Question

The ionization energy for a 1\(s\) electron in a silver atom is \(2.462 \times 10^{6} \mathrm{kJ} / \mathrm{mol} .\) a. Determine an approximate value for \(Z_{\text { eff }}\) for the \(A g 1 s\) electron. Assume the Bohr model applies to the 1 s electron. \(Z_{\mathrm{eff}}\) is the apparent nuclear charge experienced by the electrons. b. How does \(Z_{\text { eff }}\) from part a compare to \(Z\) for Ag? Rationalize the relative numbers.

Short answer

The approximate effective nuclear charge ($Z_{\text{eff}}$) for the 1s electron in a silver (Ag) atom is 55.7, which is larger than its actual atomic number (Z) of 47. This difference is due to the influence of other electrons in the atom, as they cause electron-electron repulsion, making the effective nuclear charge experienced by the 1s electron higher than the actual nuclear charge. However, this is only an approximation, since the Bohr model is not fully accurate for multi-electron atoms like Ag.

Step-by-step solution

Recall the formula for ionization energy in the Bohr model

The ionization energy (E) for an electron in a hydrogen-like atom can be calculated using the relationship: \( E = \dfrac{Z_{\text{eff}}^2 \cdot 13.6\ \text{eV}}{n^2} \) Here, Z_eff is the effective nuclear charge, and n is the principal quantum number for the electron. Since we are dealing with the 1s electron, n equals 1.

Convert ionization energy from kJ/mol to eV

The given ionization energy is in kJ/mol. To use the formula above, we need to convert it to eV. We can use conversion factors: 1 eV = 96.485 kJ/mol and 1 mol = 6.022 x 10^23 atoms. \( 2.462 \times 10^{6} \ \frac{\text{kJ}}{\text{mol}} \cdot \dfrac{1 \ \text{eV}}{96.485 \ \text{kJ/mol}} \cdot \dfrac{1 \ \text{mol}}{6.022 \times 10^{23} \ \text{atoms}} = 4.222 \times 10^{4} \ \text{eV} \)

Rearrange the formula and solve for Z_eff

Now that we have the ionization energy in eV, we can rearrange and plug the values into our formula to solve for Z_eff: \( Z_{\text{eff}}^2 = \dfrac{E \cdot n^2}{13.6 \ \text{eV}} = \dfrac{4.222 \times 10^4 \ \text{eV}}{13.6 \ \text{eV}} \) \( Z_{\text{eff}}^2 = 3103.09 \) Solve for Z_eff: \( Z_{\text{eff}} = \sqrt{3103.09} = 55.7 \) Thus, the approximate effective nuclear charge (Z_eff) for the Ag 1s electron is 55.7. #b. How does Z_eff from part a compare to Z for Ag? Rationalize the relative numbers.

Compare Z_eff to Z for Ag

The atomic number of Ag (silver) is 47. Comparing Z_eff (55.7) to Z (47) shows that Z_eff is larger than the actual atomic number for Ag.

Rationalize the difference between Z_eff and Z

The higher Z_eff value for the Ag 1s electron compared to the actual atomic number is due to the influence of other electrons in the atom. In multi-electron atoms, electrons experience both nuclear charge from the nucleus and electron-electron repulsion. Consequently, the effective nuclear charge experienced by the 1s electron is higher than the actual nuclear charge. However, we should note that this is only an approximation since the Bohr model is not fully accurate for multi-electron atoms like Ag.

Key concepts

Bohr Model

The Bohr Model is a fundamental concept in understanding atomic structure. Created by Niels Bohr in 1913, it describes the atom as a small, positively charged nucleus surrounded by electrons that travel in circular orbits. This model primarily applies to hydrogen-like atoms.

The Bohr Model introduced the idea of quantized energy levels, where electrons reside in distinct orbits at fixed radii. Each orbit corresponds to a particular energy level, with electrons jumping between levels when they absorb or emit energy. Although the model is an oversimplification, particularly for multi-electron atoms, it provides a simple way to calculate and understand the basic concept of ionization energy.

In the exercise, it is assumed that the Bohr Model applies to silver's 1s electron. Here, specific formulas from the Bohr Model help determine the effective nuclear charge experienced by these electrons.

Effective Nuclear Charge

Effective Nuclear Charge (

• The idea of effective nuclear charge explains how a nucleus can impact different electrons within an atom.
  Understanding that in multi-electron atoms, an electron feels not just the attraction to the nucleus but also repulsion from other electrons helps see why effective nuclear charge differs from actual nuclear charge.
• The formula to compute effective nuclear charge (\(Z_{ ext{eff}}\)) often considers both the atomic number and the inner electron cloud's impact. In our calculations from the exercise, we see a larger value than the actual atomic number of silver because of these factors.

The Bohr Model's calculation used in our original exercise was one way to approximate an electron's effective nuclear potential. However, it's an approximation and might not hold perfectly true due to interactions the Bohr Model doesn't account for.

Silver Atom

The silver atom, with an atomic number of 47, provides an interesting case study in the Bohr Model and effective nuclear charge. Silver is a transition metal with a complex electron configuration, but for the purposes of ionization energy calculations, focusing on the innermost 1s electron is instructive.

This electron feels a strong pull from the 47 positive charges in the nucleus, but it also has to contend with the repulsion from the other electrons in silver's electron cloud.

The computed effective nuclear charge (\(Z_{ ext{eff}}\)) for silver's 1s electron (55.7) exceeding 47 demonstrates these intramolecular forces at play. Silver's ionization energy helps illustrate the larger concept of weighting nuclear and repulsive forces in understanding atomic behavior.

Quantum Mechanics

Quantum Mechanics builds upon and vastly expands beyond the Bohr Model. It provides a more accurate and comprehensive framework for understanding atomic and subatomic particles.

Unlike the Bohr model's fixed orbital paths, quantum mechanics describes electrons as existing in probability clouds known as orbitals, not fixed orbits. This approach accounts for electron movement's inherently uncertain nature, a fundamental principle of quantum mechanics.

In the context of this exercise, while the Bohr Model offers clear calculations, it oversimplifies a complex reality. Quantum mechanics, considering electron distribution and interactions more thoroughly, would give a more nuanced picture of electron positions and energies in a silver atom.

The idea of effective nuclear charge can be more exactly explored within quantum mechanical models through methods such as Hartree-Fock calculations and Density Functional Theory. While these approaches were beyond our initial exercise, they demonstrate quantum mechanics' power in understanding atomic and molecular structures.