Question

A fundamental relationship of electrostatics states that the energy required to separate opposite charges of magnitudes \(Q_{1}\) and \(Q_{2}\) that are a distance \(d\) apart is proportional to \(\frac{Q_{1} \times Q_{2}}{d} .\) Use this relationship and any other relevant factors to explain the following: (a) The IE \(_{2}\) of He \((Z=2)\) is more than twice the \(\mathrm{IE}_{1}\) of \(\mathrm{H}(Z=1)\). (b) The IE \(_{1}\) of He is less than twice the \(\mathrm{IE}_{1}\) of \(\mathrm{H}\).

Short answer

The \(\mathrm{IE}_{2}(\mathrm{He})\) is more than twice \(\mathrm{IE}_{1}(\mathrm{H})\) because of reduced electron shielding. The \(\mathrm{IE}_{1}(\mathrm{He})\) is less than twice \(\mathrm{IE}_{1}(\mathrm{H})\) due to electron-electron repulsion in He.

Step-by-step solution

Understanding the Energy Relationship

The energy to separate opposite charges of magnitudes \(Q_{1}\) and \(Q_{2}\) at a distance \(d\) is proportional to \(\frac{Q_{1} \times Q_{2}}{d}\). The ionization energy (IE) is related to this energy.

Analyze Ionization Energy (IE)

Ionization energy (IE) is the energy required to remove an electron from an atom. For hydrogen (\(Z=1\)), it is given as \(\mathrm{IE}_{1}(\mathrm{H})\). For helium (\(Z=2\)), \(\mathrm{IE}_{1}(\mathrm{He})\) and \(\mathrm{IE}_{2}(\mathrm{He})\) correspond to the first and second ionization energies respectively.

Compare IE\(_{2}\) of He and IE\(_{1}\) of H

Compare the second ionization energy of helium (\(\mathrm{IE}_{2}(\mathrm{He})\)) with the first ionization energy of hydrogen (\(\mathrm{IE}_{1}(\mathrm{H})\)). \(\mathrm{IE}_{2}(\mathrm{He})\) is greater than twice \(\mathrm{IE}_{1}(\mathrm{H})\) because, after the first electron is removed from He, the remaining single electron in He\(^+\) experiences a stronger effective nuclear charge due to reduced electron shielding, making it harder to remove this electron.

Compare IE\(_{1}\) of He and IE\(_{1}\) of H

Compare the first ionization energy of helium (\(\mathrm{IE}_{1}(\mathrm{He})\)) with the first ionization energy of hydrogen (\(\mathrm{IE}_{1}(\mathrm{H})\)). \(\mathrm{IE}_{1}(\mathrm{He})\) is less than twice \(\mathrm{IE}_{1}(\mathrm{H})\) because the two electrons in He experience electron-electron repulsion that partially offsets the stronger nuclear attraction from the higher proton number.

Key concepts

Electrostatics

Electrostatics is the study of electric charges at rest. It focuses on the forces and energies between charged particles. The fundamental relationship of electrostatics states that the energy required to separate two opposite charges, say of magnitudes \(Q_{1}\) and \(Q_{2}\), that are a distance \(d\) apart is given by the formula \( \propto \frac{Q_{1} \times Q_{2}}{d} \). This principle helps to understand ionization energy, as it is essentially the energy needed to overcome the electrostatic attraction between an electron and the nucleus.

By applying this relationship, we can deeper dive into why it takes different amounts of energy to remove electrons from various atoms.

Ionization Energy

Ionization energy (IE) is the energy required to remove an electron from an atom in its gaseous state. For instance:

• \(\mathrm{IE}_{1} \) of Hydrogen (\( Z=1 \)) refers to the energy needed to remove its only electron.
• \( \mathrm{IE}_{1} \) of Helium (\( Z=2 \)) is the energy to remove one of its two electrons.
• \( \mathrm{IE}_{2} \) of Helium is the energy required to remove the second electron after the first one has already been removed, resulting in He⁺.

Ionization energies can vary significantly between elements due to differences in effective nuclear charge and electron shielding, which influence an electron’s attachment to the nucleus.

Effective Nuclear Charge

Effective nuclear charge (\( Z_{eff} \)) is a concept used to understand the net positive charge experienced by an electron in a multi-electron atom. It is given by the difference between the total number of protons in the nucleus (\( Z \)) and the shielding effects from inner electrons.

For example:

• In a Hydrogen atom, the single electron feels almost the full charge of the single proton, hence the IE\(_{1}\) value.
• In a Helium atom, the two electrons experience a higher positive charge but also shield each other slightly. However, once one electron is removed, the remaining electron is more strongly attracted since it feels a greater \( Z_{eff} \).

This explains that \( \mathrm{IE}_{2}(\mathrm{He}) \) is more than double \( \mathrm{IE}_{1}(\mathrm{H}) \). The reduced electron shielding in He⁺ leads to a stronger hold on the remaining electron.

Electron Shielding

Electron shielding occurs when inner electrons repel outer electrons, reducing the effective nuclear charge felt by the outermost electrons. This concept is pivotal in understanding ionization energy variations.

Consider these points:

• In Helium, the two electrons repel each other, causing electron shielding. This slightly reduces the effective nuclear charge felt by each electron, making removal less energy-intensive compared to the scenario where no shielding occurs.
• For the first electron removal (\(\mathrm{IE}_{1}\)), the experience is similar to Hydrogen, albeit with a slightly higher energy due to the additional proton in Helium.
• For the second electron removal (\(\mathrm{IE}_{2}\)), there is significantly less shielding since the electron is now alone. Consequently, it feels a more substantial effective nuclear charge compared to the original state.

This interplay between shielding and nuclear charge underlines why the ionization energies vary as observed.