Question

Given the following sets of quantum numbers, indicate those that could not occur and explain your answer. (a) \(1,2,0,+\frac{1}{2}\) (b) \(2,1,2,+\frac{1}{2}\) (c) \(3,0,0,-\frac{1}{2}\) (d) \(3,2,1,+\frac{1}{2}\) (e) 4,2,-2,0

Short answer

Question: Identify the sets of quantum numbers that can NOT occur: (a) (1, 2, 0, +1/2) (b) (2, 1, 2, +1/2) (c) (3, 0, 0, -1/2) (d) (3, 2, 1, +1/2) (e) (4, 2, -2, 0) Answer: Sets (a), (b), and (e) can NOT occur due to incorrect values for one of the quantum numbers.

Step-by-step solution

Check set (a)

We have \(n = 1\), \(l = 2\), \(m_l = 0\), and \(m_s = +\frac{1}{2}\). However, since \(l\) must be \(0, 1, 2, ..., (n-1)\), then having \(l = 2\) when \(n = 1\) violates this condition. Therefore, this set cannot occur due to the value of \(l\).

Check set (b)

We have \(n = 2\), \(l = 1\), \(m_l = 2\), and \(m_s = +\frac{1}{2}\). The principal and azimuthal quantum numbers are valid, but the magnetic quantum number \(m_l\) must be \(-l, -(l-1), ..., 0, ..., (l-1), l\), that for \(l=1\) would be \(-1, 0, 1\). Then \(m_l=2\) violates this condition, so this set cannot occur due to the value of \(m_l\).

Check set (c)

We have \(n = 3\), \(l = 0\), \(m_l = 0\), and \(m_s = -\frac{1}{2}\). All these values are plausible, since \(l\) can be \(0\) when \(n = 3\), \(m_l\) can be \(0\) when \(l = 0\), and the spin quantum number is valid. Therefore, this set is valid and can occur.

Check set (d)

We have \(n = 3\), \(l = 2\), \(m_l = 1\), and \(m_s = +\frac{1}{2}\). All these values are plausible, since \(l\) can be \(2\) when \(n = 3\), \(m_l\) is within the range of possible values for \(l = 2\), and the spin quantum number is valid. Therefore, this set is valid and can occur.

Check set (e)

We have \(n = 4\), \(l = 2\), \(m_l = -2\), and \(m_s = 0\). The principal, azimuthal, and magnetic quantum numbers are valid. However, the spin quantum number must be either \(+\frac{1}{2}\) or \(-\frac{1}{2}\) for electrons, so here=m_s = 0 is incorrect. Therefore, this set cannot occur due to the value of \(m_s\).

Key concepts

Quantum Number Rules

Quantum numbers are a fundamental part of understanding electrons within atoms. They describe characteristics of an electron in an atom and are governed by a set of rules that determine how these numbers can be combined.

• The first rule is the principal quantum number (), which specifies the electron's energy level and is always a positive integer ( = 1, 2, 3, ...).
• The second rule involves the azimuthal (or angular momentum) quantum number (), which defines the shape of the electron's orbital and can range from 0 to - 1 for each energy level.
• The third rule is for the magnetic quantum number (), which describes the orientation of the electron's orbital in space and ranges from - to +.
• Finally, there is the spin quantum number (), which represents the electron's spin direction and can be either +1/2 or -1/2.

Any set of quantum numbers that does not adhere to these rules cannot describe a real electron in an atom, as we saw with the example sets given for the exercise.

Azimuthal Quantum Number

The azimuthal quantum number, often symbolized as or , has a crucial role in defining the shape and sublevel of an electron's orbital. It can take on any integer value from 0 to -1, where is the principal quantum number.

For example, if = 3, then can be 0, 1, or 2. These values correspond to the s, p, and d subshells, respectively – with s being spherical, p being dumbbell-shaped, and d being more complex. The azimuthal quantum number is related to the angular momentum of the electron and gives rise to the different orbital shapes that help determine the chemical properties of an element.

Remember, the value of the azimuthal quantum number is directly tied to the principal quantum number; one cannot simply choose a random value for without considering the value of .

Magnetic Quantum Number

The magnetic quantum number, denoted as or , explains the orientation of the electron's orbital in a three-dimensional space. It is determined by the azimuthal quantum number and can have values that range from - to +, including zero. This range means that for a given value of , there are 2+1 possible values for .

For instance, if = 2, then the possible values for are -2, -1, 0, +1, and +2. These different values correspond to different orientations of an orbital in space. In a magnetic field, these orientations can affect how an electron's energy level splits – a concept known as magnetic quantum number splitting, which influences phenomena like the Zeeman effect.

Understanding the magnetic quantum number is vital for grasping how electrons are distributed within an atom and how they contribute to chemical bonding and the atom’s magnetic properties.