Question

For the electrons of oxygen atom, which of the following statements is correct? (1) \(Z_{\text {ett }}\) for an electron in a 2s orbital is the same as \(Z_{\text {eft }}\) 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) \(\mathrm{Z}_{\mathrm{efl}}\) for an electron in a 1 s orbital is the same as \(Z_{\text {etf }}\) for an electron in a 2 s orbital. (4) The two electrons present in the 2 s orbital have different spin quantum numbers \(\left(m_{s}\right)\).

Short answer

Statement 4 is correct.

Step-by-step solution

Find the Effective Nuclear Charge (Z_eff)

The effective nuclear charge, denoted as \(Z_\text{eff}\), is the net positive charge experienced by an electron in an atom. It can be different for electrons in different orbitals due to electron shielding.

Evaluate Statement 1

Statement 1 claims that \(Z_\text{eff}\) for an electron in a 2s orbital is the same as for one in a 2p orbital. While both orbitals are in the same energy level (n=2), the 2s electron often experiences a slightly different \(Z_\text{eff}\) due to differences in penetration and shielding effects. Thus, the effective nuclear charges are usually not exactly the same.

Evaluate Statement 2

Statement 2 asserts that an electron in the 2s orbital has the same energy as one in the 2p orbital. This statement is incorrect because 2s electrons are typically lower in energy compared to 2p electrons in the same atom. This is due to better shielding and closer penetration to the nucleus by the 2s electrons.

Evaluate Statement 3

Statement 3 suggests that \(Z_\text{eff}\) for an electron in a 1s orbital is the same as for one in a 2s orbital. This is incorrect because electrons in the 1s orbital experience a higher effective nuclear charge compared to those in the 2s orbital due to the former being closer to the nucleus with less shielding effects.

Evaluate Statement 4

Statement 4 proposes that the two electrons present in the 2s orbital have different spin quantum numbers (\(m_{s}\)). This is correct because spin quantum numbers can be either \(+\frac{1}{2}\) or \(-\frac{1}{2}\), and no two electrons in the same orbital can share the same set of quantum numbers.

Key concepts

Effective Nuclear Charge (Z_eff)

The Effective Nuclear Charge, often symbolized as \(Z_\text{eff}\), represents the net positive charge that an electron experiences within an atom. This concept is crucial for understanding electron interactions and overall atomic structure.
The nucleus is composed of positively charged protons. However, in a multi-electron atom, electrons repel each other. Outer electrons are less attracted to the nucleus compared to inner electrons because outer electrons experience a shielding effect.
The shielding effect is the reduction in the effective nuclear charge by the inner-shell electrons. Inner-shell electrons block some of the positive charge from reaching the outer electrons.
Thus, outer electrons experience a net positive charge that is less than the actual charge of the nucleus, and we call this the Effective Nuclear Charge. The formula often used to estimate it is:
\[ Z_{\text{eff}} = Z - S \]where:

• \(Z\) is the total number of protons, or the actual nuclear charge
• \(S\) is the average number of electrons between the nucleus and the electron in question, or the shielding constant

Electrons in different orbitals will experience different effective nuclear charges due to variations in their shielding and penetration capabilities.

Electron orbitals

Electron orbitals describe regions in an atom where electrons are likely to be found. These regions differ based on energy levels and shapes.
Each orbital can hold a specific number of electrons and has a unique shape and orientation.
Orbitals are organized into different shells (energy levels), with each shell containing one or more subshells: s, p, d, and f. Here are some key points to understand:

• **1s orbital:** Spherical and closest to the nucleus. Holds up to 2 electrons.
• **2s orbital:** Also spherical but larger in size. Holds up to 2 electrons in the second energy level.
• **2p orbitals:** Dumbbell-shaped and oriented in three directions (x, y, z). Each can hold up to 2 electrons, for a total of 6 across the three orbitals in the second energy level.

While 2s and 2p orbitals are in the same energy level (n=2), they differ in energy due to their shapes and orientations. 2s is typically lower in energy because it penetrates closer to the nucleus, experiencing less shielding compared to 2p orbitals.
The rules of orbital filling follow specific principles:

• **Aufbau Principle:** Orbitals are filled from lowest to highest energy levels.
• **Pauli Exclusion Principle:** No two electrons can have the same set of quantum numbers within an atom.
• **Hund’s Rule:** Electrons will fill degenerate orbitals (same energy) singly before pairing up.

These principles help describe the electron configuration and the probable locations and energies of an atom’s electrons.

Spin quantum numbers

The spin quantum number, symbolized as \(m_{s}\), is a quantum number that describes the intrinsic angular momentum (also known as spin) of an electron within an orbital.
Unlike other quantum numbers (n, l, m_l), which describe the spatial distribution and energy of an electron’s orbit, the spin quantum number specifies the direction of an electron’s spin.
The spin quantum number can have only one of two possible values:

• \( +\frac{1}{2} \) – Called 'spin up'
• \( -\frac{1}{2} \) – Called 'spin down'

This concept is vital because of the Pauli Exclusion Principle, which states that no two electrons in the same atom can have the same set of four quantum numbers (n, l, m_l, and \(m_{s}\)).
As a result, if an orbital holds two electrons, their spins must be opposite to comply with this principle. This explains why electrons in a fully-occupied orbital have opposite spins. For instance, if one electron in the 2s orbital has a spin quantum number \( +\frac{1}{2} \), the other must have \( -\frac{1}{2} \).
This balanced spinning creates a paired state, stabilizing the atom and contributing to the magnetic properties of the material.
In summary, understanding spin quantum numbers helps in comprehending electron configurations and the physical properties of elements.