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 \(3 s\) and \(3 p\) valence electrons in \(\mathrm{P}\) ? (b) Repeat these calculations using Slater's rules. (c) Detailed calculations indicate that the effective nuclear charge is \(5.6+\) for the \(3 s\) electrons and \(4.9+\) for the \(3 p\) electrons. Why are the values for the \(3 s\) and \(3 p\) electrons different? (d) If you remove a single electron from a P atom, which orbital will it come from?

Short answer

The effective nuclear charge acting on the 3s and 3p valence electrons in a phosphorus (P) atom under the given condition is 5+. Using Slater's rules, the effective nuclear charge is 7.7+ for 3s electrons and 8.7+ for 3p electrons. The difference between effective nuclear charges for 3s and 3p electrons is due to the 3s electron cloud being closer to the nucleus compared to the 3p electron cloud. If we remove a single electron from a P atom, it will come from the 3p orbital.

Step-by-step solution

Determine the electron configuration of phosphorus

Phosphorus (P) has 15 electrons. Let's find its electron configuration. The electron configuration of P is: \[ 1s^2 \ 2s^2 \ 2p^6 \ 3s^2 \ 3p^3 \].

Calculate the effective nuclear charge under the given condition

According to the statement, we assume that core electrons are completely effective at screening the valence electrons, and valence electrons provide no screening for each other. In phosphorus, there are 10 core electrons (1s², 2s², and 2p⁶) and 5 valence electrons (3s² and 3p³). Since the core electrons perfectly shield the valence electrons, the effective nuclear charge Z_eff for a phosphorus atom would be the total charge of the nucleus (Z) minus the number of core (shielding) electrons (10). Z for phosphorus is 15 (protons are equal to electrons). Z_eff = Z - shielding electrons = 15 - 10 Thus, the effective nuclear charge acting on the 3s and 3p valence electrons would be: Z_eff = 5+.

Calculate the effective nuclear charge using Slater's rules

Now we will calculate the effective nuclear charge using Slater's rules. - For a 3s electron: S(3s) = (0.35 x 2) + (0.85 x 8) = 7.3 Z_eff(3s) = Z - S(3s) = 15 - 7.3 = 7.7+ - For a 3p electron: S(3p) = (0.35 x 2) + (0.85 x 6) + (0.35 x 2) = 6.3 Z_eff(3p) = Z - S(3p) = 15 - 6.3 = 8.7+ Using Slater's rules, the effective nuclear charge on 3s electrons would be 7.7+ and on 3p electrons would be 8.7+.

Discuss the difference between effective nuclear charges for 3s and 3p electrons

According to the detailed calculations provided, the effective nuclear charge is 5.6+ for the 3s electrons and 4.9+ for the 3p electrons. The difference between these values can be attributed to the fact that the 3s electron cloud is more concentrated closer to the nucleus compared to the 3p electron cloud. Moreover, 3p electrons have more angular momentum and are farther from the nucleus on average. Due to their closer proximity to the nucleus, 3s electrons experience a higher effective nuclear charge.

Determine which orbital the electron will be removed from

In a P atom, if we want to remove a single electron, it will come from the highest energy and least tightly bound orbital. In this case, the electron configuration is: \[ 1s^2 \ 2s^2 \ 2p^6 \ 3s^2 \ 3p^3 \]. The highest energy and least tightly bound orbital is the 3p orbital. Thus, an electron will be removed from the 3p orbital if we remove a single electron from a P atom.

Key concepts

Slater's Rules

Slater's rules are a set of guidelines used to estimate the effective nuclear charge ( Z_{eff} ) felt by an electron in an atom. These rules help account for the partial shielding effect of electrons on each other within an atom. The concept was devised by John C. Slater to better understand the structure and behavior of electrons in different atomic orbits.
To understand Slater's rules:

• The electrons are divided into groups or "shells," labeled as 1 (closed), 2 (if s or p) and 3 if in n or higher s, p electrons, plus separate factors for d and f electrons.
• Electrons in the same group are multiplied by 0.35 to account for partial shielding.
• Electrons in the preceding shells provide stronger shielding by multiplying by higher values, usually 0.85 for the inner and 1 for even more inner shells.

Slater's rules allow us to approximate how the positive charge of the nucleus is mitigated by core and other electron interactions. This is crucial to determine how tightly electrons are bound to the nucleus.

Electron Configuration

Electron configuration is the systematic arrangement of electrons within orbits or shells surrounding the nucleus of an atom. It is crucial to figuring out the chemical properties and behavior of elements. For instance, the electron configuration of phosphorus (P), with an atomic number of 15, is: \[ 1s^2 \ 2s^2 \ 2p^6 \ 3s^2 \ 3p^3 \].
This notation indicates that:

• The first energy level has 2 electrons in the s subshell.
• The second energy level is filled with 2 in the s subshell and 6 in the p subshell, totally 8 electrons. This concludes the core electrons.
• The third energy level holds the valence electrons: 2 in the s subshell and 3 in the p subshell.

Electron configurations help in predicting the possible states an atom can reach when it bonds and for calculating effective nuclear charges using methods like Slater's rules.

Valence Electrons

Valence electrons are the outermost electrons of an atom that are responsible for forming bonds with other atoms. They exist in the highest occupied energy level, making them vital in determining the chemical reactivity and properties of an element. For phosphorus, the electron configuration \[ 1s^2 \ 2s^2 \ 2p^6 \ 3s^2 \ 3p^3 \] shows 5 valence electrons are in the 3s and 3p orbitals.
Understanding valence electrons:

• They dictate how atoms interact and bond with each other.
• They provide very little shielding from each other compared to core electrons.
• They are crucial for calculating the effective nuclear charge using methods like Slater's Rules, as seen in the difference in effective nuclear charges among various orbitals.

By controlling chemical bonding, valence electrons are central to the roles atoms and molecules play in chemical reactions and large biological processes.

Core Electrons

Core electrons are the electrons found within the inner shells of an atom. They remain relatively unreactive and serve the primary purpose of shielding the valence electrons from the full attractive force of the nucleus. For phosphorus, core electrons are those in the 1s, 2s, and 2p subshells of the electron configuration \[ 1s^2 \ 2s^2 \ 2p^6 \ 3s^2 \ 3p^3 \], totaling 10 electrons.
Key points about core electrons:

• They create a shielding effect, significantly reducing the attraction experienced by the valence electrons.
• Core electrons usually do not participate in bonding.
• When calculating effective nuclear charge, the shielding effect they contribute is subtracted from the total nuclear charge.

In chemical reactions, core electrons' only involvement is in how they influence the perceived nuclear charge on the valence electrons via the shielding effect. This shielding is essential for understanding why certain electrons are more loosely bound and thus more easily removed.