Each electron inside of an atom has its own ‘address’ that consists of four quantum numbers that communicate a great deal of information about that electron. In this lesson, we will be defining each quantum number and explaining how to write a set of quantum numbers for a specific electron.
Four Quantum Numbers
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How would you describe to someone exactly where you lived? I’m guessing you would start with your address. When you specify the location of a building, you usually list which country it’s in, which city and state it’s in within that country, and its street address. Just like no two buildings have the exact same address, no two electrons can have the same set of quantum numbers.
Also, there are very specific rules about quantum numbers that can exist together, just like you wouldn’t say that Wisconsin is a state in the country of Spain. A quantum number describes a specific aspect of an electron. Just like we have four ways of defining the location of a building (country, state, city, and street address), we have four ways of defining the properties of an electron or four quantum numbers.
Before starting this lesson, you should have an understanding of what an electron configuration is and how to write one for an element.
Remember that an electron configuration tells us where each electron is in an atom, and knowing the arrangement of the electrons is necessary in order to understand how an element will react and what types of molecules it will form. So let’s start with an atom of silicon. What would its electron configuration be? You should have an answer of 1s2 2s2 2p6 3s2 3p2. Silicon has a total of 14 electrons, which are all represented by that electron configuration. So what do all those numbers and letters mean? This lesson is going to crack that electron configuration code.
Before we go into great detail about those quantum numbers, it is important to note that when I say location, I mean probable location. There is really no way to know exactly where an electron is at a given time; they are very elusive. But it is possible to determine which specific three-dimensional region it is probably in. These three-dimensional boundaries where an electron is most likely found are called an atomic orbital.
Principal Quantum Number
The first quantum number that describes an electron is called the principal quantum number. It is often symbolized by the letter n. This number tells us the energy level or size of an orbital.
The higher the number, the larger the region is. So let’s take the electron configuration for silicon and look at the very last electron that was added to silicon. It should be one in the 3p orbital. That 3 indicates the principal quantum number. So for this electron, n = 3. The orbital that the last electron is going to be in will be larger than the 2p orbital because it has a higher number.
This means that the 2p electron is more likely to be found closer to the nucleus than the 3p electron. You will also hear the term ‘energy level’ a lot when dealing with electrons and their locations. The 2p electrons are located in the second energy level and the 3p electrons are located in the third energy level. They will have more energy than the electrons in the 2p orbitals. So, in silicon, how many electrons will have n = 3 as one of their quantum numbers? The answer is 4.
There are two 3s electrons and two 3p electrons. All start with 3, so all will have a principal quantum number of 3.
Angular Momentum Quantum Number
The next quantum number relates to the letters in the electron configuration. Which letters did you encounter when you wrote out electron configurations? You should have encountered s, p, d, and f. The letters represent the angular momentum quantum number.
It sounds like a mouthful, but it’s really just the shape of the orbital and is sometimes symbolized by the letter l. The s orbitals have a spherical shape, p orbitals are sort of dumbbell-shaped, d orbitals look similar to a three-dimensional four-leaf clover, and f orbitals have more of a flower shape. When assigning a number to each shape, the s-shaped orbitals have an l = 0, the p orbitals have an l = 1, the d orbitals have l = 2, and the f orbitals have l = 3. So that last electron that we added to the silicon atom in the 3p orbital will have an l = 1 and be sort of dumbbell-shaped. So for this electron the n = 3 and the l = 1. You may notice that some combinations of quantum numbers are going to be impossible.
For example, you can’t have n = 1 and l = 2 for a cluster of quantum numbers because that would mean that the electron configuration would have to be 1d, and there are no 1d electrons.
Magnetic Quantum Number
The next quantum number indicates the position of the orbital – how it’s arranged in space. It is called the magnetic quantum number and it’s sometimes symbolized by ml. No matter how many times you try to rotate a sphere, you will always end up with only one orientation. This means that there is only one magnetic quantum number possible for all the s orbitals. That is ml = 0.
The p orbitals can have three different orientations, so they’re assigned three different magnetic quantum numbers for the possible positions of the p orbitals – we have ml = -1, ml = 0, and ml = +1. The -1 means that the dumbbells are aligned along the x-axis, the 0 indicates that the dumbbells are aligned along the z-axis, and the +1 indicates that the dumbbells are aligned along the y-axis. The d orbitals and f orbitals are a little bit more complicated, but know that the d orbitals can have 5 different orientations (-2, -1, 0, +1, and +2) and the f orbitals can have 7 different orientations, so the different positions of the f orbitals are represented as -3, -2, -1, 0, +1, +2, and +3. That last electron that we put in the silicon atom would have an n = 3, an l = 1, and an ml = either -1, 0, or +1, representing the three possible positions the p orbitals can have.
Spin Quantum Number
The final quantum number doesn’t deal with the size, shape, or position of the orbital, but the actual electron itself.
The spin quantum number, often symbolized by s or ms, deals with the spin of the electron and plays a very important role in determining the magnetic properties of an atom or molecule. Each individual orbital can hold up to two electrons, and each electron will have a different spin, represented as either +1/2 or -1/2. So for our silicon electron, it would have an ms = +1/2 or -1/2. There is really no way to tell just by looking at the electron configuration which one it is.
Okay, now we are going to do a little practice. Write out the electron configuration for calcium.
You should have 1s2 2s2 2p6 3s2 3p6 and 4s2 as an answer. We are going to focus on just one of the two 4s electrons. One of these electrons should have a series of quantum numbers like this: n = 4, l = 0 (0 is the number associated with the s orbitals), ml = 0 (there is only one way to orient a sphere), and ms will equal either -1/2 or +1/2.
Whichever spin this 4s electron will have, the other will have the opposite spin.
Understanding the quantum numbers can be quite difficult because it’s such an abstract concept. You never actually see electrons in your daily life so it’s not very easy to relate to. What helps is thinking of a quantum number like a piece of an address, each getting more and more specific. The principal quantum number just indicates the size of the orbital or energy level, the angular momentum quantum number indicates the shape of the orbital, the magnetic quantum number indicates the orientation or position of the orbital, and the spin quantum number indicates the spin of the electron, represented as either +1/2 or -1/2.Remember, an orbital is just a three-dimensional location around the nucleus of an atom where an electron is probably going to be located.
After watching this lesson, you should be able to:
- Define quantum number and atomic orbital
- Identify and describe the four quantum numbers and understand how to write them based on electron configuration