Problem 35 - Chemistry and Chemical Reactivity

Question

What is the maximum number of orbitals that can be identified by each of the following sets of quantum numbers? When "none" is the correct answer, explain your reasoning. (a) \(n=3, \ell=0, m_{\ell}=+1\) (b) \(n=5, \ell=1\) (c) \(n=7, \ell=5\) (d) \(n=4, \ell=2, m_{\ell}=-2\)

Step-by-Step Solution

Answer

(a) None, (b) 3, (c) 11, (d) 1.

Step 1: Understanding Quantum Numbers

In quantum mechanics, an orbital is defined by a set of three quantum numbers: the principal quantum number \( n \), the azimuthal (or angular momentum) quantum number \( \ell \), and the magnetic quantum number \( m_{\ell} \). The number \( \ell \) can be any integer from 0 to \( n-1 \). The number \( m_{\ell} \) can take integer values ranging from \( -\ell \) to \( +\ell \).

Step 2: Assessment of Case (a): \(n=3, \ell=0, m_{\ell}=+1\)

For \( n = 3 \), \( \ell \) can only be \( 0, 1, \) or \( 2 \). For \( \ell = 0 \), the only allowable value of \( m_{\ell} \) is \( 0 \) as \( m_{\ell} \) ranges from \(-\ell\) to \(+\ell\). Since \( m_{\ell} = +1 \) is not possible when \( \ell = 0 \), this set of quantum numbers does not correspond to any valid orbital.

Step 3: Assessment of Case (b): \(n=5, \ell=1\)

For \( n = 5 \), \( \ell \) can take values of \( 0, 1, 2, 3, \) or \( 4 \). Given \( \ell = 1 \), the possible values for \( m_{\ell} \) are \(-1, 0, \) and \(+1\). Thus, there are three orbitals corresponding to the quantum numbers: one for each value of \( m_{\ell} \).

Step 4: Assessment of Case (c): \(n=7, \ell=5\)

For \( n = 7 \), \( \ell \) can range from 0 to 6. When \( \ell = 5 \), \( m_{\ell} \) ranges from \(-5\) to \(+5\). This gives a total of 11 possible values for \( m_{\ell} \), meaning there are 11 orbitals that match these quantum numbers.

Step 5: Assessment of Case (d): \(n=4, \ell=2, m_{\ell}=-2\)

For \( n = 4 \), \( \ell \) can range from 0 to 3. For \( \ell = 2 \), \( m_{\ell} \) can take the values \(-2, -1, 0, 1, \) and \(+2\). With the specific quantum numbers \( m_{\ell} = -2 \) given, it identifies exactly one unique orbital, as this value is allowed.

Key Concepts

Principal Quantum Number

The Principal Quantum Number, often represented as \( n \), is the first quantum number in defining an electron's position within an atom. It gives us a broad idea about an electron's energy level and its relative distance from the nucleus.

• Each value of \( n \) corresponds to a specific electron shell, which is roughly associated with the electron's energy level.
• The principal quantum number can take any positive integer value starting from 1 (\( n = 1, 2, 3, \ldots \)).
• The energy and size of an electron shell increase with higher values of \( n \).

In the provided exercise, we observe scenarios with various values of \( n \), such as \( n = 3 \), \( n = 5 \), \( n = 7 \), and \( n = 4 \). Each signifies a different energy level of orbitals in an atom. The principal quantum number is paramount as it sets the stage for further quantum numbers, which refine the description of electron positioning even further.Understanding the principal quantum number helps us grasp how electrons are organized in an atom, driving the structure of periodic tables and chemical properties.

Azimuthal Quantum Number

The Azimuthal Quantum Number, denoted by \( \ell \), plays a crucial role in determining the shape of an electron's orbital. It is sometimes referred to as the angular momentum quantum number.

• This quantum number can take values ranging from \( 0 \) to \( n-1 \), where \( n \) is the principal quantum number.
• Each value of \( \ell \) corresponds to a particular type of atomic orbital: \( \ell = 0 \) describes an \( s \) orbital (spherical), \( \ell = 1 \) corresponds to a \( p \) orbital (dumbbell-shaped), \( \ell = 2 \) represents a \( d \) orbital, and \( \ell = 3 \) indicates an \( f \) orbital.

In the given examples:- When \( n = 3 \) and \( \ell = 0 \), it refers to a spherical \( s \) orbital.- For \( n = 5 \) and \( \ell = 1 \), orbitals are \( p \) orbitals.- For \( n = 7 \) and \( \ell = 5 \), it depicts a complex \( d \) orbital structure.- When \( n = 4 \), \( \ell = 2 \), these are \( d \) orbitals.The azimuthal quantum number is vital for understanding how orbitals shape the electron's path and influence chemical bonding and spectral lines.

Magnetic Quantum Number

The Magnetic Quantum Number, denoted as \( m_{\ell} \), is key to determining the orientation of an orbital within a magnetic field. It provides us with the final refinement in positioning an electron within its orbital.

• The values of \( m_{\ell} \) range from \( -\ell \) to \( +\ell \), including zero. This means for any given \( \ell \), there are \( 2\ell + 1 \) possible values for \( m_{\ell} \).
• Each value of \( m_{\ell} \) corresponds to a different orbital within a subshell, distinguishing its orientation in space.

For instance, in our given questions:- \( n = 3, \ell = 0 \) with \( m_{\ell} = +1 \) is invalid because \( m_{\ell} \) should be 0.- \( n = 5, \ell = 1 \) has three valid orbital orientations: \(-1, 0, +1\).- \( n = 7, \ell = 5 \) corresponds to eleven orientations for orbitals, ranging \(-5 \) to \(+5 \).- \( n = 4, \ell = 2, m_{\ell} = -2 \) perfectly points to a specific orbital.The magnetic quantum number fine-tunes the understanding of atomic electron configurations, helping us predict an atom's magnetic properties and how its electrons will interact with an external magnetic field.