# Question 7dc85

Jan 21, 2017

See explanation.

#### Explanation:

DISCLAIMER This answer is incorrect because it completely disregards the Madelung Rule, which states that the energy of an orbital depends on both the principal quantum number, $n$, and the angular momentum quantum number, $l$.

This answer was left in place to serve as a reminder of that. So remember, the following solution is incorrect.

$\frac{\textcolor{w h i t e}{a a a a a a a a a a a a a a a}}{\textcolor{w h i t e}{a a a a a a a a a a a a a a a}} \textcolor{red}{\downarrow \underline{\text{INCORRECT SOLUTION}} \downarrow} \frac{\textcolor{w h i t e}{a a a a a a a a}}{\textcolor{w h i t e}{a a a a a a a a a}}$

The energy level on which an electron resides in an atom is given by the principal quantum number, $n$, which is the number that starts any set of four quantum numbers.

Now, electrons that are located farther away from the nucleus are higher in energy. In other words, the energy of the electrons increases as you move away from the nucleus.

This implies that the value of $n$ will increase as you move away from the nucleus. In other words, the higher the value of $n$, the higher the energy of the electrons.

The four quantum number sets given to you have $\textcolor{red}{n}$ equal to

• $\textcolor{red}{3} , 2 , 1 , \frac{1}{2} \to$ this electron is located on the third energy level

• $\textcolor{red}{4} , 2 , - 1 , \frac{1}{2} \to$ this electron is located on the fourth energy level

• $\textcolor{red}{4} , 1 , 0 , - \frac{1}{2} \to$ this electron is located on the fourth energy level

• $\textcolor{red}{5} , 2 , 0 , 0 , \frac{1}{2} \to$ this electron is located on the fifth energy level

The highest value for the principal quantum number is

$\textcolor{red}{n = 5}$

which means that the answer is (d).

Jan 21, 2017

$\textsf{\left(b\right)}$

#### Explanation:

I am assuming that the numbers refer to the four quantum numbers that describe an electron in an atom.

$n$ is the principal quantum number and tells you the number of the energy level. It takes integral values 1, 2, 3, .....etc

$l$ is the angular momentum number and tells you the sub - shell or orbital the electron is in. It can take integral values of 0 up to $\left(n - 1\right)$.

$m$ is the magnetic quantum number and tells you how the orbitals are aligned in space. It takes values of $- l$ ......0.......$+ l$

$s$ is the spin quantum number and takes values of $\pm \frac{1}{2}$.

The Pauli Exclusion Principle tell us that no electron in an atom can have all four quantum numbers the same.

The higher the value of $n$, the greater the energy of the electron.

For non - hydrogen like atoms the sub - shells are not degenerate (equal in energy). Within a particular energy level $s$ orbitals are lower in energy than $p$ orbitals which are lower in energy than $d$ orbitals.

This means that the energy of a particular electron depends on both $n$ and $l$.

A rule has been put forward which states that the energy of an electron depends on the value of $\left(n + l\right)$. This is also known as "The Madelung Rule".

If it turns out that 2 electrons have the same value of $\left(n + l\right)$ then the electron with the higher value of $n$ is said to have the higher energy.

Lets apply this to the question:

$\left(a\right)$
$n = 3$, $l = 2$, $m = 1$, $s = \frac{1}{2}$

$\therefore$ $\left(n + l\right) = 3 + 2 = \textcolor{red}{5}$

$\left(b\right)$

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

$\therefore$$\left(n + l\right) = 4 + 2 = \textcolor{red}{6}$

$\left(c\right)$

$n = 4$, $l = 1$, $m = 0$, $s = - \frac{1}{2}$

$\therefore$$\left(n + l\right) = \left(4 + 1\right) = \textcolor{red}{5}$

$\left(d\right)$

$n = 5$, $l = 0$, $m = 0$, $s = \frac{1}{2}$

$\therefore$$\left(n + l\right) = 5 + 0 = \textcolor{red}{5}$

You can see that (b) has the highest value of $\left(n + l\right) = 6$ so is highest in energy.

HEALTH WARNING

This rule is used to describe the order which sub - shells fill in an atom since we assume that they fill in order of increasing energy.

I need to stress that this is a rule and not law . A law is universally applicable but a rule may have exceptions and this is the case here.

The rule gives diagrams which you may have seen in text books like this:

Example (b) is a 4d electron. Example (d) is a 5s electron and it shows the 5s filling first as we have shown.

You may also have seen energy level diagrams like this which arise from this rule:

The problem here is that for higher values of $n$ the energy levels converge and become closer and closer in energy.

This means that the complex interactions which take place between the electrons starts to become significant, giving rise to many anomalies.

The implication is that this diagram is common to all atoms. They have different numbers of electrons with different shielding and interactive effects so this is not the case.

These effects are discussed more fully in this answer by @Truong-Son N:

https://socratic.org/questions/why-is-the-electron-configuration-of-chromium-1s-2-2s-2-2p-6-3s-2-3p-6-3d-5-4s-1295560

For example, at some point after calcium the 3d falls below the 4s. This means that for the 1st transition series the outer electrons are the 4s and not the 3d

In this example in this question the electron structure of yttritium is $\left[K r\right] 4 {d}^{1} 5 {s}^{2}$, the $5 s$ being higher in energy.

This rule, and the accompanying diagrams can only really be used with confidence up to $z = 20$

If I want the electron configuration of an element beyond this I use a reliable text such as "Cotton and Wilkinson" or a trustworthy site such as "Webelements".