The Four Quantum Numbers

Transcript

0:00Introduction

Hello everyone! Okay, one small puzzle before we start. Your home address has a few parts: state, city, street, house number, right? Put them together, and the postman finds you. An electron in an atom also gets an address, made of four numbers: the four quantum numbers. Now the puzzle. The third shell of an atom: at most, how many electrons can it hold? Not so easy, is it? By the end of this video, you will answer it in two seconds. And look, in JEE, quantum numbers often show up as quick counting questions. So, let's begin.

0:40Shell and subshell: n and l

So, the first number is n, the principal quantum number. n can be one, two, three, and so on; the three dots mean it keeps going, okay? n tells us the shell: its size, and mainly, its energy. Notice the word mainly, in brackets. Clear? Second, small l, the azimuthal quantum number. l starts from zero and goes up to n minus one. It tells us the subshell, and the shape of the orbital. Alright, quick check. If n is three, which values can l take? Zero, one and two. Yes, exactly! The double arrow means gives. And if you also wrote three, that is the classic JEE trap. l never reaches the value n itself; it stops at n minus one. Got it? Now, every value of l has a letter name. l equal to zero is s, one is p, two is d, and three is f, okay? s, p, d, f: write these letters down.

1:44Orientation and spin: ml and ms

Third, the magnetic quantum number. We write it m sub l: m, with a small l written below it. Its values run from minus l, through zero, up to plus l, okay? It tells us the orientation of the orbital: which way it points. Now you try. For l equal to one, what values of m sub l are possible? Minus one, zero and plus one. Exactly! Three values, so three orbitals: the three p orbitals. Same subshell, three different orientations. With me? And the last number: the spin quantum number, m sub s, with a small s written below. Only two values: plus half, or minus half. It tells us the spin of the electron. And these two spin states have no classical analogue: nothing in everyday physics quite matches them.

2:37Counting orbitals

Alright, now the counting rules. This is where the marks are. Shell n has exactly n subshells. Because l takes exactly n values, from zero up to n minus one. Simple, right? Next, each subshell l has two l plus one orbitals, one for each value of m sub l, okay? Count them: l values on the minus side, l values on the plus side, and one zero. l plus l plus one: two l plus one. Clear? And a whole shell n has n squared orbitals; the small two on top means squared, okay? Don't just believe me. Check it for n equal to three. Done? n three gives l zero, one and two. Their orbitals: one, three and five. One plus three plus five is nine, and nine is three squared. Perfect match!

3:32Counting electrons

Now, electrons. Each orbital takes two electrons, so a subshell holds two times, two l plus one, electrons. The brackets mean: double the whole of two l plus one, okay? So tell me. An f subshell has l equal to three. How many electrons can it hold? Fourteen! Yes, exactly. Two times, two times three plus one gives fourteen; the cross means times. And the same way: s two, p six, d ten, f fourteen. Learn this row by heart. And a whole shell? n squared orbitals, times two electrons each: two n squared electrons. Oh, wait! This cracks our puzzle; we finish it in Example two.

4:18Angular momentum

One last pair of formulas. Orbital angular momentum is the square root of, l times l plus one, times h over two pi. The tick-shaped sign is the square root, h is Planck's constant, and pi is this Greek letter. Look, only l sits inside the root, never n, okay? Put n there, and you lose marks. And spin angular momentum has the same form, with s in place of l, and s equal to half. Careful: this s is the spin, not the s subshell, okay? And that's all the theory. Now let's solve.

4:54Example 1: which quantum number sets are allowed

Alright, Example one. Which of these sets of quantum numbers are not allowed? Part a: n three, l three. Part b: n two, l one, m sub l minus one. Part c: n one, l zero, m sub l one. And part d: n four, l two, m sub l minus two, m sub s plus half. Pause the video, and check each one. Done? Okay, let's check. Part a: n three, l three. This is our trap from before. l can go only up to n minus one, that is two. The sign with a line under it means less than or equal to. So l three is not allowed. Part b: n two, so l can be at most one, and l is one; fine. And m sub l minus one lies in minus one to plus one. So, it is allowed. Clear? Now, part c: n one, l zero; so far, so good. But m sub l is one. Is that allowed? Not allowed! When l is zero, the range minus l to plus l shrinks to just zero. So m sub l can only be zero. And part d: n four, so l can go up to three, and two is fine. m sub l minus two lies from minus two to plus two, and plus half is a proper spin value. Allowed. So the answer: part a and part c are not allowed. Really easy, isn't it?

6:32Example 2: counting electrons from quantum numbers

Now a harder one, JEE level. How many electrons in an atom can have: part a, n three; part b, n four and l one; part c, n three and m sub l zero; and part d, n two and m sub s minus half? Pause and try. Hint: count the orbitals first, then the electrons. Done? Okay, let's check. Part a is our puzzle from the start: the third shell! Two n squared: two times three squared, that is eighteen electrons. See, puzzle solved, in two seconds! Part b: n four, l one; write the shell number first, then the letter, so this is the four p subshell. Two times, two times one plus one: six electrons. Part c: n three, m sub l zero. m sub l zero means one orbital, so two electrons, right? No, no, careful! This is the JEE trap. The question fixes n, but it does not fix l, okay? n three has l zero, one and two, and each of these subshells has one orbital with m sub l zero. Three orbitals, two electrons each: six, not two. Got it? Last part: n two, and only spin minus half. How many electrons? Shell two holds two n squared, that is eight electrons. Exactly half of them have spin minus half: four. So the answers are eighteen, six, six and four. See how fast that was?

8:15Quick revision: n, l, ml, ms and the counting rules

So, quick revision. n gives the shell; l, from zero to n minus one, gives the shape; m sub l, from minus l to plus l, gives the orientation; and m sub s is plus half or minus half. Orbitals: two l plus one in a subshell, n squared in a shell. Electrons: two times two l plus one in a subshell, two n squared in a shell. Now go and try the practice problems on this page. You will find them easy, I promise. All the best!