Question

A hydrogen atom orbital has \(n=4\) and \(m_{l}=-2\). (a) What are the possible values of \(l\) for this orbital? (b) What are the possible values of \(m_{s}\) for the orbital?

Short answer

(a) The possible values of \(l\) for this orbital are 2 and 3. (b) The possible values of \(m_s\) for the orbital are always +1/2 and -1/2.

Step-by-step solution

(a) Possible values of l

To find the possible values of the angular momentum quantum number l, we need to recall the relationship between n, l and m_l: \(l\) can have integer values from 0 to n-1, and \(m_{l}\) can have integer values ranging from -l to +l. In our case, n = 4, and \(m_{l} = -2\). Since m_l = -2, we know that l must be at least 2, since m_l can only take values from -l to +l. Therefore, we will try each possible value of l = 2, 3, and check if \(m_l = -2\) is a valid option. For l = 2, \(m_l\) can take values -2, -1, 0, 1, 2, which includes \(m_l = -2\). For l = 3, \(m_l\) can take values -3, -2, -1, 0, 1, 2, 3, which also includes \(m_l = -2\). So, the possible values of l for this orbital are 2 and 3.

(b) Possible values of m_s

To find the possible values of the spin quantum number \(m_s\), we recall that for an electron in an orbital, there are only two possible spin orientations available: 1. Spin-up (represented as +1/2) 2. Spin-down (represented as -1/2) So, for any electron in an orbital, the possible values of \(m_s\) are always +1/2 and -1/2.

Key concepts

Angular Momentum Quantum Number

The angular momentum quantum number, represented by the symbol \(l\), plays a crucial role in quantum mechanics, specifically in the understanding of atomic orbitals. It helps define the shape of an electron's orbital within an atom. The value of \(l\) is directly linked to the principal quantum number \(n\), which determines the electron’s energy level.

In any atom:

• \(l\) can take any integer value from 0 to \(n-1\).
• Each value of \(l\) corresponds to a different type of orbital, like \(s, p, d,\) and \(f\).

For example, if \(n = 4\), \(l\) can be 0, 1, 2, or 3. These numbers dictate the shape and complexity of the orbitals: 0 for \(s\), 1 for \(p\), 2 for \(d\), and 3 for \(f\).

In the exercise provided, since \(m_{l}=-2\), the possible \(l\) values that permit \(m_{l} = -2\) are 2 and 3. This is due to the fact that the magnetic quantum number \(m_{l}\) can only be within the range \(-l \leq m_{l} \leq +l\). Therefore, both \(l = 2\) and \(l = 3\) can have \(m_{l} = -2\).

Spin Quantum Number

The spin quantum number, denoted by \(m_{s}\), is essential in understanding the intrinsic "spin" or angular momentum of electrons within an atom. Unlike other quantum numbers which describe electron paths, \(m_{s}\) denotes the orientation of the electron's spin.

The possible values for \(m_{s}\) are unique because:

• \(m_{s}\) can only take on two possible values: +1/2 or -1/2.
• These two values represent the two possible orientations of an electron's spin - often referred to as "spin-up" and "spin-down".

In any given orbital, electrons will usually pair with opposite spins, minimizing their mutual repulsion, which is a cornerstone principle of electron configurations.

In the context of the exercise, regardless of other quantum numbers like \(n\), \(l\), or \(m_{l}\), \(m_{s}\) remains limited to these two values. This reflects the fundamental nature of electron spin, a quantum property that is fixed for all electrons.

Magnetic Quantum Number

The magnetic quantum number, symbolized by \(m_{l}\), further refines the understanding of an electron's position within an atom's electron cloud. It gives insight into the orientation of an orbital around the nucleus in space.

The following guidelines determine the possible values of \(m_{l}\):

• \(m_{l}\) can take on integer values ranging from \(-l\) to \(+l\).
• It defines the electron’s exact orientation in space within a given orbital type \(l\).

For instance, when \(l = 2\) (a \(d\) orbital), \(m_{l}\) can be -2, -1, 0, 1, or 2, showing five unique spatial orientations. Similarly, for \(l = 3\) (an \(f\) orbital), \(m_{l}\) can range from -3 to +3, offering seven orientations.

In the given exercise scenario, \(m_{l}=-2\) constrains \(l\) to those values that allow this orientation, namely 2 and 3. It provides insight into the possible spatial distribution of the electron within the hydrogen atom, showing how quantum numbers interact to define states of the electron.