Question

What is the maximum number of orbitals that can be identified by each of the following sets of quantum numbers? When "none" is the correct answer, explain your reasoning. (a) \(n=4, \ell=3\) (b) \(n=5\) (c) \(n=2, \ell=2\) (d) \(n=3, \ell=1, m_{\ell}=-1\)

Short answer

(a) 7 orbitals, (b) 25 orbitals, (c) None, (d) 1 orbital.

Step-by-step solution

Identify possible orbitals for (a)

For set (a) with quantum numbers \(n=4\) and \(\ell=3\), we need to use the magnetic quantum number \(m_{\ell}\). For \(\ell=3\), \(m_{\ell}\) can have values of -3, -2, -1, 0, 1, 2, 3. This totals 7 possible orbitals for this set of quantum numbers.

Identify possible orbitals for (b)

For set (b) with \(n=5\), the angular momentum quantum number \(\ell\) can take values from 0 to 4. For each \(\ell\), \(m_{\ell}\) has a corresponding range from \(-\ell\) to \(+\ell\). Therefore:- \(\ell=0\), \(m_{\ell}=0\): 1 orbital- \(\ell=1\), \(m_{\ell}=-1, 0, 1\): 3 orbitals- \(\ell=2\), \(m_{\ell}=-2, -1, 0, 1, 2\): 5 orbitals- \(\ell=3\), \(m_{\ell}=-3, -2, -1, 0, 1, 2, 3\): 7 orbitals- \(\ell=4\), \(m_{\ell}=-4, -3, -2, -1, 0, 1, 2, 3, 4\): 9 orbitalsAdding these together gives a total of 25 possible orbitals for \(n=5\).

Identify possible orbitals for (c)

For set (c) with \(n=2\) and \(\ell=2\), note that \(\ell\) is not valid since for \(n=2\), the maximum \(\ell\) can be 1. Therefore, with these quantum numbers, no orbitals exist. The answer is none.

Identify possible orbitals for (d)

For set (d) with \(n=3\), \(\ell=1\), and \(m_{\ell}=-1\), the given values directly point to a single valid orbital. So, there is exactly 1 orbital for this set of quantum numbers.

Key concepts

Orbitals

Orbitals are regions within an atom where electrons are likely to be found. They are defined by quantum numbers, which describe the energy level, shape, and orientation of the electron cloud. Each orbital can hold up to two electrons. Understanding orbitals requires comprehension of the following quantum numbers:

• Principal quantum number ( "): Determines the energy level. As the value of increases, the energy level and the size of the orbital increase.


• Angular momentum quantum number ("): Defines the shape of the orbital. The values range from 0 to -1.


• Magnetic quantum number ("): Describes the orientation of the orbital in space. It can take values from - to +.

Recognizing how these quantum numbers interplay will help you understand the different possible configurations of orbitals.

Magnetic Quantum Number

The magnetic quantum number (") is crucial in determining the spatial orientation of an orbital. It can have values ranging from - to +, including zero. The number of values indicates the number of orbitals available within a particular subshell defined by the angular momentum quantum number, ".

• If " is 0, the only possible value for " is 0. Therefore, there is just one orbital.


• For " = 1, " can be -1, 0, or 1, resulting in three orbitals.


• Similarly, if " = 2, " can be -2, -1, 0, 1, or 2, leading to five orbitals.

This pattern continues with increasing values of ". This quantum number explains why atoms have different orientations for their electron clouds in three-dimensional space.

Angular Momentum

Angular momentum in quantum mechanics is related to the shape of the orbital, described by the angular momentum quantum number ("). For every value of ", there is a corresponding type of subshell.

• A value of " = 0 corresponds to an s orbital, which is spherical.


• " = 1 corresponds to p orbitals, which are dumbbell-shaped.


• When " = 2, the orbitals are d-shaped and have a more complex structure.


• Finally, " = 3 means the orbitals are f-shaped, with even more intricate shapes.

These subshells allow for different numbers of spatial orientations, which are described by the magnetic quantum number ("). Understanding angular momentum helps us know why there are different types and numbers of orbitals in an atom.

Quantum Mechanics

Quantum mechanics is a fundamental theory in physics that explains the micro-world, particularly the behavior of atomic and subatomic particles. This field is centered around the concept that energy levels in atoms are quantized, meaning they can only take certain discrete values.

• Quantum mechanics introduces quantum numbers, which allow us to predict the behavior and location of electrons in an atom.


• It explains phenomena that classical physics cannot, such as electron transition between energy levels and the peculiar idea that electrons can exist in multiple states until observed.


• The principle of superposition, a key quantum mechanics principle, states that particles like electrons exist in a combination of all possible states simultaneously until measured.

This fascinating field provides not only a more profound understanding of matter at the smallest scales but also technological advancements in areas like semiconductors and MRI machines.