Shells, Subshells and Orbitals
“the addresses of electrons”
The Schrödinger equation will definitely NOT be an equation that we will solve in order to demonstrate that everything in this reading is mathematical sound and therefore a good explanation of how electrons are found in atoms. Even your own teacher doesn’t have enough math in her/his background to make a whole lot of sense of this equation. So, as all of the discussion below is made, we have to recognize that to some “math heads” this is like 2 + 2 = 4 (only a little harder).
Use the “Shells, Subshells and Orbitals” handout as you follow this reading.
The solution to the Schrödinger equation is not one number or answer. There are infinite number solutions with three main parts to each solution of the equation. When you put the solutions for solving the equation together in order to understand the atom and its electrons, you find that there are 3-dimensional regions in space that electrons may occupy. These “regions” have subdivisions and shapes and different energies. They help explain not only those line spectra from the last reading, but bonding, color, and wonderful deeper insights about life like photosynthesis and respiration that will boggle your mind as you continue your adventure learning science to appreciate and understand the universe around you.
This whole understanding is the present-day 21st century model of the atom called the Quantum Mechanical Model of the Atom. And, it’s all about where electrons are found. That is why the subtitle for this reading is “the addresses of electrons.”
When the Schrödinger equation is solved, the first three of four solutions that inform us of the location of electrons are found. Even though there are really four quantum numbers that tell us where electrons are in atoms, only the first three are from Schrödinger’s complex equation. The four quantum numbers are sometimes given the letters n, l, m and s to identify them in discussions.
The four quantum numbers identify electrons in atoms are indicated in double rectangles on the “Shells, Subshells and Orbitals” handout. Note that the first three quantum locations of electrons correspond to the three words: shells, subshells and orbitals. There will be more vocabulary as we go.
refers to the shells or energy levels in an atom. Remember Max Planck’s equation? Even electrons have only certain, specific and unique energies because they move like waves with their own frequencies. So, different electrons have different energies due to their different frequencies and this gives them different locations in an atom. That was an important sentence – read it again! Because each electron has its own energy, we say that it is quantized. This means each electron has a frequency that is unique to it in the atom. And, this way of looking at electrons is as if electrons move like waves.
I, like a lot of other scientists and textbooks, prefers the word energy levels over shells. So, I more likely to refer this quantum number as an “energy level”. Any electron in any atom can be found in any energy level of the same atom if only it is given the quantum or energy it needs to move to a higher level location in the atom. This means fire, electricity, heat or other energy can move an electron from its present location to a “higher energy level” location in an atom. According to the solutions of the Schrödinger equation, there is an infinite number of energy levels in an atom (from energy level one to infinity). In nature, there isn’t an atom with an infinite number of electrons, so we don’t have examples of atoms with gigantic numbers of energy levels. In elements of the Periodic Table (what we know about nature and the types of atoms that exist), there really are only seven (7) energy levels. For the heck of it, count the number of rows in the Periodic Table. Surprise! Finally, the larger the energy level, or the value of n, the greater the distance that that electron is from the nucleus.
The greatest number of electrons in any energy level is determined by 2n2. This is not part of the Schrödinger equation, although it is certainly helpful to calculate the number of electrons in an energy level. So, in the first energy level there are 2 electrons [2(1)2], 8 in the second energy level [2(2)2], 18 in the third [2(3)2], and 32 in the fourth level energy level [2(4)2]. Oh, this goes on into infinity but we are only most interested in atoms of common elements and these first four energy levels (first four rows of the Periodic Table) cover the most abundant stuff of Earth and beyond in our universe.
One more important piece to the first quantum number. The energy level “energy difference” decreases as electrons are further and further from the nucleus of the atom. See the energy diagram at the bottom of the first quantum number discussion of the handout. This means as electrons are further and further from the nucleus of the atom the difference between their energies gets less and less. Remember how many energy levels are possible from the solution to the Schrödinger equation – infinite! Does this mean that an atom can possibly be so large that we could see it? Nope. With this understanding, as electrons are further from the nucleus, they are closer and closer to previous energy levels and we know that this still means really small atoms.
is referred to as the sublevels or subshells of the atom. Like your home address, where there is a good probability of finding you, there are many rooms in your house where you could be found. Each energy level n has n sublevels. This means the first energy level has one sublevel; the second energy level has 2 sublevels, the third energy level has 3 sublevels and so on. Each sublevel has a letter symbol to describe it. The sequence of letters for sublevels goes: s, p, d, f …Look at the diagram at the bottom of the discussion for the second energy level. Do you see how the sublevels are symbolized in an energy level? For example, the third energy level (n=3) has three sublevels and these sublevels symbolically represents s, p, and d.
As you are reading this and trying to make sense of just what is going on, remember always that this is only about locating electrons in an atom. There are four aspects to locating electrons or finding the address of an electron. We just covered the first two. Every electron has an energy level and sublevel. Electrons are found in smaller atoms like hydrogen close to the nucleus in the first energy level. More specifically, they are located in the s sublevel found in this first energy level.
refers to the orbitals in atoms where electrons are found. Each sublevel has a fixed number of orbitals. Orbitals are finally the three dimensional region in space where electrons are found. When the Schrödinger equation is solved, the orbitals component of the solution gives the x, y and z orientation of the orbital. The numbers of orbitals in sublevels follows the “odd number progression” 1, 3, 5, 7 and so on. In other words, in the s sublevel, there is one orbital; in the p sublevel, there are three orbitals and in the d sublevel, there are five orbitals and so on.
Orbitals have three-dimensional shapes. The s orbital is always spherical. The p orbital looks like two tear drops put together tip by tip (most textbooks say this looks like a “dumbbell” – you decide). The d orbital looks like a four-leaf clover. The f orbital looks bizarre (not many books show the shapes of the seven f orbitals). See the diagram (12.3 on page 327 and 12.4 on page 328) in your book to become familiar with these shapes.
Refers to the spinning of electrons and is not from solving the Schrodinger equation. An electron is understood to be spinning on an imaginary axis, just like Earth. This helps explain a number of observation in nature at the sublevel of electrons. The spinning can be clockwise or counter-clockwise around this imaginary axis. Interestingly, we must imagine the electron as a particle (not a wave) in order to understand this fourth quantum number. We symbolize the spin directions with arrows pointing up or down ( ). “Clockwise and counter-clockwise are more challenging to keep track of and symbolize so these arrows refer to the spin direction.
Now, let’s put it all together and with a few rules see how electrons are found in atoms using the four quantum numbers. The location of an electron in an atom is called the electron configuration.
You will have to come back to the following three rules over and over before they make a lot of sense.
1. The Aufbau Principle says any electron will be in the lowest energy location of an atom. Electrons will therefore be in the lowest energy level, lowest energy sublevel and lowest energy orbital.
2. The Pauli Exclusion Principle says that an atomic orbital has only at most 2 electrons. Of course, one electron is spinning clockwise, the other spinning counter-clockwise.
3. Hund’s Rule gives the order of filling of a sublevel with more than one orbital. The rule says that when electrons occupy orbitals of equal energy, one electron enters every orbital with one spin direction (clockwise or counter-clockwise) before the “paired electron” enters filling the orbital with the total of two electrons (one with each type of spin).
Hund’s Rule can be diagrammed for the filling of a p sublevel (with 3 orbitals) like this:
Orbital xOrbital yOrbital z
______The first electron fills the first orbital with a spin.
______The second electron fills the second orbital with same spin.
______The third electron fills the third orbital with same spin.
______The fourth electron fills the first orbital with opposite spin.
______The fifth electron fills the second orbital with opposite spin.
______The sixth electron fills the third orbital with opposite spin.
Suppose you could diagram the d or f sublevel with 5 or 7 orbitals? Same pattern – just more electrons.
NO 2 ELECTRONS CAN HAVE THE SAME SET OF 4 QUANTUM NUMBERS.
How is it that the 5 electrons found in an atom of boron are different than the 79 electrons in an atom of gold? Over the years, I have created a model to help students follow this messy way that electrons of different elements are arranged in an atom. We call it the Electron Hotel. The electron hotel is not real. It has its share of problems, and it is helpful for making sense of most everything you just read. It is an analogy, so of course it is not the same as the real atom it is trying to help us understand.
Like a real Hotel you check into, you want a bed (your own), in a room, on a floor of the hotel.
First Quantum Number (energy levels): The Electron Hotel has an infinite number of floors – we’ll only draw four.
Second Quantum Number (sublevels): The Electron Hotel has the number of rooms on the floor as the floor number. So, the first floor has one room, the second floor has two rooms, the third floor has three rooms… The room will only be room s, room p, room d or room f. (Recall that the number of sublevels is equal to the energy level.)
Third Quantum Number (orbitals): The number of beds in a room depends on what room it is. An s room only has one bed. A p room has three beds; a d room has 5 beds and an f room has 7 beds. (Recall the “odd number progression” of orbitals in sublevels.)
Fourth Quantum Number (spin director): Electrons do to beds in rooms on the floor according to a particular order. The three rules: Aufbau Principle, Pauli Exclusion Principle and Hund’s Rule gives the rules of putting electrons to bed.
See if you can figure out the pattern of putting electrons to bed in the Electron Hotel.
The Electron Hotel
First look at the diagram below. Can you see the first three quantum numbers from the Schrödinger equation: 1. the 4 floors, 2. the s, p, d and f rooms and 3. the beds (represented by “____” ) in the hotel?
S ______4th floor ______
S ______
3rd floor ______
S ______
2nd floor
S ______
1st floor
The electron hotel diagram above is for hydrogen. Where it that one electron of hydrogen? It is found in the first energy level, in the s sublevel, in the only orbital, spinning in one direction or another (clockwise or counter-clockwise.) Incidentally, the nucleus is in the basement.
We symbolize hydrogen’s electron configuration: 1s1 (notice that the spin direction is not in this configuration).
1s1
oneelectron in only orbital in s sublevel (bed)
s sublevel (room)
first energy level (floor)
See if you can make sense of this example: Oxygen
S p d ______f ______4th floor ______
S p d ______
3rd floor ______
S ______p ______
2nd floor
S ______
1st floor
The Electron configuration of oxygen is: 1s2 2s2 2p4
Can you tell how these 8 electrons filled the floors, rooms and beds using the rules? The electron hotel has its faults; learn to not rely on it.