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In this video, we will learn about the quantum numbers that are used to describe an electron in an atom or ion.
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Firstly, letβs define what a quantum number is.
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The word sounds quite strange to begin with.
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On the surface, a quantum number is simply any number in a series of numbers that are all defined by the same rule.
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For instance, the positive integers are all round counting numbers, like one, two, three, four, and so on.
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But the numbers in between, like 2.5 and π, are not positive integers, so theyβre not quantum numbers within this rule.
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We can also think of quantum numbers as quantized numbers.
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There is an infinite number of numbers between zero and 10.
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But we could quantize the range by moving in increments of two, having zero, two, four, six, eight, and 10 only.
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Many things are quantized like shoe sizes.
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We could get a 9, 9 and half, or 10, but itβll be very hard to get a 9.2 shoe.
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Now, how do quantum numbers relate to electrons?
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Electrons behave in a way that is very complex.
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A lot of maths is required to model how an electron behaves.
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It would take many years to discuss all the maths that goes into quantum physics.
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However, there are a few simple parts that we can break down together.
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Imagine a row of hotels on a street where each hotel is unique.
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You can think of quantum numbers like the door number on the street, the room number in the hotel, and so on.
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Of course, the number doesnβt tell you every detail of the hotel, but you do need to understand the meaning of the number to know where to go.
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Itβs the same with quantum numbers for electrons.
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Each quantum number is a part of a much more complicated equation.
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If we know the meaning of the quantum number, we can use it to determine the shell, subshell, orbital, and spin of an electron.
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Letβs imagine we have an atom of helium with two electrons in the 1s subshell.
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On the surface, we canβt distinguish between the two electrons in the 1s subshell.
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However, on closer inspection, they are slightly different.
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Electrons have a property called spin.
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The spin of an electron is either up or down.
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Itβs beyond the scope of this video to explain exactly what spin is.
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But you can remember that a spin up electron is represented by an upward pointing arrow and a spin down electron is represented by a downward pointing arrow.
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Itβs convention to use a fishhook arrowhead when representing electrons.
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So, letβs return to our helium atom.
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We can represent the orbital in the 1s subshell with a line or with a box.
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By convention, the first electron in the orbital is spin up and the second one is spin down.
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The electrons in the 1s orbital can now be distinguished.
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Theyβre in the same shell, same subshell, and the same orbital, but one is spin up and the other is spin down.
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That solves the problem of describing electrons in the same orbital.
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But what about electrons in different orbitals but in the same subshell?
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Here is the electron configuration for an atom of neon, and there are six electrons in the 2p subshell.
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How do we go about labeling these electrons uniquely?
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To start with, the 2p subshell consists of three separate orbitals.
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Theyβre called the 2p π₯, 2p π¦, and the 2p π§ orbitals.
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So, we can describe each electron in a unique way.
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For instance, an electron in the 2p subshell could be in the 2p π₯ orbital and be spin up.
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The other pieces of information we need are already given to us.
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The two in 2p indicates weβre looking at the second electron shell, and the p tells us weβre in a p-type subshell.
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Therefore, the full address of this electron is spin-up, 2p π₯ orbital, 2p subshell, second electron shell.
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This is a great format for a chemist.
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We talk about this type of thing a lot, but itβs hard for a physicist or mathematician to plug these into an equation.
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This is where quantum numbers come in.
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The first quantum number we need, which is equivalent of the number of the hotel, is called the principal quantum number and has the symbol π.
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The principal quantum number describes which electron shell the electron is in.
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The rule for the principal quantum numbers is they are all positive integers.
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Just like addresses, the lowest principal quantum number is number one.
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After that, we count up using just the integers.
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The second quantum number is called the subsidiary quantum number, and it has the symbol π.
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Itβs also known as the as azimuthal quantum number, the orbital angular momentum quantum number, or the subshell quantum number.
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Itβs the subsidiary quantum number that tells us which subshell an electron is in.
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The rule for π actually depends on which subshell the electron is in.
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So, it depends on the value of π.
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For the first electron shell, where π is one, the value of π can only be zero.
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For the second electron shell, π can be either zero or one.
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And from the third electron shell, we start to see a pattern.
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π can be zero, one, or two.
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If we extend this to any value of the principal quantum number π, the values of π we can have are zero, one, two, and so on until we reach the value π minus one.
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Each value of π corresponds to a certain type of subshell.
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A value of zero for the subsidiary quantum number indicates an s-type subshell.
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One corresponds to a p-type subshell, two to a d-type subshell, and three to an f-type subshell.
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We could go higher, but itβs not that relevant.
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The letters s, p, d, and f correspond to features of a spectrum that correspond to these subshells.
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The descriptions are sharp, principal, diffuse, and fundamental.
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Itβs not really important to know what s, p, d, and f mean, but it is interesting.
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We can see from the general rule that the higher the value of π, the more types of subshell theyβll be in that electron shell.
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We can see from the pattern that the number of types of subshell in an electron shell is equal to π.
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The first electron shell has one type of subshell, the s shell, and the third electron shell has three types of subshell: an s, a p, and a d.
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Weβll look in detail about the nature of these subshells when we get to the magnetic quantum number.
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The third quantum number is called the magnetic quantum number, and it has the symbol π subscript π.
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You could call it the orbital quantum number since it identifies the orbital our electron is in.
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The potential values of the magnetic quantum number depend on the value of π, the subsidiary quantum number, which gives us the subshell.
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If π is zero, then π π can be only zero.
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If π is one, π π can be negative one, zero, or positive one.
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If π is two, then π π can be negative two, negative one, zero, positive one, or positive two.
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So, our general rule for the value of π π, the magnetic quantum number, is that itβs equal to zero, plus or minus one, plus or minus two all the way up to plus or minus π.
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For each magnetic quantum number we throw into our formulas, weβd get an orbital.
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Unlike with π, itβs harder to connect magnetic quantum numbers to specific features of specific orbitals.
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But we can say for a p-type subshell where the value of the subsidiary quantum number is one, there are three different orbitals.
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For a d-type subshell, there are five orbitals.
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And for a generic value of π, the number of orbitals we get is two π plus one.
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The important thing is to remember the general formula so we can apply it to any value of π.
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Now weβve got down to the orbital level, the last thing we need to account for is spin.
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The fourth and final quantum number relative to electrons is called the spin quantum number, and it has the symbol π subscript π .
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This quantum number identifies whether the electron is spin up or spin down.
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There are only two potential values for the spin quantum number in this context, positive half or negative half.
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This allows for the fact that when we see an orbital, the maximum number electrons we can see in that orbital is two.
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Thatβs four quantum numbers that uniquely identify an electron in an atom or ion.
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Letβs do a quick run through of more.
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At the top, we have the principal quantum number, the shell number.
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For the first electron shell, thereβs only one value of the subsidiary quantum number allowed and thatβs zero.
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And for the first electron shell and the s-type subshell, we can only have one value of the magnetic quantum number and that value is zero.
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And finally, the value of the spin quantum number can either be negative a half or positive a half.
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This allows for two electrons, one spin up and one spin down, in one orbital in the single s-type subshell of the first electron shell, which gives us two electrons in total in the first electron shell.
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Letβs move on to the second shell.
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The values of π allowable for the second shell are zero and one since weβre counting up to π minus one.
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Just like in the first shell, when π equals zero, π π can only equal zero.
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But for π equals one, π π can equal negative one, zero, or positive one.
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And, as always, π π is either equal to positive a half or negative a half.
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So, the s-type subshell in the second electron shell contains a maximum of two electrons, and the p-type subshell contains a maximum of six electrons.
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Altogether, the second electron shell can contain a maximum of eight electrons.
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Letβs do one more, the third electron shell.
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For π equals three, we can have π equal to zero, one, or two.
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As the value of the subsidiary quantum number π increases, the number of values we can get for the magnetic quantum number π π increases as well.
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But the spin quantum number is reliable as always.
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So, the s-type subshell in the third electron shell can contain a maximum of two electrons.
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The p-type subshell can contain a maximum of six electrons.
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And the d-type subshell can contain a maximum of 10, giving us a grand total of 18 electrons for the maximum occupancy of the third electron shell.
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The last thing weβre going to study is how to convert between quantum numbers and the notation we use too.
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This is the electron configuration of a fluorine atom.
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We can imagine that the electrons occupy the two p orbitals 2p π₯, 2p π¦, and 2p π§ according to this pattern.
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And weβre going to figure out the quantum numbers that describe this electron.
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We could do it in any order, but Iβm going to do it π, π, π π, π π .
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We can find the value of π by looking at the number in the subshell notation.
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2p means that π is equal to two.
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Weβre in the second shell.
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What about π?
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In order to work out π, we need to convert p back to π notation. p corresponds to a value of the subsidiary quantum number of one.
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Now the magnetic quantum number.
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Determining the magnetic quantum number from the orbital itself is a lot more complicated.
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It just so happens that the 2p π§ orbital corresponds with a magnetic quantum number of zero.
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And lastly, we have the spin quantum number π π .
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The electron is spin up.
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Therefore, the spin quantum number for that electron is positive half.
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Normally, you wouldnβt need to know beyond π and π, but itβs interesting to know the others as well.
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So, itβs about time we had some practice.
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How many electrons in total can have the quantum numbers π equals two and π equals one?
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Quantum numbers are numbers we assign to electrons to describe where they are, shell, subshell, and so on.
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π is the symbol given to the principal quantum number.
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If π equals one, weβre looking at the first electron shell.
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If it equals two, weβre looking at the second.
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In this example, π equals two.
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So, weβre looking at the second electron shell.
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π is the symbol used for the subsidiary quantum number, also known as the orbital angular momentum quantum number or the subshell number.
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Each value of π corresponds with a type of subshell, like an s-type or a p-type subshell.
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With π equal to one, weβre dealing with a p-type subshell.
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The question is asking us, how many electrons in total can have these particular quantum numbers?
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The key here is that if weβre looking at the same atom or ion, no two electrons can have exactly the same set of quantum numbers.
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So, is the answer one, since we have quantum numbers π and π fixed at values of two and one, respectively?
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Well, no, there are two other quantum numbers that can uniquely define an electron.
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Thereβs another quantum number that defines the orbital the electron is in, and thatβs known as the magnetic quantum number.
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The possible values of the magnetic quantum number are zero, plus or minus one, plus or minus two until we reach plus or minus π, which for a value of one would give the values of π π of negative one, zero, and one.
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This means we have three orbitals in total.
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The fourth quantum number we need is the spin quantum number π π , which can either be positive a half or negative a half.
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This gives us two possible spin states.
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We can now calculate the number of possible sets of quantum numbers and therefore the number of electrons we can possibly have.
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There are two possible values for the spin quantum number, three values for the magnetic quantum numbers, so thatβs two electrons per orbital, giving us six electrons in total.
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You could have done this question simply by remembering that a p-type subshell contains a maximum of six electrons.
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But this way, weβve proved it.
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Letβs finish off with the key points.
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There are four quantum numbers that uniquely describe electrons on the same atom or ion.
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The first, the principal quantum number, has the symbol π, and it describes the shell.
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The principal quantum number can take the value of any positive integer, starting with the number one.
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Next is the subsidiary quantum number, also known as the orbital angular momentum quantum number, which uniquely identifies the subshell.
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π can take a value of zero, one, two up until π minus one. s subshells match up with the value of π of zero, p one, d two, and f three.
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Meanwhile, magnetic quantum numbers of a subshell correspond roughly with the orbitals.
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π π can take any integer value between negative π and positive π.
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And lastly, we have the spin quantum number π π .
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The value of the spin quantum number can either be positive a half or negative a half corresponding to spin up or spin down.
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Together, these four quantum numbers allow us to efficiently describe the state of an electron.