Hi everybody, my name is Iman. Welcome back to my YouTube channel. Today we're going to be continuing our lecture on the electronic structure of atoms and we've left off on Objective 5, which is titled the quantum mechanical model of an atom.
Now to begin and motivate this, we're going to continue discussing Bohr's model. Now at first, Bohr's model appeared to be very promising. The energy levels calculated by Bohr closely agreed with the values that were obtained from the hydrogen emission spectrum.
However, when Bohr's model was applied to atoms other than hydrogen, it did not work at all. And there were some attempts that were made to adapt the model, but it was concluded that Bohr's model is fundamentally incorrect. And now you're probably like, whoa, you monchil girl, what are you talking about?
I thought we liked Bohr, now we're dissing him. And yeah, maybe a little bit. But listen, while Bohr's model marked a significant advancement, in the understanding of the structure of atoms, his model ultimately proved to be inadequate to explain the structure and behavior of atoms that had more than one electron. Now these limitations, they were ultimately addressed by the development of quantum mechanics, which provided a more comprehensive and broadly applicable framework for understanding atomic and subatomic systems. The wave mechanical model of the atom developed by Schrodinger and Heisenberg, among others, very much superseded Bohr's model by treating electrons as wave-like entities with probabilistic distributions around the nucleus rather than particles in fixed orbit.
Now the most important difference between Bohr's model and the modern quantum mechanical model is that Bohr postulated that electrons follow a clearly defined circular path or orbit. at a fixed distance from the nucleus, whereas modern quantum mechanics has shown that this is not the case. Rather, we understand that electrons move rapidly and they're localized within regions of space around the nucleus called orbitals.
Orbitals, not orbits. An orbital is a region of space around the nucleus that's defined by the probability of finding an electron in that region. Now, quantum mechanics is heavily based in mathematics, and we really don't have to worry about the math, but it will help us understand where we get our description of atomic orbitals from. So I'm just going to briefly work you through the idea, all right? While we don't need to know any of the math in general chemistry, I just want to talk a little bit about where we get atomic orbitals.
So in quantum mechanics, we have a wave equation. equation. A wave equation describes the behavior of electrons and particles.
Then we have wave functions. Wave functions are the solutions to the wave equation. We can get information about allowed energy from this. Now, if we take the absolute square of the wave function, this gives us the probability of finding an electron in a location.
We said quantum mechanics is probabilistic in nature, so we can only get a probability of finding an electron. But in the 3D plot of that probability, which is the square of the wave function, that is going to generate an image of our atomic orbitals. That is where our atomic orbitals come from. Now, in the current quantum mechanical model, we have orbitals.
Because it's impossible to pinpoint exactly where an electron is at any given moment in time. And this is best expressed by the Heisenberg uncertainty principle. It says that it is impossible to simultaneously determine with perfect accuracy the momentum and the position of an electron. If we want to assess the position of an electron, the electron has to stop.
And when it does that, we remove its momentum. If we want to assess its momentum, then the electron has to be moving, which means it's changing its position. And so this is the expression that demonstrates and expresses the sentiment of the Heisenberg uncertainty principle.
Main takeaway is that it is impossible to simultaneously determine with perfect accuracy the momentum and the position of an electron. And that is all that this expression is saying. Now, to better understand the distribution of electrons in an atom, quantum numbers were developed.
These numbers describe the energy, the shape, and the orientation of an orbital, allowing us to predict the electron configuration of an atom and its chemical behavior. Now, modern atomic theory postulates that any electron in an atom can be completely described by the four quantum numbers. And to properly understand this concept, we're going to have to learn a couple of things in a specific order. So here's our workflow, guys.
First, we're going to actually need to know about Aufbau's principle, Pauli exclusion principle, and Hund's rule. Once we define those, we're going to keep them in the back of our mind, and then we're going to talk about the four quantum numbers. After we talk about the four quantum numbers, then we're going to begin to visualize the differences between shells, subshells, and orbitals. Then we're going to connect all those topics together, and we'll make the most sense of this information by talking about electron configuration. Now, all these topics are extremely interrelated, so let's just get started.
Let's start talking about these things, and then we'll connect them as we go. Let's start with Aufbau's principle. This says that the lowest energy orbital is filled first. Then we have the Pauli exclusion principle. This says that each orbital can accommodate a maximum of two electrons that have opposite spins.
And then last, we have Hund's rule, which says one electron is placed in each degenerate orbital first. before electrons are paired up. So we're going to keep these definitions in the back of our mind, we're going to come back to them a little bit later. But what we want to do now is move into our sixth objective and discuss the four quantum numbers, principle, angular momentum, magnetic, and spin.
Let's start with the principle quantum number. This defines the size and energy level of the orbital where an electron is likely to be found. It's denoted as lowercase n.
And it can take on any positive integer value. The larger the value of n, the higher the energy level and the larger the orbital. Within each shell, there is a capacity to hold a certain number of electrons, and that is given by the equation 2n squared. This will make a little more sense later. Then we have our angular momentum quantum number denoted by L, and this defines the shape of the orbital, and it can take on any integer value from 0 to n-1.
Each value of L corresponds to a specific subshell dictating the shape of the orbital. When L equals 0, we're talking about a subshell that's going to be called S. When L equals 1, this gives us the P subshell. When L equals 2, this is the D subshell. And when L equals 3, that gives us the F subshell.
Next, we have the magnetic quantum number, ML. This defines the orientation of the orbital in space, and it can take on integer values anywhere between minus L to positive L, including 0. This quantum number arises due to the orientation of orbitals within a magnetic field. Then last, but certainly not least, we have the spin quantum number, denoted ms, and this only has two possible values, plus one half or minus one half, and this reflects the two possible orientations of an electron spin.
We're not going to get into electron spin, because that is a very complicated topic, and we don't need to know that. All we need to know is that there's two possible values for the spin quantum number, plus one half and minus one half. Now we're going to take our first pause here. All right, we've defined some terms, off-bouw, poly, hand, and now we've gone over quantum numbers.
What's the connection here? The connection is that the arrangement of electrons within these orbitals is governed by the fundamental principles we covered. Now, this next topic is really going to help us frame these ideas together even better. That topic is what are shells?
subshells, and orbitals. Let's first start with shells. Shells are the primary energy levels of an atom and they're defined by the principal quantum number n. This figure here shows two such shells.
We have the innermost shell, n equals 1, which is the closest to the nucleus and it has the lowest energy. And then we also have the next shell, n equals 2, which is at a higher energy level and farther from the nucleus. Each shell can hold a larger number of electrons as n increases, accommodating them within subshells and orbitals that exist at that energy level.
That helps us transition into subshells. Now, subshells are categorized by the angular momentum quantum number L, and they specify the shape of the space where electrons are likely to be found. So for our shell... n equals 1. If n equals 1, l can only have one possible value, and that's 0, because remember, l is equal to 0 all the way up to n minus 1. But for n minus 1, the only possible value here is going to be 0. Now, l equals 0 refers to the s subshell. Now, if we look at the n equals 2 shell, when n equals 2, L can be 0 or 1, where 0 refers to our S subshell and 1 refers to our P subshell.
And we see both of those subshells here, all right, our S, 2S, and our 2P subshells. Next, we have orbitals. Orbitals are the individual regions within a subshell where electrons are most likely to be found.
All right, and it's actually kind of shown here by the different... shapes of the P subshell in the n equals 2 shell. Each orbital can hold up to two electrons with opposite spins.
So let's go back to the start. If we have an n equals 1 shell, we only have one kind of subshell, that's the S subshell. The S subshell is just circular, it's spherical, indicating an equal probability of finding an electron at any point around the nucleus. All right.
The 1s orbital then can fit just two electrons of opposite spin. Then we have our n equals 2 shell, which has s and p subshells. The s subshell, it can hold, it's also spherical, it can only hold two electrons of opposite spin. So in the 2s orbital, we have two electrons. Then we have our p. p subshell and again like we said there are three orbitals in the p subshell we have 2px 2py 2pz this refers to the different orientations in space the 2p orbitals are dumbbell shaped and they're oriented along the x y and z axis showing that electrons at this energy level have a higher probability of being found in certain directions relative to the nucleus you And each of these p orbitals, px, py, pz, can fit two electrons in opposite directions.
With that, here's our second pause. Let's go back to the page with our quantum numbers and start working through some examples and start visualizing things a little bit better. So we're going to erase so we can make some space. And then let's talk through these quantum numbers again and do an example or examples.
All right, so our principal quantum number, again, this describes the energy level of an electron and the size of the orbital. The larger the value of n, the higher the energy level and the larger the orbital size. And it can take on any positive integer value. Then our angular momentum quantum number describes the shape.
and the number of subshells within a given principal energy level or shell. It has important implications for chemical bonding and bond angles, and for any given value of n, the range of possible values for L is 0 to n-1. The subshells, remember, are designated by the letters SPDF for L values 0, 1, 2, 3, respectively.
Then we have our quantum number, our magnetic quantum number. This describes the orientation of an orbital in three-dimensional space. And the possible values of ML are the integers between minus L and positive L, including zero. Now, the number of orbitals in a subshell is going to actually be equal to 2L plus 1. Alright, then last we have our spin quantum number.
This describes the spin of an electron in an orbital, and they can have values of plus one half or minus one half. Now, whenever two electrons are in the same orbital, they have to have opposite spins. Alright, so that's an important thing to remember. Now let's go through different shells. Alright, let's pretend that n is equal to one.
If n is equal to one, then l can only be zero. And if L is equal to 0, we're referencing the S subshell. All right? The S subshell. Here then, ML is anywhere from minus L2 plus L, but we just have 0 here.
So for ML, this is also going to be equal to 0. And that makes sense because what does ML tell us? What does magnetic quantum number tells us? It tells us the orbital orientation.
But we know that... when L equals zero, we have the S subshell. This is a spherical subshell. There's really only one orientation for a S orbital anyway, AKA one orbital.
All right. It's just a sphere. And then MS refers to spin, and that can be either plus one half or minus one half.
Now if we're in the N equals one shell, all right, our subshell can only be the S subshell, and there's only one orientation for that subshell. And in that orbital, all right, that 1s orbital, we can house two electrons and they have to have opposite spins, one with a plus one half value, one with a minus one half value. Okay, that was simple and easy for the n equals one.
Let's continue. Let's do n equals two and make sure that we really understand this, okay? n equals two. When that's the case, l can be zero or it can be one.
Zero refers to the s subshell. One refers to the p. subshell. Then our values for ML, they can be anywhere from minus L to positive L, which means they could be minus one, zero, and positive one.
Now we just discussed that for an S orbital, there is only one orientation, all right, because it's spherical. And that means that we can house in this S orbital, just two electrons of opposite spin. But for our P orbitals, Our p orbitals can have three different orientations, x, y, and z. And this minus 1, 0, and plus 1 refers to the three orbitals we can fill in the p subshell.
Each of these orbitals houses two electrons of opposite spins. Wonderful. All right, and then ms again, plus 1 half, minus 1 half.
Those are the possible values. Okay, let's get just a little bit more complicated. All right, just a little bit more complicated. N equals 3. If N equals 3, then L can be 0, 1, and 2. 0 is for the S subshell, 1 is for the P subshell, and 2 is for the D subshell.
Now, for our values of ML, it can be anywhere from minus L to positive L. So we can have values of minus 2, minus 1, 0, plus 1, and plus 2. two. Now, again, we have to use our logic here, right? So for our S subshell, it's spherical.
There's only one orientation and this orbital can hold two electrons. We just talked about our P orbital. It has three possible orientations and each orbital can fit two electrons.
All right. So we have these three orbitals that house two electrons. Each our D orbitals are a little more complicated.
Our d orbitals can have five different orientations. Each orbital can house two electrons. That is where those ML values of minus two, minus one, zero, plus one, plus two come into play in describing the d orbital and the possible orientations for the d subshell. There are five orientations, or in other words, five orbitals.
In short, We have our four quantum numbers, all right, and our principal quantum number gives us information about our shells and energy levels. Our angular momentum quantum number tells us information about our subshells and their shapes, and then our magnetic quantum number tells us about the orbitals and the electron probability regions. Now when we get into electron configuration, which is the next step here, all right, We're going to use these quantum numbers to predict an atom's electron configuration, which is going to tell us the number of electrons in each orbital and the number of valence electrons.
And in our discussion of electron configuration, what's really going to be important is our reference back to the three principles we talked about. Aufbau's principle, Pauli exclusion principle, and Hund's rule. For a given atom or ion, the pattern by which subshells are filled, as well as the number of electrons within each principal energy level and subshell, are designated by its electron configuration. Electron configurations use spectroscopic notation, where the first number denotes the principal energy level, the letter designates the subshell, and then the subscript gives us the number of electrons in that. subshell.
Now to write out the atom's electron configuration, we need to know the order in which subshells are filled. Electrons, they fill from lower to higher energy subshells according to Aufbau's principle, and each subshell will fill completely before electrons begin to enter the next one. Now here we see a list of the ordering from lower to higher energy subshells.
We have 1s, then 2s, then 2p, 3s followed by 3p, then 4s, then 3d, then 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s, so on and so forth. Now, this is a lot to memorize, but an easier way to approach electron configuration is through simply reading the periodic table. So what I'm going to do is I'm going to scroll down here, all right?
And we saw an image earlier about which blocks refer to which subshells. So this right here is our S shell. Here is where our P subshells. Here are our D subshells.
And these refer to our F subshells. I'm going to go back to it just to show it to you again, all right? Right here are... S block, P block, D block, and F block. And if we just remember that, then we can read electron configuration just by looking at our periodic table.
If we're interested in an element that is like right here, then what we can do to write its electron configuration is read the periodic table like a sentence. We go through one S and we fill both of those. So one S2, because there's two of them. Then the next thing we fill is the... 2s subshell and we fill both of them here so 2s2 all right and remember in an s subshell you can only fit in two electrons then we make it to the p block all right and here we pass through one two three four five six elements and that is the most electrons that you can fit in a p subshell so we write 2p6 then this is followed by the 3s subshell 3s2 and then we make it to this element right here this mystery element, all right?
And that is the p block, the 3p, all right? But we're only filling in two of these, so 3p2. And just like that, we can read off of the periodic table the electron configuration.
So this is the electron configuration for this unknown element right here. Now, keeping that in mind, let's just go over... some tips and tricks to keep in mind to write an electron configuration.
First, you want to start with the element and then add electrons one by one following the subshell order that's indicated by that periodic table. Remember, use the off-bou principle to determine the subshell filling order 1s, 2s, 2p, then 3s, then 3p, then 4s, then 3d, and so on and so forth. Make sure to apply the Pauli exclusion principle.
to ensure that each orbital gets at most two electrons with opposite spins, and then follow Hund's rule by filling all orbitals in a subshell singly before pairing electrons in one orbital. Now with that, let's do a couple of example problems to make sense of that workflow, okay? So this first problem says, which will fill first, the 5D subshell or the 6S?
sub shell. So we can easily look at our ordering here, or we can also just work through it mentally. So we can look at our 5D, that's right here. All right. And then our 6s is right here.
All right. 6s comes before we make it to 5d. All right.
So 6s subshell has lower energy and it will fill first. Now we can also work through it like this. So for the 5d orbital, our n is going to be equal to 5 and l is going to be equal to 2 because that's what refers to the d subshell.
Now what we can do here. is we can add the principal quantum number and we can add the angular momentum quantum number together, n plus l, and this gives us a value of 7. Okay, keep that in mind. We're going to talk about this little trick right here. Then we can do that for 6s. The n here is equal to 6, and this is the s subshell, so l is going to be equal to 0. Let's add up the principal quantum number and the angular momentum quantum number, this gives us a value of six.
All right. Six is less than seven. So this is a lower energy shell and it will be filled up first.
So the six S subshell has lower energy and it will fill first. So this is a quick trick that you can do to figure that out. If you forget the ordering, figure out the quantum number, the angular momentum, quantum number, sum them up. And the one with a smaller value is going to fill first because it is lower energy.
Okay, let's do some more problems. This one is really fun. This says, what is the electron configuration for nitrogen? And then according to Hund's rule, what is the orbital diagram?
Okay, so let's find where nitrogen is. Nitrogen is right here. All right, I'm going to point to it in black. And we're going to write the electron configuration for nitrogen.
We're just going to read it off of the periodic table. All right. We start off with 1s, 1 and 2. We fill up the 1s orbital with the maximum two electrons. Then we make it through the 2s part and we fill both electrons in that orbital as well.
Now we've made it to the p block, the 2p block, and we make it through 1, 2. three elements before we get to nitrogen. So this is going to be 2p3. That is the electron configuration for nitrogen. Wonderful.
Now what we want to do is we want to draw the orbital diagram for nitrogen according to Hund's rule. So what we need here is we need to remember that nitrogen has atomic number of seven and its electron configuration we just figured out is 1s2 2s2 2p3. According to Hund's rule, the 2s orbitals are filled completely. So if we draw a 1s2 orbital, this has both electrons in it.
And if we draw the 2s2 orbital, this has also both electrons in it. Now we can draw the 2p orbital. This has three orbitals, each of which can have two electrons.
This is where we have to really remember Hund's rule, which says that you will fill each orbital with one electron before you double up. All right? So this is our 2p orbital. We have three electrons that we need to fit in here.
We're going to put them in each separate orbital. We will not draw them like this. This breaks Hund's rule. We have to draw each electron as a single electron in each orbital before we pair up any electrons. So...
This is the electron configuration for nitrogen, and this is how we would draw the orbital diagram for nitrogen following Hund's rule. Now there's one more thing I want to talk about before we end the chapter. In the realm of electron configurations, there are a few exceptions to the predicted order of filling orbitals, particularly within the transition metals here.
These exceptions arise because of subtle differences, subtle energy differences between orbitals which can lead to more stable arrangements when orbitals are either half-filled or fully filled. Now there are two key exceptions that we should know, chromium and copper. So for chromium we can write the electron configuration as so 1s2 2s2 2p6 3s2 3p6 4s2 and 3d4, 3d4, because chromium is right here.
All right, so 1s2, 2s2, 3p6. 3s2, 3p6, 4s2, and then 3d4. Now also by the way, sometimes notations can be simplified since this is so long. So sometimes what people will do, they will write the electron configuration in a simple way. They take the most recent noble gas that you would pass before you get to your element of interest here, that would be argon, and then they would write argon in brackets, and then they would just write the following electron configuration.
after argon, which would be 4s2 3d4. Now here for chromium, according to the rules established, this would be the electron configuration. However, actually moving one electron from the 4s subshell to the 3d subshell will allow the 3d subshell to be half filled.
So if one electron moved from here to here, and we would have 4s1 3d5. this would actually result in a half-filled orbital. Remember, d orbitals have five, the d shell has five orbitals, five orientations. And if you had five electrons to fit into these orbitals, you would simply put one in each based off of Hund's rule.
And having this half-filled shell is extremely desirable, so much so that the actual electron configuration for chromium is 4s1 3d5. instead of what you would assume following the convention we talked about. All right, so chromium is an exception. Another exception here that we want to talk about is copper.
So copper is right here. If we were to write the electron configuration for copper, keeping our easy notation, we can write argon, and this is going to be 4s2 3d9. So close to having a full... D subshell.
And actually, this would not be the correct electron configuration. You would have one electron that moves over here to satisfy a full shell. So it's going to be 4s1 3d10.
All right. And that's because it's just more energetically favorable to have a full D shell than it would be to have an S shell here. So these are the two exceptions to keep in mind. copper, and chromium. There are more, but for general chemistry, these are the main ones that we should commit to memory.
With that, we've completed everything that we wanted to for this chapter. Please let me know if you have any questions, comments, or concerns. Other than that, good luck, happy studying, and have a beautiful, beautiful day.