Question

For the electrons of oxygen atom, which of the following statements is correct? (1) \(Z_{\text {etl }}\) for an electron in a 2 s orbital is the same as \(\mathrm{Z}_{\text {elt }}\) for an electron in a 2p orbital. (2) An electron in the 2 s orbital has the same energy as an electron in the \(2 p\) orbital. (3) \(Z_{\text {ett }}\) for an electron in a 1 s orbital is the same as \(Z_{\text {eff }}\) for an electron in a 2 s orbital. (4) The two electrons present in the 2 s orbital have different spin quantum numbers \(\left(\mathrm{m}_{\mathrm{s}}\right)\).

Short answer

Statement (4) is correct.

Step-by-step solution

Understanding Effective Nuclear Charge \(Z_{\text{eff}}\)

Effective Nuclear Charge, represented as \(Z_{\text{eff}}\), is the net positive charge experienced by an electron in a multi-electron atom. It accounts for both the total positive charge of the nucleus and the shielding or screening effect of other electrons present.

Evaluating Statement (1)

Statement (1) suggests that \(Z_{\text{eff}}\) for an electron in a 2s orbital is the same as for an electron in a 2p orbital. However, due to differences in shielding and penetration abilities, \(Z_{\text{eff}}\) for a 2s orbital is generally higher than for a 2p orbital in the same atom. Therefore, statement (1) is incorrect.

Evaluating Statement (2)

Statement (2) suggests that an electron in the 2s orbital has the same energy as an electron in the 2p orbital. Nevertheless, it is known that 2s electrons usually have lower energy (are more tightly bound to the nucleus) than 2p electrons due to their greater penetration and lesser shielding. Hence, statement (2) is incorrect.

Evaluating Statement (3)

Statement (3) suggests that \(Z_{\text{eff}}\) for an electron in a 1s orbital is the same as for an electron in a 2s orbital. In reality, electrons in the 1s orbital experience a greater effective nuclear charge because they are closer to the nucleus with minimal shielding by other electrons compared to 2s electrons. Thus, statement (3) is incorrect.

Evaluating Statement (4)

Statement (4) suggests that the two electrons present in the 2s orbital have different spin quantum numbers \(m_{s}\). According to the Pauli Exclusion Principle, no two electrons in the same orbital can have the same set of quantum numbers. Therefore, within the 2s orbital, one electron must have \(m_{s} = +\frac{1}{2}\) and the other must have \(m_{s} = -\frac{1}{2}\). Hence, statement (4) is correct.

Key concepts

Effective Nuclear Charge

Effective Nuclear Charge, often represented as \( Z_{\text{eff}} \), is a crucial concept in understanding atomic structure. It indicates the attractive charge of the nucleus felt by an electron, after accounting for the shielding effects of other electrons in the atom.

The formula for \( Z_{\text{eff}} \) is given by: \( Z_{\text{eff}} = Z - S \), where:

• **Z** is the total number of protons (atomic number).
• **S** is the shielding constant, representing the repulsion by other electrons.

Electrons closer to the nucleus generally experience a higher effective nuclear charge compared to those further away because they have fewer inner electrons repelling them. Understanding this concept helps explain many properties of atoms, such as ionization energies and atomic radii. The higher the \( Z_{\text{eff}} \), the more tightly an electron is held by the nucleus.

Electron Shielding

Electron Shielding, also known as screening effect, occurs when inner electrons shield the outer electrons from the full attractive force of the nucleus. This effect explains the differences in \( Z_{\text{eff}} \) across different orbitals.

Here’s how it works:

• Inner-shell electrons repel outer-shell electrons.
• This repulsion reduces the net nuclear charge felt by the outer electrons.

Shielding is more effective with s-orbital electrons compared to p-orbital electrons. This is because s-electrons are closer to the nucleus and more effective at repelling outer electrons. Therefore, electrons in a 2s orbital experience a higher effective nuclear charge than electrons in a 2p orbital, affecting their energy levels and chemical behavior.

Pauli Exclusion Principle

The Pauli Exclusion Principle is a fundamental rule in quantum mechanics that states no two electrons in an atom can have the same set of four quantum numbers.

The four quantum numbers are:

• **n** (principal quantum number): denotes the energy level
• **l** (azimuthal quantum number): denotes the shape of the orbital
• **m_l** (magnetic quantum number): denotes the orientation of the orbital
• **m_s** (spin quantum number): denotes the spin of the electron

In any given orbital, an electron can have a spin quantum number \( m_{s} = +\frac{1}{2} \) or \( m_{s} = -\frac{1}{2} \). Therefore, two electrons in the same orbital must have opposite spins. This principle ensures that each electron within an atom has a unique quantum state, thus determining the electronic structure and chemical properties of the elements.

Electron Orbital Energy Levels

Electron Orbital Energy Levels describe the specific energy states that electrons can occupy in an atom. These levels depend on the type of orbital (s, p, d, f) and the principal quantum number (n) of the electron.

In general:

• **s-orbitals** are spherical and have the lowest energy within a given level.
• **p-orbitals** are dumbbell-shaped and have higher energy than s-orbitals of the same principal quantum number.

For example, in oxygen:

• The 2s orbital electrons experience a higher effective nuclear charge than 2p orbital electrons because of less shielding and better penetration to the nucleus.
• As a result, 2s electrons are lower in energy and more tightly bound to the nucleus than 2p electrons.

Knowing the energy levels of orbitals is essential for understanding the arrangement of electrons in atoms and their behavior in chemical reactions.