Atomic Structure and Quantum Mechanics

ELECTROMAGNETIC RADIATION

Electromagnetic radiation is one of the ways energy travels through space. This includes X-rays, gamma rays, ultraviolet light (or UV), infrared light, microwaves and radio waves. Visible light also makes up a small fraction of what is known as the electromagnetic spectrum. The electromagnetic spectrum is the spectrum of all possible types of electromagnetic radiation.

The Visible Spectrum

Although gamma rays may be vastly different from visible light, they both exhibit the same type of wave behavior. All types of electromagnetic radiation travel at the speed of light in a vacuum too.
The three main characteristics of a wave are wavelength, frequency, and speed. Wavelength is the distance between two consecutive peaks or troughs. It is symbolized by the Greek letter lambda, l . Frequency is the number of waves (cycles) that pass a given point in one second. It is symbolized by the Greek letter nu, n . Speed is defined as cycles per second, or hertz, abbreviated Hz.
Wavelength and frequency are inversely related. This is shown by the formula

c = (l)(n)

In which n is the frequency, l is the wavelength, and c is the speed of light (2.9979 ´ 10⁸ m/s).

THE NATURE OF MATTER

Around 1900, everyone believed that energy and matter were two different things altogether. Matter was thought to be made of particles (atoms) and energy was thought to be made of light (electromagnetic radiation). Particles had mass while energy occurred in waves and was massless.
That changed in 1900 when Max Plank, a German physicist, studied the emission of radiation from solid objects. He observed that the physics of his day could not describe the results he was obtaining. He postulated the theory that energy can be only gained or lost in whole number multiples of the quantity hn , where h is Plank’s constant. Experiments determined the constant to be equal to 6.626 ´ 10^-34 J× s. Therefore, the change in energy for a system can be shown by the equation

DE = nhn

where n is an integer (1, 2, 3, …).
With this, a new set of thinking came about. Before it was thought that a transfer of energy was continuous. Now it was realized that energy could only occur in "packets" called quantum.
Albert Einstein was next, proposing that even electromagnetic radiation was quantized. He suggested that electromagnetic radiation could be viewed as a stream of particles called photons. The energy of a photon could be described by the equation

E_photon = hn = hc/l

where h is Plank’s constant, n is the frequency, and h is the wavelength.
In his special theory of relativity, he showed that energy has mass. This is shown in the formula

E = mc²

where c is the speed of light, m is the mass, and E is the energy.
The proof of energy having mass is shown if you rearrange the formula in the following form:

m = E / c²

Since you know that

m = E / c² =  (hc/l) / c² = h / lc

one can clearly see de Broglie’s equation,

l = h / mv

v = velocity, not frequency
This allows us to calculate the wavelength for a particle. It also proves that particles do behave as waves.
In summary, it was now found that energy is quantized, meaning that it can occur only in discrete units called quanta. It was also discovered that energy does contain mass. This is known as the dual nature of light.

BOHR’S ATOMIC MODEL

In 1913, Niels Bohr, a Danish physicist, developed a quantum model for the hydrogen atom. By combining classical physics and some theories of his own Bohr was able introduce a revolutionary model for all atoms.
Previous atomic models had the negatively charged electrons orbiting the positively charged nucleus. However, this is inaccurate because this would cause the electrons to eventually "fly off" the atom. The man who invented this model of the atom, Ernst Rutherford, knew that this was true, but couldn’t think of a better way to describe an atom.
Using the spectrum of hydrogen, Bohr found that when a prism diffracted the light color was displayed at only four discrete increments.

He saw that these results would fit his model if he assumed that the angular momentum of the electron could only occur in certain increments. Angular momentum equals the product of mass, velocity, and orbital radius. Using this he assigned the hydrogen atom energy levels consistent with the hydrogen emission spectrum.
This gave way to Bohr historic equation for the energy levels available to the electron in the hydrogen atom. The n is an integer. The larger the value of n, the larger is the orbital radius. Z is the nuclear charge.

Bohr’s equation was able to accurately calculate the energy levels for the hydrogen atom. Each energy level pertains to an electron in an exited state. It will move up to higher energy levels, but when it goes back to its ground state (where n=1) or to a lower level orbital it emits energy. That is what Bohr viewed when he saw the spectrum of hydrogen.
To find the total change in energy once subtracts the energy in the initial state minus the energy in the final state.

DE = E_final - E_initial

If a negative sign appears as the result, that means that the atom lost energy and thus is in a more stable state. However, if you insert this value into a separate equation, use the absolute value of the change in energy.
The next equation’s purpose is derived from Bohr’s formula. It determines the energy used for an electron moving from one level (n_initial) to another level (n_final).

-2.178 * 10^-18 J ( 1/(n_final)²) - (1/(n_initial)²))

However, it was soon found that Bohr’s model does not work for other atoms. Later models, however, would explain atoms other than hydrogen.

QUANTUM MECHANICS

It was now known that Bohr’s model would work for hydrogen and hydrogen only. A new model was needed. This became known as quantum mechanics. As de Broglie’s equation showed that electrons acted as waves. Many physicists now tried to attack the problem of atomic structure by using the wave properties of an electron. To them the electron bound to the nucleus seemed similar to a standing wave.
Take a stringed instrument. The string vibrates to produce a musical tone. The waves are "standing," because they are stationary; the waves do not travel along the string. Now put this into the hydrogen atom, with an electron acting as a standing wave. One physicist, Erwin Sgrödinger, used the formula

H y = E y

to illustrate the wave properties of an atom. The formula is in that form because the math is too complicated to be detailed here. The y is called the wave function. It is a function of the x, y, and z points on a three-dimensional graph. The H represents a set of mathematical operations called an operator. The E represents the total energy of the atom. When this equation is analyzed in its entirety, each solution consists of a wave function (y ) that is characterized by a particular value of E. A specific wave function is often referred to as an orbital.
To show this, let us concentrate on the hydrogen atom. The wave function corresponding to the lowest energy is called the 1s orbital. Remember that an orbital IS NOT a Bohr orbit. Th electron is the 1s orbital is not moving around the nucleus in a circular motion.
Here is a quick point. According to Werner Heisenberg "there is a fundamental limitation to just how precisely we can know both the position and the momentum of a particle at a given time. This is known as the Heisenberg uncertainty principle. In the formula

Dx * D(mv) > h/4p

where Dx is the uncertainty in a particles position and D(mv) is the uncertainty of a particle’s momentum. This all means that one cannot simultaneously find a particle's position and momentum.
Using this it was determined that for the 1s orbital, one has the greatest probability of finding an electron near the nucleus. That is because of the smaller radius and volume of the atom as a whole. Thus, the farther away from the nucleus you look, the less of a chance there is to find an electron.
To better understand electron orbitals as a whole, electron numbers have been made to identify the electrons of an atom.
The principal quantum number (n) has values of 1,2,3… It pertains to the period that the atom is in. Periods are represented by rows. For example, period two has a principal quantum number of 2.
The angular momentum quantum number (l) has values of 0 to n-1. This is related to the shape of an orbital. If l=0, the letter is s; l=1 is called p; l=2 is called d; and l=3 is known as f. This is explained later in this reading.
The magnetic quantum number (m_l) has integral values between l and –l, including zero. For example, if the value of l is 2, then the possible values of m_l are -2, -1, 0, 1 and 2.
The electron spin quantum number (m_s) can have one of two values: +(1/2) and –(1/2). This is because only two electrons can occupy any orbital, and they must have opposite spins.
But what do these have to do with anything? There will be more on that next.

AUFBAU PRINCIPLE

The s orbitals are represented by the Alkali and Alkali Earth Metal groups (groups 1A and 2A, respectively). The p orbitals are the other representative elements (3A through 8A). The d orbitals are the transition metals, excluding the inner transition metals. Finally, the f orbitals are the inner transition metals.
According to the rules stated previously, hydrogen is in the 1s orbital. This is because it is in the s orbital of the first period. And since it is the first atom of the 1s orbital, it is given the distinction 1s¹. Thus helium, even though it is technically a Noble Gas, is classified as 1s². That is because helium is the second atom of the 1s orbital.
Lithium is in the 2s orbital. It is identified as 2s¹. Since you have to add on the previous orbitals to make it correct, the correct from is 1s²2s¹. Beryllium is 1s²2s². Are you beginning to see the pattern? Boron, which is the p orbital is also the first atom in the 2p group. Thus it is 2p¹. However, you must add the previous orbitals on to make it right, so the correct form is 1s²2s²2p¹. Carbon is in the form 1s²2s²2p². Neon, which is at the end of the 2p orbital, is 1s²2s²2p⁶.
The d orbital is somewhat different. Since the transition metals begin at the fourth period, the first orbital is the 4s orbital. However, when writing the transition metals, one must start with 3d. Why isn’t it 4d? This is because the 3d orbitals are part of the same orbital as the 3s and 3p orbitals, but they are at a higher energy level than 4s. That is why you have to fill the 4s orbitals before the 3d orbitals.
Therefore, the configuration of scandium is 1s²2s²2p⁶3s²3p⁶4s²3d¹. After filling the 3d orbitals, the 4p orbitals are filled. The two exceptions to the rule are chromium and copper. Instead of normal configurations, they are 4s¹3d⁵ and 4s¹3d¹⁰ respectively. No one really knows why this happens.
Also, a shortcut to writing these configurations is substituting the highest filled Noble Gas for its configuration. For example, the configuration of potassium is normally 1s²2s²2p⁶3s²3d¹⁰4s¹. However one can substitute the configuration of argon with [Ar]. Thus the shorthand way to write an electron configuration for potassium is [Ar]4s¹.
For the f orbitals the rules change again. The f orbitals represent the inner transition metals. Since it is often very difficult to explain, use the chart below to figure out the filling order for all electrons.

Take the chart below.

--Image not done yet--

You can clearly see that the 3d orbital is higher than 4s. That is a visual representation of the fill order. The first two electrons, representing hydrogen and helium, respectively. It now looks like this:

--Image not done yet--

Notice that there are two arrows used. The up arrow describes the +(1/2) electron and the down arrow describes the –(1/2) electron.
The 2s elements are lithium beryllium. Add them and you get.

--Image not done yet--

Then fill up the 2p orbitals one by one. Remember that you fill from left to right. Therefore, there shouldn’t be any 2p orbitals with two electrons until oxygen. When you reach neon, the whole 2p orbital is filled. That is why the Noble Gasses are so unreactive. Their orbital subshells are completely filled, thus leaving no spots for elements to bond with it.
This process goes on and on. Use the fill chart given above to remember what order the electrons fill in.

HYBRIDIZATION

Hybridization is the mixing of normal atomic orbitals to form special orbitals for bonding. It is a type of covalent bonding. For instance, the first type of hybrid orbital is the sp³ orbital. It is named that way because it is a combination of one 2s and three 2p orbitals. From the figure below, you can see how the hybridization occurs.

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From the figure below for carbon we see that there are the 2s and 2p orbitals to work with. Since carbon is sp³ hybridized, you take both the 2s and 2p orbitals and create a new set of orbitals, sp³. This means you now have a set of four orbitals. Since you all four electrons with you, you now have an electron in each orbital (fill left to right). Thus, carbon can bond with up to four other elements that can donate an electron to fill each orbital. An example of such an element is hydrogen. The electron of hydrogen can bond with the lone electron for each of carbon’s four hybridized orbitals. Thus, CH₄ (methane) is formed.

Ethylene (C₂H₄) is an example of sp² hybridization. The two carbons are double bonded to each other. For each carbon atom three sp² hybrid orbitals are created. Because of this, one 2p orbital must be left behind. If you looked at this geometrically, you would see the three sp² orbitals coming out of hydrogen with the remaining p orbital perpendicular to the sp² orbitals. See the figure below. The sp² orbitals, when bonded, share a pair of electrons. As a result, two sp² orbitals are each bonded to a hydrogen atom and the other produces a carbon-carbon bond. These bonds are known as sigma (s) bonds.

However, carbon is double bonded. That means that another bond needs to be made to form the double bond. This job is left to the remaining 2p orbital. As shown in the figure below, the p orbital creates what is known as a pi (p ) bond.

For sp hybridization, let us look again at carbon. You should already know by now that sp means that there are two hybridized orbitals. This also means that there are two unchanged 2p orbitals remaining. Let us take the CO₂ molecule. We know that the carbon is double bonded to both oxygen atoms and that each oxygen atom contains two lone pairs of electrons. As stated with the sp² hybridization, sigma bonds must link up the carbon and oxygen atoms. However, since oxygen has two lone pairs of its own, it must undergo hybridization of its own. Oxygen is sp² hybridized. Then, since carbon has two unchanged 2p orbitals, it creates two pi bonds, one with each oxygen atom. And since oxygen is sp² hybridized, it will form a p bond with the carbon atom.
By now you should realize that every double bond is made of a sigma and a pi bond.
Another example of sp hybridization is the N₂ molecule. The nitrogen molecule has a triple bond, with each atom containing a lone pair of electrons. Thus, it must have a sigma bond and two pi bonds. The unused sp orbital contains the lone pair of electrons.
The octet rule states that an atom must always have eight electrons in its valence shell to be stable. However, there are exceptions to this rule. One exception is phosphorous pentachloride (PCl₅). When looking at the molecules’ structure, you notice that there are ten electrons in the valence shell. To do this, the molecule must be dsp³ hybridized. This contains a one d orbital, one s orbital, and three p orbitals. Each dsp³ orbital creates a sigma bond with a sp³ orbital from the chlorine atom. The other three sp³ orbitals create the atom’s lone pairs.
The last example is d²sp³ hybridization. Sulfur hexafluoride (SF₆) is an example. By now, you should pretty much know what will happen.

BOND ORDER & PARAMAGNETISM

Bond order is an indicator of bond strength. It is the number of bonding electrons minus the number of antibonding electrons, all divided by two.

For any molecule, for example O₂, you take the total number of electrons involved, which is 16, and place them across the chart starting at the bottom. Remember to go left to right. Then total up the bonding and antibonding electrons and calculate the bond order. The greater the bond order, the greater the bond strength. Also, while bond order increases, bond energy increases and bond length decreases.
Paramagnetism is when a substance is attracted into an inducing magnetic field. Diamagnetism is when the substance is repelled by an inducing magnetic field.
Looking at the stack chart can identify paramagnetism. If the chart shows that the molecule has any unpaired electrons, as shown above with O₂, then it is paramagnetic.

Atoms, Molecules and Compounds

Law of Multiple Proportions

The Law of Multiple Proportions simply states: When two elements form a series of compounds, the ratios of the masses of the second element that combine with 1 gram of the first element can always be reduced to small whole numbers.
For those of you who are lost, it just means that only multiples of an amount of one element can react with a constant amount of another in a two element compound. For example, here is a table comparing three compounds of nitrogen and oxygen.

Compound	Amount of Nitrogen that reacts with 1 gram of oxygen
Compound A	1.750 g
Compound B	.8750 g
Compound C	.4375 g

See the smallest number, .4375? It divides into the other numbers with no remainder. That is, B/C is 2, and A/C is 4. This just shows that molecules are made up of finite sized units, or atoms. You can't have 2.3 atoms combine with 1 atom. Same thing here.
There is no way to know from this data alone exactly the ratios of the molecules. You could take the base unit to be any one of the numbers. Let's do two examples.

Using .4375 and ratio 1N : 1O (Compound C):
Let's call .4375 the ratios of the molecules 1N : 1O. Since Compound B contains twice as much N, you could say that the ratio there is 2N : 1O. Since Compound A contains four times as much N as Compound C, you could say that the ratio is 4N : 1O.

Using .8750 and ratio 2N : 2O (Compound B):
Now you say that .8750 is your base number, and the ratio 2N : 2O. What will the ratio of Compound C be? You know that there is half as much Compound C as in Compound B, so just reduce the N by one-half; 1N : 2O. And since you know there is 4 times as much in Compound A, it's 4N : 2O. It's that easy.
As you can guess, there is an infinite number of situations that satisfy the table.

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Dalton's Atomic Theory

These are pretty straightforward and require no explanation. Although you don't use these on tests or anything, it's good stuff to know (not to mention our whole understanding of chemistry is based on this):

• Each element is made up of tiny particles called atoms.

• The atoms of a given element are identical; the atoms of different elements are different in some fundamental way or ways.

• Chemical compounds are formed when atoms combine with each other. A given compound always has the same relative numbers and types of atoms.

• Chemical reactions involve reorganization of the atoms--changes in the way they are bound together. The atoms themselves are not changed in a chemical reaction.

These ideas form the basis of what a lot of AP Chem is all about; chemical reactions. It's nothing more than breaking some bonds and making some bonds, to put the atoms in a different order to create new things.


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Intro to Atomic Structure

An atom consists of a nucleus and electrons orbiting around it. The nucleus is made up of positively charged particles called protons, and neutral particles called neutrons. The electrons are negatively charged particles.
Here is a simplified picture:

This atom has two protons (red), so it is a helium atom.
One thing that pictures like the above cannot express is the relative sizes of the nucleus and the atom. If the atom was really that big (diameter 108 pixels), then the nucleus would be 1/100th of a pixel big! Or on a bigger scale, if the nucleus was the size of a pea, then the atom would be as wide as a football field. Another thing is mass. Virtually all the mass is concentrated in the nucleus. The 'pea' would weigh 250 million tons!
There are two numbers to know about any atom. One is the atomic number, which is the number of protons (and electrons in a neutral atom), and the atomic mass number, which is how much the thing weighs. One proton's weight is close to one neutron's weight which is close to one. Electrons weigh virtually nothing when compared to the huge protons, so they don't affect the total mass. So the atomic mass number is simply the number of protons plus the number of electrons.
The notation of writing atoms is like this:

The top number represents the mass number.
The bottom number represents the atomic number.
Some things to remember while solving these types of problems:
(Note: Problems of this simplicity will not be on the AP test, I guarantee that. We are not so lucky. But you will be tested in class on this stuff.)

• The bottom number is always the number of protons.
• The bottom number is also usually the number of electrons, but only in a neutral element. Ions come later.
• The number of neutrons is always the top number minus the bottom number. All you're doing is taking the total mass, taking away the portion that is due to the protons, and what you got left is the neutrons. Likewise, if you know number of neutrons and protons, to find this number all you got to do is add.

Example:

Sample Problem Here is a sample atom. How many electrons, protons, and neutrons does it got? Solution The bottom number is always the number of protons you have. So there are 9 protons. It's a neutral atom, by the fact that there are no plus or minus signs anywhere, so you can assume there are 9 electrons as well. There are 19 total nucleons (protons and neutrons) in the nucleus, and 9 are protons. So there are 10 neutrons. Easy.

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Molecules and Ions

Compounds are made up of two or more atoms held together in some way. Although I'll go way more in depth later, atoms can be bonded together by both sharing electrons or one taking electrons away from the other.
Those compounds that share form a covalent bond, and the resulting formation is a molecule. An example is water, H--O--H. Each H shares its electron with the O, so that the O has two shared electrons, completing its valence shell. Another is methane, CH₄, which has each H sharing an electron with C, and C sharing a total 4 electrons with the H's.

A molecule acts as a unit by itself. It moves together. To represent a molecule, you can use either a ball-and-stick model or a space filling model. Examples of H₂O are shown below.

Ball-and-stick model	Space filling model

As a general rule, a covalent bond results when two non-metals are put together.
Let's consider the case of NaCl. Chlorine has a higher affinity for electrons than sodium, and so it takes one away, shown by the following:

Cl + e^- -> Cl^-

This shows that the electron joins up with the chlorine, and becomes a negatively charged ion--also known as a anion. But where did this electron come from? From the sodium, of course:

Na -> Na⁺ + e^-

This shows that the normal sodium atom gives off an electron--and becomes a positively charged ion in the process--also known as a cation.
Since these charges have opposite signs, they attract. The positive Na⁺ and the negative Cl^- form a ionic solid, also a salt. (Keep in mind that a salt refers to any ionic solid, not just our example of table salt. Ions don't have to be just one atom; they can be polyatomic, or consisting of many atoms. Some examples are the anion C₂H₃O₂^- and the cation NH₄⁺. A general rule, not to be followed strictly; if a compound has a metal and non metal in it, it's ionic. One of many exceptions is NH₄Cl; I guess the best rule to follow is to look for those ions!
One of the most annoying things in chemistry is memorizing ions. Once every week, my class would have to memorize ten ions and have an Ion Quiz. Although you might forget them, it is a good idea to review them while preparing for the test. I have included an ion table in the Tables section.

Compound Naming

I hated this part in Chem I. But it's not too bad. On a test, a question might be to mix dinitrogen pentoxide with something else. No problem, but what's that made of? This section will show you how to take a chemical structure and give a name to it, and vice versa.
Binary Ionic Compounds
These are just compounds made up of two different ions, like sodium chloride. All you do is: Put the name of the cation, followed by the name of the anion. That's it.

Example:

Compound	Name	Ions
KI	Potassium Iodide	K+, I-
CaS	Calcium Sulfide	Ca2+, S2-
MgO	Magnesium Oxide	Mg2+, O2-
Li3N	Lithium Nitride	Li+, N3-

For single one-atom ion compounds such as above, you keep the name of the cation, and change the ending of the anion to -ide. Other examples are hydrogen => hydride, carbon => carbide, Fluorine => Fluoride, you get the idea.
One more thing to consider is the charge on the metal atom. If it can have only one possible value (such as +1 for sodium and +2 for magnesium above), then the normal name is fine. If the metal can form cations with different charges, like iron and tin, then you must put the Roman Numeral of it in parenthesis right after the name of the cation. Here are some examples:

Metal Ion	First half of compound name
Fe2+	Iron (II)
Fe3+	Iron (III)
Pb2+	Lead (II)
Pb4+	Lead (IV)

How do you know if to use roman numerals or not? Generally, if the metal is a transition metal (belonging to the three row skinny middle part of the periodic table up to the metalloids), then you need to use it. Some exceptions are aluminum (+3), zinc (+2), silver (+2), and cadmium (+2). Even though these are transition metals, they have only one possible ion. These metals and the metals in the first two columns need not use it, in fact, don't use them.
Another thing; mercury is really weird here. The two possible ions with mercury are Hg²⁺2²⁺. The difference when writing this is that Mercury (II) for the first, and Mercury (I). Because there is a total of +2, and there are two.
Nothing to it. Before I get to the sample problems, there is one more important point to all this: Compounds don't have charge. So when you add up all the charge from the ions, they must equal zero. And if they don't, you must balance it. Note that in the first table, the first three ions, when charge is added up, give zero. The last one is different. It still must add up to zero, but note that N is -3, but Li is only +1. It's common sense that you need three Li atoms to cancel out the -3 in N, hence the formula.

Example:

Give the name of these compounds:

a.) CuCl b.) NaF c.) MnO₂ d.) FeBr₂ e.) Fe₂O₃

Answers

1. By the rules, this is Copper Chloride. But copper can have different charges, so you must figure what the charge on this one is. Let's see: you know chloride can have only one value, and that's a -1. So if they must add up to zero, Cu must be +1. The answer is copper (I) chloride.
2. This is sodium and fluorine, so the name is sodium fluoride. Sodium can only be +1, so there is no need for roman numerals.
3. The name is Manganese Oxide, but a roman numeral is necessary, since it's a transition metal. In this case, manganese is a transition metal, so it needs a roman numeral. Let's start with oxygen: Each oxide ion has a charge of -2, and there are 2 of them. So the total charge due to oxide is -4. The single manganese ion must compensate for this, so its charge must be +4. The name is manganese (IV) oxide.
4. Same here. It needs a roman numeral. Br is -1, and 2 of them, so Fe must be +2. Name is Iron (II) Bromide.
5. e.This one's a little bit different, but the same concepts apply. You start with the oxygens. 3 of -2 ions make it a total of -6 charge due to oxide. So the two irons must compensate. If two of them must equal +6, each must be...+3!!! It should be no surprise. So the name of this is Iron (III) Oxide.