Problem 41 - Chemistry and Chemical Reactivity

Question

Write a complete set of quantum numbers \((n, \ell,\) and \(m_{\ell}\) ) that quantum theory allows for each of the following orbitals: (a) \(2 p,\) (b) \(3 d,\) and (c) \(4 f\)

Step-by-Step Solution

Answer

(a) 2p: (2,1,-1), (2,1,0), (2,1,1); (b) 3d: (3,2,-2) to (3,2,2); (c) 4f: (4,3,-3) to (4,3,3).

Step 1: Understanding the Quantum Numbers

Quantum numbers are used to describe the properties of atomic orbitals and the electrons in those orbitals. The principal quantum number, \( n \), defines the energy level or shell. The angular momentum quantum number, \( \ell \), defines the shape of the orbital and ranges from 0 to \( n-1 \). The magnetic quantum number, \( m_{\ell} \), defines the orientation of the orbital and ranges from \( -\ell \) to \( +\ell \).

Step 2: Determine Quantum Numbers for 2p Orbital

For a \( 2p \) orbital:- The principal quantum number \( n = 2 \) since it is the second shell.- The orbital type \( p \) corresponds to \( \ell = 1 \).- Therefore, \( m_{\ell} \) can be \( -1, 0, \) or \( +1 \) since \( m_{\ell} \) ranges from \( -\ell \) to \( +\ell \). The complete set of quantum numbers for \( 2p \) is \((2, 1, -1), (2, 1, 0), (2, 1, +1)\).

Step 3: Determine Quantum Numbers for 3d Orbital

For a \( 3d \) orbital:- The principal quantum number \( n = 3 \) since it is the third shell.- The orbital type \( d \) corresponds to \( \ell = 2 \).- Therefore, \( m_{\ell} \) can be \( -2, -1, 0, +1, \) or \( +2 \) because \( m_{\ell} \) ranges from \( -\ell \) to \( +\ell \). The complete set of quantum numbers for \( 3d \) is \((3, 2, -2), (3, 2, -1), (3, 2, 0), (3, 2, +1), (3, 2, +2)\).

Step 4: Determine Quantum Numbers for 4f Orbital

For a \( 4f \) orbital:- The principal quantum number \( n = 4 \) since it is the fourth shell.- The orbital type \( f \) corresponds to \( \ell = 3 \).- Therefore, \( m_{\ell} \) can be \( -3, -2, -1, 0, +1, +2, \) or \( +3 \) because \( m_{\ell} \) ranges from \( -\ell \) to \( +\ell \). The complete set of quantum numbers for \( 4f \) is \((4, 3, -3), (4, 3, -2), (4, 3, -1), (4, 3, 0), (4, 3, +1), (4, 3, +2), (4, 3, +3)\).

Key Concepts

Principal Quantum Number

The principal quantum number, denoted by \( n \), is a vital concept in understanding atomic structure. It describes the main energy level that an electron occupies in an atom, or essentially the shell. Each shell corresponds to a specific energy level, and the higher the value of \( n \), the higher the energy and the larger the average distance from the nucleus.

For example:

• In the \( 2p \) orbital, the principal quantum number \( n = 2 \), indicating it is in the second energy level.
• The \( 3d \) orbital has \( n = 3 \), meaning the electron resides in the third energy level.
• Similarly, the \( 4f \) orbital with \( n = 4 \) is positioned in the fourth energy level.

It’s important to note that \( n \) must be a positive integer (1, 2, 3, etc.). As \( n \) increases, the energy difference between successive energy levels decreases.

Angular Momentum Quantum Number

The angular momentum quantum number, represented by \( \ell \), is integral to determining the shape of an electron's orbital. This number is crucial for defining the subshells within an energy level. The possible values are integer numbers ranging from 0 to \( n-1 \), where each value corresponds to a specific type of orbital shape.

Understanding \( \ell \):

• \( \ell = 0 \) represents an \(s\) orbital, which is spherical.
• \( \ell = 1 \) corresponds to a \(p\) orbital, characterized by a dumbbell shape. This is seen in the \( 2p \) orbital.
• \( \ell = 2 \) signifies a \(d\) orbital, commonly found in the \( 3d \) energy level.
• \( \ell = 3 \) denotes an \(f\) orbital, as in the \( 4f \) subshell.

The angular momentum quantum number essentially dictates the shape and complexity of the electron cloud associated with each orbital, playing a key role in electron configuration.

Magnetic Quantum Number

The magnetic quantum number, \( m_{\ell} \), intricately defines the orientation of an orbital in space relative to the three axes. It can take on a range of integer values from \(-\ell\) to \(+\ell\), encompassing a total of \(2\ell + 1\) possible orientations for a given subshell.

For clarification:

• In a \( 2p \) orbital, where \( \ell = 1 \), \( m_{\ell} \) can be \(-1, 0, \) or \(+1\).
• For a \( 3d \) orbital with \( \ell = 2 \), \( m_{\ell} \) spans from \(-2\) to \(+2\), yielding five possible values.
• In the case of \( 4f \), where \( \ell = 3 \), \( m_{\ell} \) ranges from \(-3\) through \(+3\), resulting in seven orientations.

This component of quantum numbers adds layers to understanding how electrons populate orbitals, as each distinct \( m_{\ell} \) value correlates with a unique orbital orientation.