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Chapter 4: Problem 119

Some of the following sets of quantum numbers are correct for a \(4 \mathrm{~d}\) electron. Which are correct sets ? (a) \(4,3,2,+\frac{1}{2}\) (b) \(4,2,1,0\) (c) \(4,2,-2,+\frac{1}{2}\) (d) \(4,2,1,-\frac{1}{2}\)

Short Answer

Expert verified
Options (c) and (d) are correct sets for a 4d electron.

Step by step solution

01

Understand Electron Quantum Numbers

Quantum numbers describe the properties of an electron in an atom. There are four quantum numbers: - Principal quantum number ( "): Indicates the energy level/shell and can have positive integer values (e.g., 1, 2, 3, ...). - Azimuthal quantum number ( "): Denotes the subshell shape (s, p, d, f, ...) and ranges from 0 to (n-1). - Magnetic quantum number ( "): Represents the orientation of the orbital and ranges from - "). - Spin quantum number ( "): Represents the electron's spin and can be " or - ".
02

Identify 4d Subshell Conditions

For a 4d electron, the principal quantum number " = 4, and the azimuthal quantum number " = 2 (where d corresponds to "). The possible magnetic quantum number values for d orbitals range from -2 to +2 ( "). The electron spin quantum number ( ") can be either + " or - ".
03

Evaluate Option (a)

The given set is 4, 3, 2, +陆: - " = 4 is correct as it corresponds to the 4th energy level. - " = 3 is incorrect since it does not correspond to the d subshell (should be "). Therefore, this set is incorrect for a 4d electron.
04

Evaluate Option (b)

The given set is 4, 2, 1, 0: - " = 4 is correct for the 4th energy level. - " = 2 is correct for a d subshell. - " = 1 is within the correct range for a d orbital. - " = 0 is incorrect for an electron's spin; it must be +陆 or -陆. Thus, this set is incorrect for a 4d electron.
05

Evaluate Option (c)

The given set is 4, 2, -2, +陆: - " = 4 is correct. - " = 2 is correct for a d subshell. - " = -2 is within the range for d orbitals. - " = +陆 is a valid spin. This set correctly represents a 4d electron.
06

Evaluate Option (d)

The given set is 4, 2, 1, -陆: - " = 4 is correct. - " = 2 is correct for a d subshell. - " = 1 is within the range for d orbitals. - " = -陆 is a valid electron spin. This set also correctly represents a 4d electron.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Principal Quantum Number
The principal quantum number, often symbolized as \( n \), serves as the foundational quantum number in the hierarchy. It essentially tells us about the electron's energy level in an atom, similar to the rung of a ladder. The higher the value, the higher the energy level and the farther it is from the nucleus.

Here's a breakdown of its characteristics:
  • It has positive integer values: \( n = 1, 2, 3, \ldots \).
  • As \( n \) increases, the atomic radius typically increases, meaning electrons in higher energy levels are found further from the nucleus.
  • It dictates the size of the orbital 鈥� larger \( n \) values mean larger orbitals.
Understanding \( n \) is critical as it sets the stage for all other quantum numbers and the possible energies and positions of electrons within the atom.
Azimuthal Quantum Number
The azimuthal quantum number, represented by \( l \), is key to defining an electron's subshell or sublevel. It provides insight into the shape of the orbital where the electron is likely to be found. The value of \( l \) depends on the principal quantum number \( n \) and can range from 0 to \( n-1 \).

A concise explanation includes:
  • For \( n = 4 \), \( l \) can be 0, 1, 2, or 3. These correspond to the s, p, d, and f subshells, respectively.
  • Each value of \( l \) determines a specific shape: \( l = 0 \) is spherical (s), \( l = 1 \) is dumbbell-shaped (p), \( l = 2 \) has a double dumbbell shape (d), and so on.
  • The azimuthal quantum number also informs us about the angular momentum of an electron within that subshell.
This quantum number intricately links the electron's energy and spatial distribution within the atom, shaping how we perceive electron clouds.
Magnetic Quantum Number
The magnetic quantum number, designated as \( m_l \), delves deeper into the orientation of an electron's orbital within a subshell. It arises due to the magnetic field effects, which influence how these orbitals align naturally.

Critical points about \( m_l \) include:
  • For every azimuthal quantum number \( l \), the magnetic quantum number \( m_l \) can vary from \( -l \) to \( +l \), including zero. For instance, if \( l = 2 \) (d orbital), \( m_l \) can be -2, -1, 0, +1, or +2.
  • This number determines the number of orientations an orbital can have within a subshell.
  • An electron's specific \( m_l \) value dictates the precise shape and orientation of its orbital, which is integral for complex structures like multi-electron systems.
The magnetic quantum number thus provides a multidimensional view of possible electron presence across different orientations.
Spin Quantum Number
The spin quantum number, noted as \( m_s \), brings us into the realm of an electron's intrinsic spin, a fundamental property akin to angular momentum. This quantum number distinguishes electrons even further by indicating their specific spin direction within an orbital.

Key aspects to consider:
  • It can take one of two possible values: \( +\frac{1}{2} \) or \( -\frac{1}{2} \).
  • This property allows electrons to pair within an orbital, each having opposite spins, contributing to the stability of atoms.
  • The spin states are critical in defining phenomena like magnetism, as aligned spins generate a magnetic moment.
Understanding \( m_s \) helps to decipher the behavior of electrons under external influences and their subsequent reactions.

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Most popular questions from this chapter

Radial nodes present in \(3 \mathrm{~s}\) and \(2 \mathrm{p}\) orbitals are respectively (a) 0,2 (b) 2,0 (c) 2,1 (d) 1,2

The mass of an electron is \(\mathrm{m}\). Its charge is e and it is accelerated from rest through a potential difference \(\mathrm{V}\). The velocity acquired by the electron will be (a) \(\sqrt{\mathrm{V} / \mathrm{m}}\) (b) \(\mathrm{VeV} / \mathrm{m}\) (c) \(\sqrt{2 \mathrm{e}} \mathrm{V} / \mathrm{m}\) (d) none

The increasing order (lowest first) for the values of \(\mathrm{e} / \mathrm{m}\) (charge/mass) for electron (e), proton (p), neutron (n) and alpha particle (a) is (a) \(\mathrm{n}, \mathrm{p}, \mathrm{a}, \mathrm{e}\) (b) \(\mathrm{n}, \mathrm{p}, \mathrm{e}, \mathrm{a}\) (c) \(\mathrm{n}, \mathrm{a}, \mathrm{p}, \mathrm{e}\) (d) e, p, n, a

If the nitrogen atom has electronic configuration \(1 \mathrm{~s}^{7}\), it would have energy lower than that of the normal ground state configuration \(1 \mathrm{~s}^{2} 2 \mathrm{~s}^{2} 2 \mathrm{p}^{3}\), because the electrons would be closer to the nucleus. Yet \(1 \mathrm{~s}^{7}\) is not observed because it violates (a) Heisenberg uncertainty principle (b) Hund's rule (c) Pauli's exclusion principle (d) Bohr postulates of stationary orbits

For a d-electron, the orbital angular momentum is (a) \(\sqrt{6} \mathrm{~h}\) (b) \(\sqrt{2 h}\) (c) \(\mathrm{h}\) (d) \(2 \mathrm{~h}\)
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