# Sandbox 5

\[ \begin{array} {lrrrr} n & f(n) & D_1(n) & D_2(n) & D_3(n) & \ldots \\ 1 & 4 & -1 & 2 & 0 \\ 2 & 3 & 1 & 2 & 0\\ 3 & 4 & 3 & 2 & 0\\ 4 & 7 & 5 & 2 & 0\\ 5 & 12 & 7 & 2 & 0\\ 6 & 19 & 9 & 2 & 0\\ \vdots \\ \end{array} \]

\[ \begin{array} {lrrrr}
n & f(n) & D_1(n) & D_2(n) & D_3(n) & \ldots \\

1 & 4 & -1 & 2 & 0 \\
2 & 3 & 1 & 2 & 0\\
3 & 4 & 3 & 2 & 0\\
4 & 7 & 5 & 2 & 0\\
5 & 12 & 7 & 2 & 0\\
6 & 19 & 9 & 2 & 0\\
\vdots \\
\end{array} \]

Warm greetings from the moderators, ladies and gentlemen of Brilliant. Here's the July edition of the newsletter, and we hope you'll enjoy reading this.

P.S. Light your fireworks, Americans! Feel the freedom!

#### Contents

## New Features To Be Excited About

Our Logic section is continuously growing, and more Level 4-5 Logic problems are being posted, thanks to Pi Han Goh, Nihar Mahajan, Satyen Nabar, Vignesh Raut, Agnishom Chattopadhyay and many more!

The Brilliant homepage was re-designed to have the new "Problems of the Day." If you have any feedback, please take a look and comment at Sir Lin's note.

You can now effortlessly link a wiki page to your problem or note!

Also, you can find recently posted top solutions on problems. If you're into looking some inspiration on how a very good solution must be written, try to take a peek on it by going to the homepage.

## Challenging Problems

With the thousands of problems being posted on Brilliant each month, the challenging ones prove what you can really do. How large can you make it?, Are you, like, a crazy person?, Calvin hates his Homework, and Moo are just some of the problems that trains our brains.

There are easy problems with a tiny twist. Mind your C's and D's, You're not a cat, stop playing with strings, and Loves comparing! are some problems that you need to read twice, lest being deceived by their words.

## New and Active Members

As time goes on, more and more people join the community, and a few of them are very active in contributing to the community by posting problems, solutions, wikis, notes, and pretty much everything else to the point that we should honor them as hardworking members. Let's give a round of applause to Majed Musleh, Galen Buhain, Arulx Z, Alex Li, Siddharth Bhatnagar, and Robin Plantey.

## Who to Follow

Give these members a big appreciation by clicking the "Follow" button!

## Popular Posts

These posts are on FAYAAAAAAH!

As Brilliant continuously features our fellow members, the second part features Daniel Liu. You can learn some interesting facts about him, you know!

If you're looking for something that will make you brainstorm for a while, try this. Make sure to put an answer and an explanation!

A survey about your favorite mathematician is shared in this community. If you still haven't taken that survey yet, answer it here!

If you want to enjoy your time playing chess with other people online, or a really avid fan of chess, you're invited! See here for more details.

Our fellow member Agnishom has created a tool that can give you laughter or tears of joy... Try it out, and you'll find it fascinating.

So, I guess this ends this month's newsletter. Stay tuned for the next one!

Cheers, The moderation team

## QM Model

## Concept of Atomic Orbital

Main Article: orbitals and quantum numbers

When Heisenberg put forward his Uncertainty principle, which said that, at any one time, it is impossible to calculate both the momentum and the location of an electron in an atom; it is only possible to calculate the probability of finding an electron within a given space.

And thus the Quantum Mechanical Model redefined the way electrons travel, according to this approach, we cannot simply say that the electron exists at a particular point in space. Instead of defining a particular path, it proposed some region in space around the nucleus, called an **orbital**, where the probability of finding an atom is maximum. **Thus the electron doesn't always remain at a definite distance from the nucleus.**

**Energy level and Sub-Energy levels**

The Energy levels classify the Orbitals based of their proximity towards the nucleus, these are represented as \[\text{K }(n=1), \text{L }(n=2), \text{M }(n=3), \text{N }(n=4), \text{O }(n=5), \cdots\] The lowest energy level is K or 1, the next being L or 2 and so on. Thus for an electron the energy level describes the path of the electron and the energy of the electron given by the equation;

\[\begin{align} E_n = - hcR_\infty\dfrac{Z^2}{n^2} &= -\dfrac{2\pi^2mZ^2e^4}{n^2h^2}\text{kJ/mol}\\ &\approx -\dfrac{2.18\times 10^{-19}Z^2}{n^2}\text{J/atom}\\ &\approx -\dfrac{13.6Z^2}{n^2}\text{eV/atom} \end{align}\]

Where \(h\) is the plank's constant, \(c\) is the speed of light, \(R_\infty\) is the Rydberg's constant, \(Z\) is the atomic number of the element and the electron is present in the \(n^{th}\) energy level.

The energy level are sub-divided into sub-shells, which are designated as \(s\), \(p\), \(d\) and \(f\), and the number of sub-shells in each energy level is given by the number itself, for example: the K-energy level has only \(1\) sub-shell \(s\), and the L-energy level has \(2\) sub-shells \(s\), and \(p\).

The sub-shells are precisely defined with the help of quantum numbers, which govern the number of orbitals in each sub-shell. The **Magnetic Quantum Number**\((m_l)\) visualizes the behavior of an electron under the influence of a magnetic field(like earth). We know that the movement of electric charge can generate a Magnetic field, and under the influence of an external magnetic field the electrons tend to orient themselves in certain regions around the nucleus(called orbitals), which is why this quantum number gives the number of orbitals in a particular sub-shell.

The values of the Magnetic Quantum Number depends upon the **Azimuthal Quantum Number**\((l)\), for example if the Azimuthal Quantum Number of an atom is \(l\), then the Magnetic Quantum Numbers range from:

\[m_l= -l,( -l+1), . . . , 0 , . . . (l-1), l\]

So, there are \(2l+1\) values of \(m_l\) for a given value of \(l\), i.e. there will be \(2l+1\) orbitals. According to the Magnetic Quantum Number, the number orbitals in:

\(s\) sub-shell is \(1\), because \(l=0 \ \ \ \therefore m_l = 0\).

\(p\) sub-shell is \(3\) because \(l=1 \ \ \ \therefore m_l = -1, 0, 1\).

\(d\) sub-shell is \(5\) because \(l=2 \ \ \ \therefore m_l = -2, -1, 0, 1, 2\).

\(f\) sub-shell is \(7\) because \(l=3 \ \ \ \therefore m_l = -3, -2, -1, 0, 1, 2, 3\).

**Note:** We see that each sub-shell can have many orbitals, but all orbitals are assumed to have equal energy as their energies differ by very small negligible values.

**Shape of Atomic Orbitals**

The shape of these orbitals are defined by the wave equation, and they get pretty weird as we reach higher orbitals.

## Hund's Rule, Pauli's Exclusion and Afbau's Principle

Format: \(\ce{_8Ne}:\boxed{\uparrow\downarrow}\quad\boxed{\uparrow\downarrow}\boxed{\uparrow\downarrow}\boxed{\uparrow\downarrow}\)