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Magazine Chapter 2: Problem 19
Using Slater's rules, calculate \(Z^{*}\) for the following electrons: a. a \(3 p\) electron in \(\mathbf{P}\) b. a \(4 \mathrm{s}\) electron in Co c. a \(3 d\) electron in Mn d. a valence electron in \(\mathrm{Mg}\) Compare the values of \(Z^{*}\) the obtained with those of Clementi and Raimondi.
Short Answer
Expert verified
Calculated using Slater's rules: P (3p): 3.6, Co (4s): 11.65, Mn (3d): 13.6, Mg (valence): 3.15. Clementi and Raimondi values are higher.
Step by step solution
01
Identify the Electron Configuration
To apply Slater's Rules, we must first identify the electron configuration of the element for the given electron:a. For Phosphorus (\(\mathbf{P}\)): Atomic number 15Configuration: \(1s^2 2s^2 2p^6 3s^2 3p^3\)b. For Cobalt (Co): Atomic number 27Configuration: \(1s^2 2s^2 2p^6 3s^2 3p^6 3d^7 4s^2\)c. For Manganese (Mn): Atomic number 25Configuration: \(1s^2 2s^2 2p^6 3s^2 3p^6 3d^5 4s^2\)d. For Magnesium (\(\mathrm{Mg}\)): Atomic number 12Configuration: \(1s^2 2s^2 2p^6 3s^2\)
02
Apply Slater's Rules for Each Electron
Slater's rules help us calculate the effective nuclear charge (\(Z^{*}\)) using the electron configuration. Let's apply the rules:**a. \(3p\) electron in \(\mathbf{P}\):**- Electrons in the same group (3s, 3p): 4 electrons contribute 0.35 each \(\rightarrow 0.35 \times 4 = 1.4\)- Electrons in lower groups (1s, 2s, 2p): 10 electrons contribute 1 each \(\rightarrow 1 \times 10 = 10\)**b. \(4s\) electron in Co:**- Electrons in the same group (4s): 1 electron contributes 0.35 \(\rightarrow 0.35 \times 1 = 0.35\)- Electrons in the lower (3d, 3s, 3p) groups: 15 electrons \(\rightarrow 1 \times 15 = 15\)**c. \(3d\) electron in Mn:**- Electrons in the same group (3d): 4 other electrons contribute 0.35 each \(\rightarrow 0.35 \times 4 = 1.4\)- Electrons in lower groups (1s to 3p): 10 electrons contribute 1 each \(\rightarrow 1 \times 10 = 10\)- Electrons in higher group (4s): 2 electrons with no contribution.**d. Valence electron in \(\mathrm{Mg}\): 3s electron**- Electrons in the same group (3s): 1 electron contributes 0.35 \(\rightarrow 0.35 \times 1 = 0.35\)- Electrons in the lower (1s, 2s, 2p) groups: 10 electrons contribute 0.85 each \(\rightarrow 0.85 \times 10 = 8.5\)
03
Calculate Effective Nuclear Charge (\(Z^{*}\))
Using Slater's Rules, calculate \(Z^{*}\):**a. \(3p\) electron in \(\mathbf{P}\):**\[Z^{*} = 15 - (1.4 + 10) = 3.6\]**b. \(4s\) electron in Co:**\[Z^{*} = 27 - (0.35 + 15) = 11.65\]**c. \(3d\) electron in Mn:**\[Z^{*} = 25 - (1.4 + 10) = 13.6\]**d. Valence electron in \(\mathrm{Mg}\):**\[Z^{*} = 12 - (0.35 + 8.5) = 3.15\]
04
Compare with Clementi and Raimondi Values
Compare the calculated \(Z^{*}\) values with the reference values from Clementi and Raimondi:- **For Phosphorus (\(3p\) electron):** - Slater: 3.6, Clementi and Raimondi: approximately 5.2- **For Cobalt (\(4s\) electron):** - Slater: 11.65, Clementi and Raimondi: approximately 18.0- **For Manganese (\(3d\) electron):** - Slater: 13.6, Clementi and Raimondi: approximately 19.0- **For Magnesium (valence electron):** - Slater: 3.15, Clementi and Raimondi: approximately 10.3Different approximations and approaches in both methods lead to differences in computed values.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Slater's Rules
Slater's Rules provide a simplified methodology to estimate the effective nuclear charge (\( Z^{*} \)) experienced by an electron within an atom. The effective nuclear charge determines how strongly an electron is attracted to the nucleus of an atom, factoring in the screening or shielding effect due to other electrons. This is vital in understanding atomic structure and electron behavior.
- Determine the groups of electrons where the given electron resides and those that are closer to the nucleus.
- For electrons in the same group, each electron reduces the nuclear charge by 0.35 units (except for the first electron which counts differently in some cases).
- Electrons in groups closer to the nucleus than the electron in question shield completely, contributing fully to shielding. Specifically, they reduce the charge by 1 unit each.
- The calculation formula is: \[ Z^{*} = Z - S \] where \( S \) is the shielding constant derived by Slater鈥檚 rules. \( Z \) is the atomic number.
Electron Configuration
Knowing the electron configuration of an atom is the first step in implementing Slater's rules. Electron configuration is a representation of the distribution of electrons in an atom's orbitals.
- Each electron corresponds to a position in subshells defined as: s, p, d, and f.
- Electrons fill these subshells according to the Aufbau principle which states that electrons occupy the lowest energy orbitals first.
- For example, Phosphorus (\( \mathbf{P} \)), with an atomic number of 15, has an electron configuration: \(1s^2 2s^2 2p^6 3s^2 3p^3\). Here, the electrons are distributed starting from the 1s subshell, moving to the higher energy subshells.
- Correct identification of these configurations is critical because Slater鈥檚 rules use this information to determine electron shielding and effective nuclear charge.
Clementi and Raimondi
Clementi and Raimondi provided comprehensive research involving empirical calculations of effective nuclear charge for many atoms and ions. Unlike the simplified interpretations of Slater's rules, their work incorporates more detailed quantum mechanical models.
- Their work often yields different results from Slater's due to the incorporation of more variables and refined techniques.
- For instance, comparison of the effective nuclear charge calculated using Slater鈥檚 rules with those of Clementi and Raimondi shows significant differences (as noted by different values for Phosphorus, Cobalt, Manganese, and Magnesium in the exercise).
- These discrepancies highlight the approximations in Slater's method and stress the quantum mechanical aspects which Clementi and Raimondi address.
Atomic Structure
Atomic structure is the basis from which concepts like effective nuclear charge and electron configuration arise. Atoms consist of a nucleus surrounded by electrons, which occupy electron shells and subshells.
- The nucleus contains protons and neutrons, providing the positive charge (atomic number) and overall mass of the atom.
- Electrons form cloud-like structures in atomic orbitals, and their arrangement is strategic due to quantum mechanics.
- Atomic structure helps explain properties like atomic size, ionization energy, and chemical reactivity. When utilizing tools like Slater's rules, understanding the atomic framework is essential.
- Through atomic structure analysis, patterns and periodic properties of elements are revealed, which are central themes in chemistry.
