Problem 52 - Chemistry The Central Science

Question

How many possible values for \(l\) and \(m_{l}\) are there when (a) \(n=3 ;\) (b) \(n=5 ?\)

Step-by-Step Solution

Answer

(a) For \(n=3\), the possible values of \(l\) are 0, 1, and 2. (b) For \(n=5\), the possible values of \(l\) are 0, 1, 2, 3, and 4.

Step 1: Calculate possible values of \(l\)#

For a given principal quantum number \(n\), \(l\) can take integer values from 0 to \(n-1\). We are asked to find the number of possible values of \(l\) for (a) \(n=3\) and (b) \(n=5\). For each case, we will list the possible values of \(l\).

Key Concepts

Principal Quantum Number

The principal quantum number, denoted as n, plays a vital role in the quantum mechanics of atoms. Think of it as the address for an electron's energy level or shell.

The value of n starts from 1 and increases in positive integer steps (1, 2, 3, ...). As n increases, it represents electrons that are further from the nucleus, at higher energy levels, and with greater potential energy. Also, as n gets higher, the difference in energy between neighboring energy levels decreases.

For example, if an electron is in an atom with a principal quantum number of 3, it is in the third energy level. This not only affects its energy but also the number of orbital shapes available for electrons, which leads us to the next concept: the azimuthal quantum number.

Azimuthal Quantum Number

Following the address analogy, the azimuthal quantum number, usually notated as l, can be seen as specifying the 'street' or the shape of the orbital within the electron's energy level.

The value of l ranges from 0 to (n-1), where n is the principal quantum number. Each value of l corresponds to an orbital shape: 0 for s-orbital, 1 for p-orbital, 2 for d-orbital, and 3 for f-orbital. So with each increase in the principal quantum number, you get a new shape available for the electron to inhabit.

For instance, when n=3, the possible values for l are 0, 1, and 2, corresponding to s, p, and d orbitals. Having a grasp of how l is determined by n is crucial in understanding the distribution of electrons in an atom.

Magnetic Quantum Number

Drilling down further into electrons' addresses, the magnetic quantum number, designated as m_l, indicates the 'house number' or the orientation of an orbital within a subshell.

This number ranges from -l to +l, including zero. Therefore, for each azimuthal quantum number, there are 2l+1 possible magnetic quantum numbers. So if you were looking at the p orbital, which has an l value of 1, your m_l values could be -1, 0, and 1 – three possible orientations for an electron to exist in that orbital.

Each orientation corresponds to a different spatial direction magnetic field lines would pass through an electron, which is significant when considering how these electrons affect each other's presence and the overall electronic structure of the atom.