Question

The \(\mathrm{H}^{-}\) ion and the He atom have two \(1 \mathrm{~s}\) electrons each. Which of the two species is larger? Explain.

Short answer

\(\mathrm{H}^-\) is larger than He due to its lower effective nuclear charge and electron-nucleus attraction.

Step-by-step solution

Understand the context

Both the \(\mathrm{H}^{-}\) ion and the He atom have two electrons in the \(1\mathrm{s}\) orbital. Despite this similarity, their sizes depend on their nuclear charge and the effective nuclear charge experienced by these electrons.

Analyze nuclear charges

The \(\mathrm{H}^-\) ion has a nuclear charge of +1, while the He atom has a nuclear charge of +2. This means that the electrons in the He atom are attracted more strongly towards the nucleus compared to those in the \(\mathrm{H}^-\) ion.

Consider electron-electron repulsion and effective nuclear charge

In \(\mathrm{H}^-,\) there are two electrons with only one proton in the nucleus, which leads to a lower effective nuclear charge per electron compared to He. He has two protons for its two electrons, resulting in a higher effective nuclear charge.

Analyze shielding effects

In both species, each electron experiences repulsion from the other electron, but in \(\mathrm{H}^-\), the repulsion relative to the attraction from a single proton reduces the effective nuclear charge more significantly compared to He, where two protons attract the electrons.

Conclusion about size

Given the lower effective nuclear charge in \(\mathrm{H}^-\), the electrons are held less tightly to the nucleus, making \(\mathrm{H}^-\) larger than He, where electrons are more strongly attracted to the nucleus.

Key concepts

Effective Nuclear Charge

Effective nuclear charge is an essential concept when understanding atomic structure. It gives an idea of how strongly an electron is attracted to the nucleus. In simple terms, it is the net positive charge experienced by an electron in an atom. This charge is less than the actual nuclear charge due to the presence of other electrons.
The more protons in the nucleus, the higher the nuclear charge. However, other electrons that are 'between' the nucleus and an electron of interest can shield the electron from the full power of the nucleus. The effective nuclear charge (\(Z_{ ext{eff}}\)) is calculated as: \(Z_{ ext{eff}} = Z - S\), where \(Z\) is the nuclear charge, and \(S\) is the number of electrons shielding the nucleus.

• In helium (He), with two protons and two electrons, the effective nuclear charge on each electron is significant.
• In hydrogen anion (\( ext{H}^-\)), one fewer proton means less attractive force on the electrons due to a lower effective nuclear charge.

This difference plays a key role in determining the size of atoms or ions.

Electron-Electron Repulsion

Electron-electron repulsion occurs when electrons, which both have a negative charge, repel each other due to their like charges.
In multi-electron atoms, this repulsion is a key factor in the size and energy of the electron's orbitals. The presence of additional electrons increases the repulsion forces within an atom or ion. This generally causes the electrons to be further from the nucleus than if the repulsions were not present.

• In \( ext{H}^-\), there is higher repulsion because there are two electrons but only one proton's attraction.
• This means that the less tightly the electrons are held, the larger the ion or atom is.

Understanding electron-electron repulsion helps explain why \( ext{H}^-\) is larger than He.

Shielding Effect

The shielding effect is when electrons closer to the nucleus shield outer electrons from the full impact of the nuclear charge. This effect influences the effective nuclear charge exerted on outer electrons and contributes significantly to the properties of multi-electron systems.
Electrons in lower energy levels act as a buffer, reducing the pull from the protons on the outer electrons.
In \( ext{H}^-\), the single proton's ability to attract both electrons is significantly reduced due to each electron helping to shield the other from the nucleus.

• In helium, two protons decrease the overall impact of shielding by maintaining a stronger pull on the electrons.
• This allows the electrons in helium to be held more tightly to the nucleus, resulting in a smaller atomic size.

This shielding discrepancy is key to understanding the difference in sizes between ions and atoms like \( ext{H}^-\) and He.

Nuclear Charge

Nuclear charge is simply the total positive charge of an atom's nucleus, determined by the number of protons it contains. It is a fundamental concept in atomic physics, providing insight into how the electrons in an atom are influenced.
In hydrogen anion (\( ext{H}^-\)), the nuclear charge is +1, as there is only one proton. In helium (He), the nuclear charge is +2, reflecting its two protons.
Higher nuclear charge generally means a stronger attraction of electrons toward the nucleus, resulting in smaller atomic or ionic radii.

• The lower nuclear charge in \( ext{H}^-\) leads to weaker attraction for the electrons, making the species larger.
• In contrast, helium's higher nuclear charge draws its electrons closer, contributing to a smaller size.

Understanding nuclear charge is fundamental to grasping why, despite similar electron configurations, the sizes of \( ext{H}^-\) and He differ significantly.