# Which set(s) of of quantum numbers describe a 4d orbital?

Jun 28, 2016

A total of $10$ sets of quantum numbers can be used here.

#### Explanation:

As you know, we use four quantum numbers to describe the position and spin of an electron in an atom. Each electron has its unique set of quantum numbers, which means that two electrons can share one, two, or even three quantum numbers, but never all four.

Now, you are given a $\textcolor{red}{4} d$ orbital and asked to find how many sets of quantum numbers can describe an electron located in such an orbital, or, in other words, how many electrons can occupy a $\textcolor{red}{4} d$ orbital.

So, the principal quantum number, $n$, describes the energy level on which the electron is located. In this case, you have

$n = \textcolor{red}{4} \to$ the electron is located on the fourth energy level

The subshell in which the electron is located is described by the angular magnetic quantum number, $l$, which for the fourth energy level takes the following values

• $l = 0 \to$ the s-subshell
• $l = 1 \to$ the p-subshell
• $l = 2 \to$ the d-subshell
• $l = 3 \to$ the f-subshell

Since you're looking for the d-subshell, you will need $l = 2$.

The specific orbital in which the electron is located is given by the magnetic quantum number, ${m}_{l}$. For any d-subshell, the magnetic quantum number can take the values

${m}_{l} = \left\{- 2 , - 1 , \textcolor{w h i t e}{-} 0 , + 1 , + 2\right\}$

Each of these five values describes one of the five d-orbitals available in a d-subshell. Finally ,the spin quantum number, ${m}_{s}$, can only take two values, $- \frac{1}{2}$ for an electron that has spin-down and $+ \frac{1}{2}$ for an electron that has spin-up.

Now, since each orbital can hold a maximum of two electrons, one with spin-up and one with spin-down, it follows that the d-obitals can hold a total of

${\text{2 e"^(-)"/ orbital" xx "5 orbitals" = "10 e}}^{-}$

Each of these ten electrons will have its unique set of four quantum numbers.

• all the ten electrons will share the principal and angular momentum quantum numbers

$n = \textcolor{red}{4} \text{ }$ and $\text{ } l = 2$

• five electrons will share the spin quantum number

${m}_{s} = - \frac{1}{2} \text{ }$ or $\text{ } {m}_{s} = + \frac{1}{2}$

• two electrons will share the magnetic quantum number

${m}_{l} = - 2 \text{ }$ or $\text{ "m_l = -1" }$ or $\text{ "m_l = color(white)(-)0" }$ or $\text{ "m_l = +1" }$ or $\text{ } {m}_{l} = + 2$

You will thus have $10$ sets of quantum numbers that can be used to describe an electron located in one of the five d-orbitals

$n = \textcolor{red}{4} , l = 2 , {m}_{l} = - 2 , {m}_{s} = + \frac{1}{2}$

$n = \textcolor{red}{4} , l = 2 , {m}_{l} = - 2 , {m}_{s} = - \frac{1}{2}$

$n = \textcolor{red}{4} , l = 2 , {m}_{l} = - 1 , {m}_{s} = + \frac{1}{2}$

$n = \textcolor{red}{4} , l = 2 , {m}_{l} = - 1 , {m}_{s} = - \frac{1}{2}$

$n = \textcolor{red}{4} , l = 2 , {m}_{l} = \textcolor{w h i t e}{-} 0 , {m}_{s} = + \frac{1}{2}$

$n = \textcolor{red}{4} , l = 2 , {m}_{l} = \textcolor{w h i t e}{-} 0 , {m}_{s} = - \frac{1}{2}$

$n = \textcolor{red}{4} , l = 2 , {m}_{l} = + 1 , {m}_{s} = + \frac{1}{2}$

$n = \textcolor{red}{4} , l = 2 , {m}_{l} = + 1 , {m}_{s} = - \frac{1}{2}$

$n = \textcolor{red}{4} , l = 2 , {m}_{l} = + 2 , {m}_{s} = + \frac{1}{2}$

$n = \textcolor{red}{4} , l = 2 , {m}_{l} = + 2 , {m}_{s} = - \frac{1}{2}$