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 Ag \(1 s\) 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 effective nuclear charge (Z_eff) experienced by a 1s electron in a silver (Ag) atom is approximately 46.72, which is very close to the atomic number (Z) of Ag, which is 47. This is because the 1s electron is close to the nucleus and experiences minimal shielding from other electrons, causing the effective nuclear charge to be nearly equal to the actual nuclear charge.

Step-by-step solution

Understand the concepts

To find the effective nuclear charge (Z_eff), we will use Bohr's model, which describes the behavior of electrons in hydrogen-like atoms. This model states that the ionization energy (IE) is related to the effective nuclear charge through the following formula: IE = \(\frac{Z_{eff}^2 * e^2}{2 * a_0 * n^2}\), where e is the elementary charge, a_0 is the Bohr radius, and n is the principal quantum number for the electron. The ionization energy for the given electron in the Ag atom is provided, and we know that the electron is in the 1s orbital, which means n = 1.

Rearrange the equation to find Z_eff

First, we need to isolate Z_eff in the equation. By multiplying both sides of the equation by \(\frac{2 * a_0 * n^2}{e^2}\), we get: Z_eff^2 = \(\frac{2 * IE * a_0 * n^2}{e^2}\) Now, by taking the square root of both sides of the equation, we get: Z_eff = \(\sqrt{\frac{2 * IE * a_0 * n^2}{e^2}}\)

Substitute the known values and solve for Z_eff

Now that we have the equation for Z_eff, we can plug in the values for IE, a_0, and n. The given ionization energy for Ag is \(2.462 * 10^6 \frac{kJ}{mol}\), which we need to convert to Joules by multiplying by \(10^3\). The values for the other constants are: a_0 = \(5.2918 * 10^{-11} m\), e = \(1.6022 * 10^{-19} C\), n = 1. Substituting these values into the equation, we get: Z_eff = \(\sqrt{\frac{2 * 2.462 * 10^3 * 5.2918 * 10^{-11} * (1)^2}{(1.6022 * 10^{-19})^2}}\) Calculating this expression gives: Z_eff ≈ 46.72

Compare Z_eff to Z for Ag and provide a rationalization

The atomic number of Ag (silver) is 47. Our calculated value for Z_eff for the Ag 1s electron is approximately 46.72. These numbers are very close, which indicates that the 1s electron experiences an effective nuclear charge that is very similar to the actual nuclear charge of Ag. The small difference between Z and Z_eff can be rationalized by considering the electron shielding effect. In a multi-electron atom like Ag, inner electrons shield the outer electrons from the full nuclear charge. Since the 1s electron is the closest electron to the nucleus, it experiences minimal shielding from other electrons, resulting in a Z_eff value that is very close to Z. Now we have found an approximate value for Z_eff for the Ag 1s electron, and the result matches our expectations based on the shielding effect.

Key concepts

Bohr Model

The Bohr Model is a crucial concept in understanding atomic structure, particularly in hydrogen-like atoms. It suggests that electrons orbit the nucleus in defined circular paths or shells. Each shell corresponds to a specific energy level. This model helps explain the behavior of electrons when atoms absorb or release energy. By predicting when electrons can move between shells, the Bohr Model set the foundation for modern quantum mechanics.

In the Bohr Model, the energy required to remove an electron from an atom—its ionization energy—is tied to the electron's distance from the nucleus. It's also influenced by the effective nuclear charge (\(Z_{eff}\)), which is a measure of the charge felt by an electron after accounting for the presence of other electrons in the atom. The relation between ionization energy and \(Z_{eff}\) can be expressed mathematically, providing insights into atomic structure.

Effective Nuclear Charge

Effective Nuclear Charge \(Z_{eff}\) is a concept that helps determine the net positive charge felt by an electron in an atom. This notion considers both the actual nuclear charge (\(Z\), the number of protons) and the shielding effect caused by the other electrons. Electrons closer to the nucleus can shield outer electrons, reducing the net positive force experienced.

\(Z_{eff}\) is particularly essential for understanding why ionization energy values differ for electrons in various shells. It explains why electrons residing in inner shells, like the 1s orbital, often experience a higher effective nuclear charge. Thus, they are less shielded and require more energy for removal, resulting in higher ionization energies.\(Z_{eff}\) provides a valuable tool for predicting chemical behavior and reactivity.

Electron Shielding

Electron Shielding is a phenomenon where inner-shell electrons partially block the nuclear charge from reaching outer electrons. It plays a significant role in defining the effective nuclear charge an electron experiences.

In atoms with multiple electrons, the force exerted by the nucleus on an electron is lessened due to the presence of intervening electrons. These intervening electrons create a 'shield' that reduces the effective pull of the nucleus on outer electrons. Thus, electrons that are further from the nucleus 'feel' a lower nuclear charge than those closer to it, affecting their ionization energies.

Understanding electron shielding helps in explaining trends across the periodic table, like atomic size and ionization energies. It highlights how the distribution of electrons within an atom impacts the net attractive force from the nucleus.

Ionization Process

The Ionization Process involves removing an electron from an atom, turning it into a positively charged ion. The energy needed for this process is known as ionization energy. Ionization energy provides crucial insights into the atom’s structure and reactivity.

A key player in the ionization process is the effective nuclear charge. As described earlier, \(Z_{eff}\) influences how tightly an electron is bound to its nucleus. Higher \(Z_{eff}\) means a stronger attraction to the nucleus, thus requiring more energy to remove the electron. This explains why ionization energy values can differ significantly among elements.
Understanding this concept allows us to predict how an element will behave in chemical reactions. Elements with low ionization energies tend to lose electrons easily, often participating in chemical reactions as reducing agents. The ionization process thus serves as a gateway to understanding and predicting the behaviour of elements in various chemical contexts.