Question

For each of the questions, four choices have been provided. Select the correct alternative. Which of the following sets of quantum numbers is correct for an electron in \(3 \mathrm{~d}\) orbital? (a) \(3,0,0,+1 / 2\) (b) \(3,1,1,-1 / 2\) (c) \(3,2,1,+1 / 2\) (d) \(3,2,3,-1 / 2\)

Short answer

Answer: The correct set of quantum numbers for an electron in a 3d orbital is (c) \(3,2,1,+1 / 2\).

Step-by-step solution

Principal Quantum Number (n)

For a 3d orbital, the principal quantum number (n) must be equal to 3.

Angular Momentum Quantum Number (l)

For d orbitals, the angular momentum quantum number (l) must be equal to 2.

Magnetic Quantum Number (m_l)

The magnetic quantum number (m_l) can have integer values ranging from -l to +l, inclusive. For d orbitals (l=2), m_l can be -2, -1, 0, 1, or 2.

Electron Spin Quantum Number (m_s)

The electron spin quantum number (m_s) can have two values: +1/2 and -1/2.

Compare and Select the Correct Set of Quantum Numbers

Compare each set of given quantum numbers to the values derived for a 3d orbital: (a) \(3,0,0,+1 / 2\) - l is incorrect (b) \(3,1,1,-1 / 2\) - l is incorrect (c) \(3,2,1,+1 / 2\) - n, l, m_l, and m_s are all within the correct range (d) \(3,2,3,-1 / 2\) - m_l is incorrect Only option (c) has the correct values for all four quantum numbers. Therefore, the correct set of quantum numbers for an electron in a 3d orbital is: (c) \(3,2,1,+1 / 2\)

Key concepts

Principal Quantum Number

The principal quantum number, often denoted as \( n \), plays a vital role in determining the energy level and size of an electron's orbit around the nucleus. Think of \( n \) as the "address" of the electron within an atom, helping to pinpoint which orbital shell the electron occupies.

• Values of \( n \): This quantum number can take positive integer values such as 1, 2, 3, and so on.
• Significance of \( n \): The larger the value of \( n \), the further the electron is from the nucleus and the higher its energy level.

In a 3d orbital, the principal quantum number \( n \) is equal to 3, indicating that the electron is in the third energy level of the atom. As \( n \) increases, the orbitals become larger and more spaced out, allowing for more complex and varied electron configurations.

Angular Momentum Quantum Number

The angular momentum quantum number, represented by \( l \), determines the shape of the electron's orbital. It essentially describes the "angular momentum" or "shape" of the electron cloud around the nucleus.

• Possible values for \( l \): The value of \( l \) ranges from 0 to \( n-1 \), where \( n \) is the principal quantum number of the orbital.
• Shapes of orbitals: When \( l = 0 \), the orbital is spherical (s orbital); \( l = 1 \) refers to polar (p orbital); \( l = 2 \) is cloverleaf shaped (d orbital), and so forth for higher values.

For a 3d orbital, \( l \) must be 2, indicating a d-type shape, which can hold up to 10 electrons. These shapes and configurations help define the unique chemical properties of elements.

Magnetic Quantum Number

The magnetic quantum number, denoted as \( m_l \), specifies the orientation of the electron's orbital in space. It is crucial for understanding how electrons behave in magnetic fields and defines the specific "sub-orbital" an electron may occupy.

• Range of \( m_l \): This quantum number can take integer values from \(-l\) to \(+l\), including zero.
• Sub-orbitals: For example, for a d orbital where \( l = 2 \), \( m_l \) can be \(-2, -1, 0, 1, 2\), offering five possible orientations or sub-orbitals.

In the 3d orbital with \( l = 2 \), \( m_l \) values allow the electron to occupy any one of five sub-orbitals, influencing properties such as an atom's magnetic behavior.

Electron Spin Quantum Number

The electron spin quantum number, symbolized as \( m_s \), refers to the intrinsic spin of an electron within an atom. Unlike the other quantum numbers, which describe the electron's energy and spatial distribution, \( m_s \) is concerned with the electron's intrinsic "twist."

• Possible values: The electron spin quantum number can be either \(+1/2\) or \(-1/2\). These correspond to the two possible spin states of an electron, often described as "spin-up" or "spin-down."
• Significance of \( m_s \): These values are crucial for understanding the electron pairing and overall spin state of atoms, influencing magnetic and optical properties.

Each orbital can hold two electrons with opposite spins. For a 3d orbital, the electron in option (c) has an \( m_s \) of \(+1/2\), fitting perfectly into one of its sub-orbitals and contributing to the stable, balanced electron configuration in the atom.