Question

Determine whether each of the following sets of quantum numbers for the hydrogen atom are valid. If a set is not valid, indicate which of the quantum numbers has a value that is not valid: $$\begin{array}{l}\text{ (a) }n=4,l=1,m_l=2,m_s=-\frac{1}{2}\\ \text{ (b) }n=4,l=3,m_l=-3,m_s=+\frac{1}{2}\\ \\ \text{ (c) }n=3,l=2,m_l=-1,m_s=+\frac{1}{2}\\ \text{ (d) }n=5,l=0,m_l=0,m_s=0\\ \text{ (e) }n=2,l=2,m_l=1,m_s=+\frac{1}{2}\end{array}$$

Short answer

The result is: (a) Invalid, $m_l$ is not valid. (b) Valid. (c) Valid. (d) Invalid, $m_s$ is not valid. (e) Invalid, $l$ is not valid.

Step-by-step solution

Analyze Set (a)

Here we have $n=4$, $l=1$, $m_l=2$, and $m_s=-\frac{1}{2}$. Principal quantum number (n): $n=4$ which is valid. Angular momentum quantum number (l): $l=1$, as $0\le l\le 3$, which is valid. Magnetic quantum number (m_l): $m_l=2$, but the restriction was $-1\le m_l\le 1$. Therefore, $m_l=2$ is invalid. Spin quantum number (m_s): $m_s=-\frac{1}{2}$ which is valid.

Analyze Set (b)

Here we have $n=4$, $l=3$, $m_l=-3$, and $m_s=+\frac{1}{2}$. All quantum numbers in this set follow the rules and are valid.

Analyze Set (c)

Here we have $n=3$, $l=2$, $m_l=-1$, and $m_s=+\frac{1}{2}$. All quantum numbers in this set follow the rules and are valid.

Analyze Set (d)

Here we have $n=5$, $l=0$, $m_l=0$, and $m_s=0$. Principal quantum number (n): $n=5$ which is valid. Angular momentum quantum number (l): $l=0$, as $0\le l\le 4$, which is valid. Magnetic quantum number (m_l): $m_l=0$ which is valid. Spin quantum number (m_s): $m_s=0$, but the allowed values are $-\frac{1}{2}$ or $+\frac{1}{2}$. Therefore, $m_s=0$ is invalid.

Analyze Set (e)

Here we have $n=2$, $l=2$, $m_l=1$, and $m_s=+\frac{1}{2}$. Principal quantum number (n): $n=2$ which is valid. Angular momentum quantum number (l): $l=2$, but the restriction was $0\le l\le 1$. Therefore, $l=2$ is invalid. The result is: (a) Invalid, $m_l$ is not valid. (b) Valid. (c) Valid. (d) Invalid, $m_s$ is not valid. (e) Invalid, $l$ is not valid.

Key concepts

Principal Quantum Number

The principal quantum number, denoted by $n$, is a key player in quantum theory, providing information about the energy level or shell of an electron within an atom. The value of $n$ is always a positive integer: $n=1,2,3,\dots$. These values indicate the electron's distance from the nucleus:

• A smaller $n$ means the electron is closer to the nucleus and has lower energy.
• A larger $n$ implies that the electron is further away and possesses higher energy.

In the context of the exercise, a valid $n$ was found across the various examples provided, signifying the energy levels were accurately chosen according to quantum rules.
Understanding this number helps in determining the size of the electron cloud and is essential for grasping the atom's electronic configuration.

Angular Momentum Quantum Number

Known as the azimuthal quantum number, the angular momentum quantum number is represented as $l$ and defines the electron's subshell. It indicates the shape of the electron's orbital. For a given principal quantum number $n$, the values of $l$ range from $0$ to $n-1$.

• For example, if $n=3$, then $l$ can be $0,1,$ or $2$.
• This value explains the subshells (s, p, d, f) where the electrons are found: $l=0$ for s, $l=1$ for p, $l=2$ for d, and so on.

The exercises showed proper identification except in one where an incorrect $l$ value was noted for the given $n=2$. This error reflects a common misunderstanding of subshell boundaries, which illustrates the importance of mastering this concept.

Magnetic Quantum Number

The magnetic quantum number, shown by $m_l$, gives insight into the orientation of an electron's orbital within a particular subshell. The value of $m_l$ is dependent on $l$: for each $l$, $m_l$ can take any integer value from $-l$ to $+l$.

• For instance, if $l=1$, then $m_l$ can be $-1,0,$ or $1$.

This quantum number is crucial in determining how electrons arrange themselves in space and how they interact with magnetic fields. In the exercise example, there was a case where $m_l$ was improperly set outside the valid range, demonstrating the significance of accurately applying these rules in practical scenarios.

Spin Quantum Number

Lastly, the spin quantum number, $m_s$, is an intrinsic property of electrons, describing their angular momentum as either $+\frac{1}{2}$ or $-\frac{1}{2}$. These values account for the two possible directions of an electron's spin, akin to spinning in opposite directions.

• This quantum number doesn't depend on the previous quantum numbers.
• It ensures that no two electrons in the same atom can have identical sets of quantum numbers, in accordance with the Pauli exclusion principle.

Understanding spin is essential for delving into magnetic behaviors of atoms and electron pairing. In the exercises given, an example displayed incorrect $m_s$ value, reminding us that maintaining accurate spin representations is necessary for correctly describing electron states.