Quantum Mechanics Worksheets

Wave-Particle Duality

A wave has definite characteristics. A wave has a definite frequency, wavelength, and amplitude. A wave must be a least one wavelength long, and thus is spread out in space. A wave travels with a certain velocity and produces effects like diffraction (bending around objects in its path), and interference (combing with other waves arriving at the same point in time).
A Particle has its own characteristics. A particle has mass, volume, kinetic energy, and momentum. Particles do not show diffraction and interference effects.
Light and other electromagnetic waves have been shown to have properties of both a wave and a particle. In 1905 Einstein used this explanation to account for the way photoelectrons are ejected from metal as a result of light hitting it. Light was later shown to have the (particle) property of momentum.
The French physicist Louis-Victor de Broglie (di broy-lee) suggested in 1923 that maybe particles could also have wave properties. He then developed an equation to calculate the wavelength of a moving particle. Two years later electrons were shown to have the (wave) property of diffraction.
Since the work of Einstein, de Broglie, and others, many chemists, physicists and philosophers have tried to come up with a complete picture of atomic structure. Using the particle nature of the electron the German physicist Werner Heisenberg suggested that it is impossible to know both the exact position and the exact momentum of an object at the same time. When trying to measure the position of an object, the momentum would be altered and vice versa. This fact is called the Heisenberg uncertainty principle.
Chemists and physicists struggled with this problem until the dual wave-particle nature of the electron was accepted and the wave nature of the electron was investigated. The Austrian physicist, Erwin Schrödinger treated the electron as a wave and developed a mathematical equation to describe its wave-like behavior. In Schrödinger's equation each electron within an atom can be described by a unique set of four quantum numbers. Quantum numbers represent different electron energy states. These four quantum numbers are used in describing electron behavior. We will study the four quantum numbers used in Schrödinger's equation but the actual equation involves mathematics with which you are probably not familiar with and so it will not be given.
Bohr pictured electrons orbiting the nucleus the way the planets orbit the sun. The modern model of an atom differs from the Bohr model in that the electrons do not orbit the nucleus like planets. Instead an electron occupies a three dimensional volume to form a cloud of negative charge.

Principal Quantum Number (n)
Quantum numbers represent different electron energy states. They are sometimes referred to as "shells" and are the quantum theory equivalent of the Bohr model. The first or principal quantum number, n, refers to the energy levels 1 – 7 (corresponding to the row numbers on the periodic table). As n increases, the distance of the main energy levels from the nucleus increases and their energy increases. You will recall that the maximum number of electrons found at each of these levels is: 2, 8, 18, 32, 50, 98. An easy way to remember this is 2(n)2 where n stands for the energy level.

The Second Quantum Number (l)
The "azimuthal quantum number" (l) defines the shape of the orbital and is generally designated by the letters s,p,d, and f. It has been shown that an energy level is actually made of many energy states closely grouped together. We call these states sublevels. An electron effectively occupies all the space around a nucleus by occupying a series of energy sublevels within an energy level. The second quantum number, l, identifies the energy sublevel.
The number of sublevels in each energy level is equal to the value of the principal quantum number. Within the 2nd energy level there are two sublevels (s &p), three in the 3rd (s, p, & d), 4 in the fourth (s, p, d, & f) and so on. The maximum number of electrons in the sublevels, s, p, d, and f, are 2, 6, 10, and 14. The numerical values for each sublevel are:
s, l = 0
p, l = 1
d, l = 2
f, l = 3
1. Name the four types of electron sublevels found in atoms and give the maximum population of each sublevel.
2. For the first four electron energy levels or shells (n) of atoms, determine the maximum population of the level, the sublevels that make up the level, and the electron population of each sublevel.
3. What are the n and l quantum numbers for a 3s electron? A 4d electron? Electron Configuration
We have learned that electrons fill the energy levels beginning with the lowest energy states first. Electron configuration is a method of labeling where the electrons are at the ground state. For example the two electrons of helium will choose the sublevel of the first energy level, 1s. A superscript of 2 is used to indicate the number of electrons found in the sublevel. The total electrons of an energy levels sublevels cannot exceed the maximum for that energy level.

EXAMPLE: Write the configuration for beryllium : 1s2,2s2
EXAMPLE: Write the configuration for phosphorus
1s2,2s2,2p6,3s2,3p3
4. Write the symbol, then the complete electron configuration for elements 1 - 18.

Overlapping Sublevels

With all these levels and sublevels some overlapping does occur. This happens when we reach the third and the fourth levels. There is even more over lapping when we reach the fifth, sixth and seventh.
Example: write the electron configuration for 28Ni
1s2 2s2 2p6 3s2 3p6 4s2 3d8
*note the 4s sublevel is filled before the 3d due to lower energy and greater stability.
Example: write the configuration of 88Ra
1s2 2s2 2p6 3s2 3p6 4s2 3d10 4p6 5s2 4d10 5p6 6s2 4f14 5d10 6p6 7s2

The Periodic Table and Configurations
The periodic table was originally constructed by placing elements with similar properties in a column. We now know that an atom's chemical properties are determined by its electron configuration. By reversing the procedure in which the table was constructed, the table may be used to "read" the configuration of an element. The written configuration of any element in Group IA will end in s1. The outer energy level is easily found from the table because the number of the period indicates the energy level. For example potassium's configuration would end with 4s1. The same procedure can be used for Groups IIA through VIIIA. There the endings, instead of s1, are s2, p1, through p6. For Groups IIIB through IIB, the ending are d1 through d10. The energy level is always one less than the period. For the lanthanides and actinides, the endings are f1 through f14. The energy level is always two less than the period. There are a number of exceptions to this arrangement.

The Octet Rule
One of the most basic rules in chemistry is that an atom with eight electrons in its outer level is particularly stable. Atoms react chemically in order to obtain this stable configuration. Atoms gain and lose electrons to become "isoelectronic" with a noble gas. For example we have learned that sodium, with eleven electrons, will lose one electron for a total of ten giving it the same electron configuration as neon (1s2, 2s2, 2p6). The noble gases do not react chemically due to the fact that they already have a full outer level. Although the helium atom has only two electrons in its outer level, it, too, is one of these stable elements. Its outer level is the first level and can hold only two electrons. Thus, it has a full outer level. For the transition and innertransition elements in the d and f blocks a pseudo-octet rule applies. These elements cannot obtain eight electrons in the outer most energy level, here a full or half full sublevel is particularly stable. An abbreviated configuration can be written using the stable noble gases. This notation uses the previous noble gas plus the additional electrons of the specified atom.

Example: Write the Nobel gas abbreviated notation for oxygen
O: [He] 2s2 2p4
Example: Write the Nobel gas abbreviated notation for magnesium
Mg: [Ne] 3s2

4. Write the complete electron configuration, noble gas abbreviated configuration and then assign quantum numbers n, and l for each of the electrons of lithium.
5. Predict the oxidation numbers of the following elements: Ar, P, Te, Cd, V, Sr, Cs, Cu.
6. Manganese has four oxidation states. Predict at least two of the four.Orbitals
The sum of all electron clouds in any sublevel (or energy level) is a spherical cloud. However, each sublevel is actually made up of characteristically shaped "orbitals". An orbital is a region in space occupied by one pair of electrons. We have learned that the s sublevel can hold two electrons. The shape of the s orbital is a sphere. The p orbital which holds a maximum of 6 electrons is actually 3 dumbbell shaped orbitals each holding two electrons. The d sublevel is comprised of 5 orbitals, and the f sublevel is made up of seven orbitals. Orbitals which are alike in size and shape and differ only in direction have the same energy. Orbitals of the same energy are said to be degenerate. We will only be concerned with the s and p orbital shapes.

The Third Quantum Number (ml)
The third quantum number, ml, defines each orbital more precisely by indicating its direction in space. For the p sublevel there are three possible values for m. The numbers would indicate the orbitals aligned along the x, y, and z axes. For example, the 6 electrons in a 2p orbital would be assigned m quantum numbers of -1, 0, and 1. The value of m is always -l to l. For the electrons in the 2p the l quantum number is 1 (recall p = 1), so the m quantum number can be any integer between -1 and 1. This gives three possible m numbers: -1, 0, and 1. -1 would represent the px orbital; 0 would represent the py orbital; and 1 would represent the pz orbital. For the electrons in the 4f the l quantum number is 3 (recall f = 3), and there are seven orbitals. The ml quantum number can be any integer between -3 and 3. This gives seven possible m numbers, (-3, -2, -1, 0, 1, 2, 3) one number for each of the seven orbitals.
Questions:
1. Draw the shape of an orbital in the sublevels s, p.
2. A s orbital can best be described as a _________________.
3. A p orbital can best be described as a _________________.
4. The sum of all orbitals for any particular sublevel would form a_______________.
5. Where would an electron with n, l, and ml quantum numbers of 3, 0, 0 be found? 4, 2, -1? 2, 1, 1? 3, 2, -2? 2, 0, 0?

The Fourth quantum Number (ms)
Since electrons are negatively charged particles and protons are positively charged it makes sense that the electrons are attracted to the positively charged nucleus. What seems illogical is the fact that the protons are all grouped together and the electrons occupy orbitals together. What is it that allows this to occur?
Electrons spin about their axes. If two electrons occupy an orbital together they will always have opposite spins. This difference in spin produced by a moving charge creates a magnetic force field that keeps the electrons in the orbital together.
The protons are held in the nucleus by a force called "binding energy". Binding energy is mass which has been converted to energy when the atom was formed. Consider the oxygen -16 atom. It contains eight protons, eight electrons, and eight neutrons. We can think of it as eight hydrogen atoms and eight neutrons. Each hydrogen atom has a mass of 1.007 825 2 amu. Each neutron has a mass of 1.008 665 2 amu. So the total mass of an oxygen-16 atom should be 16.131 923 2 amu. But the actual mass of the oxygen-16 atom is 15.994 915 0 amu. The difference between these two masses is called the mass defect. For an oxygen-16 atom, the mass defect is 0.137 008 2 amu. This mass (the mass defect) has been converted to energy, the binding energy which holds the nucleus together. This energy is very powerful and makes it extremely difficult to separate the nucleus.
The fourth quantum number, ms, depicts the difference in spin for two electron occupying the same orbital. The values for s are +1/2, for clockwise, and -1/2, for counterclockwise.
No two electrons will ever have the same four quantum numbers. This behavior was first observed and stated by Wolfgang Pauli and is called the Pauli exclusion principle. The quantum numbers n, l, and ml describe relative cloud size (n), the shape of the cloud (l), and direction of the cloud (ml). The fourth quantum number, ms, describes the spin of the electron.