The Quantum Mechanical Model
So, now let’s look at how do we write electronic configurations. So how do we write addresses? How to do we find the exact location of these ughhh…electrons using these orbitals. So that’s when we need to use something called as quantum numbers. So, now let me explain quantum numbers; let’s get into that. What are these quantum numbers? Actually, quantum numbers came out as a solution for Schrodinger’s wave equation and these are actually parameters explaining the electron behavior around the nucleus. In simpler terms, these quantum numbers kind of gives us the address of an electron around the nucleus.
Now I just want to show you something. So, what do you seen on your screen? Ya, that’s my driver’s license, so what do you see in that? You have my photograph. You have my name. You have my address. Now, if I ask you one question, that you have to find me, you know, locate me. What would you do? Ya, I know. You’ll go to that address but one thing, would you find me there? Most likely not! But then, what else would you do? You would go to BYJU’S classes and say, “Where is the hairy Chemistry Professor?” right? And what would they say? They say, they might say, I’m there, I’m not there. But, are you hundred percent sure that you can find me there? No, right? So, just like that, the principal quantum numbers and the other quantum numbers, which we are going to talk about, are actually going to help us, finding the address of that electron. And that address is not always accurate but the most probabilistic location of finding electron around an atom.
Now, let’s talk about principal quantum number. So what is this principal quantum number? Like we discussed, the principal quantum number gives us the energy level at which the electron is present. And this is denoted by the letter ‘n’. Now, that you know that the electron cannot exist around the nucleus without any energy. Can it exist? No, right? Because the energy levels are quantized. So, it needs some minimum amount of energy to say there, to stay there. So, just like that the value of ‘n’ for an electron present in the lowest energy state, the value of ‘n’ is 1. So, as we progress in energy states, energy levels, the value of ‘n’ increases such as 1, 2, 3, 4, 5 and so on. So, that basically gives you an idea of the principal quantum numbers. And in fact this is something which you’ve already learnt before. K shell, L shell, M shell, right? Very familiar, right? Ya, now just forget all those. Here onwards we are going to call them as letter n=1, 2, 3 so now that I’ve explained you the principal quantum number. Can you tell you me what would happen for an electron whose principal quantum number was increasing? What would happen? It would actually move away from the nucleus which means, higher the principal quantum number, the electron is located at a larger distance from the nucleus. So, the principal quantum number so actually indicates the size of an atom.
So, the next quantum number we’re going to discuss is called the Azimuthal quantum number. Pretty complicated name, right? No, it was also, you can also call it as the angular momentum quantum number. So, this actually came out from the angular momentum of the electrons around the nucleus. So now, when you look at the Radial Probability distribution graph and interpret that in three dimensions what did we get? We actually got an orbital.
So, what is an orbital? An orbital is a region around the nucleus where I have the most likely probability of finding electron and that probability is almost 95%. So, if I just join all those dots around the nucleus, I get an orbital. So, for the first energy level, for hydrogen the orbital shape is almost spherical and we call this as the ‘s’ orbital. So, we get different shapes as we go higher in energy levels. So those orbitals are called as, we have ‘s’ orbital and we have ‘p’ orbital which actually looks like this. Then we have the ‘d’ orbital which actually could look like this…like this…like this…like this and we have the ‘f’ orbital. Ufff! So many complicated shapes. So, these are different regions where, around the nucleus, you have a very great chance of finding an electron and these are called orbitals.
The Azimuthal quantum number represented by ‘l’ actually ranges from the values 0 to (n-1). So, what is ‘n’? ‘n’ is the principal quantum number. Now, let’s examine the values of ‘l’, how it varies with respect to ‘n’. Now, if we consider for the first energy level n=1 and what’s the value of ‘l’? l=0, which means in the first energy level only ‘s’ orbital can exist. Now for the second energy level n=2 and we know that ‘l’ takes values from 0 to (n-1). So we have l=o and n=1. So, in the second energy level we have both the ‘s’ orbital and the ‘p’ orbital. So similarly, for n=3, energy level three what do we have? l=0, 1 and 2, which means that in third energy level we have three different kinds of orbitals. We have ‘s’, ‘p’ and ‘d’ and, and you can just extrapolate and you know, go ahead for the other orbitals as well.
The next quantum number is called the magnetic quantum number, indicated by the letter ‘m’. So, what does this quantum number indicate? It indicates the orientation of these orbitals around the nucleus. It means that, in how many different positions can these orbitals, you know, arrange themselves around the nucleus? That’s what is indicated by the magnetic quantum number. And this magnetic quantum number ranges from the values -l to +l, so what is -l here? It is, the Azimuthal quantum number.
So now, if I take for the energy level one what happens? (n=1), (l=0) and what would be the magnetic quantum number? -0 to +0 which is actually 0, which means that for an ‘s’ orbital you only have one possible orientation which means that, very intuitive right? So ‘s’ orbital is spherical. So how many different ways can a sphere orient around itself? In one possible way. So that’s what the magnetic quantum number indicate.
Now, for the second energy level let’s calculate the magnetic quantum number. So, if the second energy level n=2 what’s the value of ‘l’ from 0 to (n-1) which is 0 and 1. Now, for l=1 what is the magnetic quantum number? m=-1, 0 and +1. So, there are actually three different angular orientations for l=1 which means that the ‘p’ has three different orbitals. So, for our visualization how do we represent them? We represent the ‘p’ orbitals such as ‘px’ which is oriented along the ‘x’ axis; we have ‘py’ which is oriented along the ‘y’ axis and ‘pz’ which is oriented along the ‘z’ axis.
So now, let’s examine for the energy level three. So for energy level three, n=3. So what’s ‘l’? ‘l’ is 0, 1, 2. So now, what’s the magnetic quantum number? You know the orientations for l=0 and l=1 for l=2 the magnetic quantum number is -2, -1, 0, +1, and +2. So, there are five different ‘d’ orbitals oriented in space around the nucleus. So, what are these? So, we have ‘dxy’ in which the two dumbles are oriented between the ‘x’ and ‘y’ axis, then we have ‘dyz’ where the dumbles are oriented between the ‘y’ and ‘z’ axis, then we have ‘dxz’ where the dumbles are oriented between the ‘x’ and ‘z’ axis and then we have the x2-y2 where the dumbles are oriented along the ‘x’ and ‘y’ axis and dz2 where the dumble, one dumble is oriented along ‘z’ axis with a little donat at the node.