Question

a. What information is given by the magnetic quantum number? b. How many orbital orientations are possible in each of the s, p, d, and f sublevels? c. Explain and illustrate the notation for distinguishing between the different p orbitals in a sublevel.

Short answer

a. The magnetic quantum number indicates orbital orientation. b. s:1, p:3, d:5, f:7 orbitals. c. p_x, p_y, p_z indicate different orientations.

Step-by-step solution

Understanding the Magnetic Quantum Number

The magnetic quantum number, represented by the symbol m_l, provides information about the orientation of an orbital around the nucleus. It indicates the number of orbitals and their orientations within a subshell. This quantum number can have integer values ranging from -l to +l, where l is the azimuthal quantum number.

Determine the Number of Orbital Orientations for Sublevels

To find the number of orbital orientations for each sublevel (s, p, d, and f), consider the possible m_l values:- For the s sublevel ( l = 0), m_l = 0; hence, 1 orbital.- For the p sublevel ( l = 1), m_l can be -1, 0, +1; hence, 3 orbitals.- For the d sublevel ( l = 2), m_l can be -2, -1, 0, +1, +2; hence, 5 orbitals.- For the f sublevel ( l = 3), m_l can be -3, -2, -1, 0, +1, +2, +3; hence, 7 orbitals.

Notation for Distinguishing Different p Orbitals

Each p orbital in a sublevel can be distinguished by their m_l value. The notation used is as follows:- p_x for m_l = -1.- p_y for m_l = 0.- p_z for m_l = +1.These notations indicate the axis along which the electron density is oriented.

Key concepts

Magnetic Quantum Number

The magnetic quantum number, symbolized as \(m_l\), plays an important role in quantum chemistry. It tells us about the orientation of an orbital in space around the nucleus. This means it specifies in which direction the orbital points.

The possible values for \(m_l\) range from \(-l\) to \(+l\), where \(l\) is the azimuthal (or angular momentum) quantum number. For example, if \(l = 1\), then \(m_l\) can be -1, 0, or +1. Each of these values corresponds to a different orientation for the orbital within the subshell.

Orbital Orientation

Orbital orientation describes how orbitals are arranged in space. It’s significant because electrons reside in these orbitals.

- For the **s** sublevel (\(l = 0\)), there is only one possible orientation since \(m_l\) can only be 0.
- In the **p** sublevel (\(l = 1\)), there are three possible orientations: \(m_l = -1, 0,\) and \(+1\).

- The **d** sublevel (\(l = 2\)) has five orientations because \(m_l\) can be -2, -1, 0, +1, and +2.

- Finally, the **f** sublevel (\(l = 3\)) boasts seven orientations: \(-3, -2, -1, 0, +1, +2, +3\). Understanding these orientations is key to determining how many orbitals can fit in each sublevel.

Sublevels in Orbitals

Sublevels help organize orbitals within atoms. They are designated by the azimuthal quantum number \(l\), and each sublevel corresponds to a different type of orbital.

- The **s** sublevel has \(l = 0\), so there's just one s orbital.

- The **p** sublevel has \(l = 1\), leading to three p orbitals.

- The **d** sublevel corresponds to \(l = 2\), resulting in five d orbitals.

- The **f** sublevel, with \(l = 3\), contains seven f orbitals.

These sublevels tell us how electrons fill an atom and help predict an element's behavior in reactions.

Notation for p Orbitals

Each p orbital in a sublevel can be identified by its magnetic quantum number (\(m_l\)). There are three p orbitals, and we label them based on their orientation.

- **p_x** corresponds to \(m_l = -1\).

- **p_y** corresponds to \(m_l = 0\).

- **p_z** corresponds to \(m_l = +1\).

This notation signifies the axis along which the p orbital is aligned. For example, the p_x orbital is aligned along the x-axis, p_y along the y-axis, and p_z along the z-axis. This helps in visualizing the spatial distribution of electrons within these orbitals.