Problem 42 - Chemistry and Chemical Reactivity

Question

A given orbital has a magnetic quantum number of \(m_{e}=\) -1. This could not be a (an) (a) forbital (c) \(p\) orbital (b) \(d\) orbital (d) s orbital

Step-by-Step Solution

Answer

The orbital cannot be an "s" orbital.

Step 1: Understanding the Magnetic Quantum Number (\(m_e\))

The magnetic quantum number \(m_e\) is associated with the orientation of an orbital in space. It can take integer values ranging from \(-l\) to \(+l\), where \(l\) is the azimuthal or angular momentum quantum number.

Step 2: Relate \(m_e\) Values to Orbital Types

Determine the possible \(m_e\) values for each type of orbital: - For \(s\) orbitals, \(l = 0\) therefore \(m_e = 0\). - For \(p\) orbitals, \(l = 1\) therefore \(m_e = -1, 0, +1\). - For \(d\) orbitals, \(l = 2\) therefore \(m_e = -2, -1, 0, +1, +2\). - For 'f' orbitals, \(l = 3\) therefore \(m_e = -3, -2, -1, 0, +1, +2, +3\).

Step 3: Identify the Orbital Type That Cannot Have \(m_e = -1\)

Examine the possible \(m_e\) values for each type of orbital. - For the 's' orbital, the only potential value for \(m_e\) is 0, so it cannot have an \(m_e = -1\). Since the problem states that \(m_e = -1\), it eliminates 's' orbitals as a possibility.

Key Concepts

Magnetic Quantum Number

The magnetic quantum number, denoted as \(m_e\), is an essential part of quantum mechanics that describes the orientation of an orbital in three-dimensional space. This quantum number is one of the four quantum numbers used to describe the unique quantum state of an electron in an atom. It determines the number of orbitals and their orientation within a subshell. To fully understand \(m_e\), we must first know that it is defined based on the azimuthal quantum number \(l\). The possible values for \(m_e\) range from \(-l\) to \(+l\). Therefore, for each value of \(l\), there are \(2l + 1\) possible values of \(m_e\). This explains why different orbitals can assume different orientations:

• For \(s\) orbitals \( (l = 0)\), \(m_e = 0\)
• For \(p\) orbitals \( (l = 1)\), \(m_e = -1, 0, +1\)
• For \(d\) orbitals \( (l = 2)\), \(m_e = -2, -1, 0, +1, +2\)
• For \(f\) orbitals \( (l = 3)\), \(m_e = -3, -2, -1, 0, +1, +2, +3\)

This pattern demonstrates how \(m_e\) helps in describing the spatial disposition of each electron and in turn, clarifies why an 's' orbital cannot have a \(m_e = -1\).

Orbital Types

In quantum chemistry, orbitals represent regions in an atom where an electron is likely to be found. Understanding these orbitals—\(s\), \(p\), \(d\), and \(f\)—is fundamental in explaining the behavior of electrons around a nucleus.- **s Orbitals**: - Spherical in shape. - Have only one orientation, with \(l = 0\) resulting in \(m_e = 0\).- **p Orbitals**: - Dumbbell-shaped and oriented along the x, y, and z axes. - With \(l = 1\), the possible \(m_e\) values are \(-1, 0, +1\).- **d Orbitals**: - More complex in shape, often described like four-leaf clovers or double doughnuts. - Possess \(l = 2\), leading to \(m_e\) values of \(-2, -1, 0, +1, +2\).- **f Orbitals**: - Even more intricate shapes. - Have \(l = 3\), with \(m_e\) ranging from \(-3\) to \(+3\).These variations in shapes and orientations of orbitals serve a critical role in predicting and explaining chemical bonding and molecular geometry. Knowing which orbitals exist for given quantum numbers helps in discerning which energy levels an electron may occupy.

Azimuthal Quantum Number

The azimuthal quantum number, symbolized as \(l\), is a crucial component in quantum mechanics that specifies the subshell or shape of an orbital. It is also known as the angular momentum quantum number, as it relates to the angular momentum of electrons. The values of \(l\) determine the orbital's shape and energy within a principal energy level.Here are the possible values for \(l\):

• \(l = 0\) for s orbitals
• \(l = 1\) for p orbitals
• \(l = 2\) for d orbitals
• \(l = 3\) for f orbitals

The value of \(l\) is always a positive integer from 0 up to \(n-1\), where \(n\) is the principal quantum number. As \(l\) increases, the complexity of the shapes and the number of nodal planes of the orbitals increases.The azimuthal quantum number is integral not only in defining the shape of an orbital but also in determining its energy hierarchy within an atom. By illustrating how electron configurations fill up orbitals, \(l\) plays a key role in predicting the magnetic properties of atoms and their interaction with external fields.