielding the outer electrons.
• However, valence electrons do not effectively shield each other due to their similar distances from the nucleus.

For phosphorus, calculations that assume complete electron shielding show a dramatic difference from those incorporating detailed corrections. This demonstrates shielding's impact, as the spatial arrangement and number of electrons determine how much of the nuclear charge is 'felt' by the valence electrons. Therefore, understanding electron shielding is essential for predicting behaviors such as ionization energies and chemical reactivity.

Valence Electrons

Valence electrons are the outermost electrons in an atom and play a significant role in chemical bonding and reactions. These electrons are found in the highest energy level or shell of an atom, making them the most accessible for participation in chemical interactions. In general, valence electrons determine the reactivity and bonding pattern exhibited by the element.

For phosphorus, which has the electron configuration 1s² 2s² 2p⁶ 3s² 3p³, there are five valence electrons:

• Two in the 3s orbital.
• Three in the 3p orbitals.

The 3s and 3p valence electrons are vital as they are the first to interact and form bonds, influencing the element's chemical properties.
When assessing aspects such as ionization energy, valence electrons are the primary focus since removing one affects the chemical identity of the element. For phosphorus, given a scenario where an electron is removed, it’s usually a valence electron that is lost. This change can affect its effective nuclear charge and its ability to bond with other elements, highlighting the importance of valence electrons in chemistry.