Thursday, 3 November 2011

Atomic Structure and Quantum Mechanics


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.
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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).


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
Ephoton = 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.


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 = Efinal - Einitial
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 (ninitial) to another level (nfinal).
-2.178 * 10-18 J ( 1/(nfinal)2) - (1/(ninitial)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.


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 (ml) has integral values between l and –l, including zero. For example, if the value of l is 2, then the possible values of ml are -2, -1, 0, 1 and 2.
The electron spin quantum number (ms) 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.


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.
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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:
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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.
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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 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 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.

Tuesday, 1 November 2011

Non-Ideal Behaviour of Gases

Deviations from Ideal Behavior 

All real gasses fail to obey the ideal gas law to varying degrees

The ideal gas law can be written as:

For a sample of 1.0 mol of gas, n = 1.0 and therefore:

Plotting PV/RT for various gasses as a function of pressure, P:
  • The deviation from ideal behavior is large at high pressure
  • The deviation varies from gas to gas
  • At lower pressures (<10 atm) the deviation from ideal behavior is typically small, and the ideal gas law can be used to predict behavior with little error
Deviation from ideal behavior is also a function of temperature:
  • As temperature increases the deviation from ideal behavior decreases
  • As temperature decreases the deviation increases, with a maximum deviation near the temperature at which the gas becomes a liquid
Two of the characteristics of ideal gases included:
  • The gas molecules themselves occupy no appreciable volume
  • The gas molecules have no attraction or repulsion for each other

Real molecules, however, do have a finite volume and do attract one another

  • At high pressures, and low volumes, the intermolecular distances can become quite short, and attractive forces between molecules becomes significant
    • Neighboring molecules exert an attractive force, which will minimize the interaction of molecules with the container walls. And the apparent pressure will be less than ideal (PV/RT will thus be less than ideal).
  • As pressures increase, and volume decreases, the volume of the gas molecules becomes significant in relationship to the container volume
    • In an extreme example, the volume can decrease below the molecular volume, thus PV/RT will be higher than ideal (V is higher)
  • At high temperatures, the kinetic energy of the molecules can overcome the attractive influence and the gasses behave more ideal
    • At higher pressures, and lower volumes, the volume of the molecules influences PV/RT and its value, again, is higher than ideal
The van der Waals Equation

  • The ideal gas equation is not much use at high pressures
  • One of the most useful equations to predict the behavior of real gases was developed by Johannes van der Waals (1837-1923)
  • He modified the ideal gas law to account for:
    • The finite volume of gas molecules
    • The attractive forces between gas molecules
van der Waals equation:
  • The van der Waals constants a and b are different for different gasses
  • They generally increase with an increase in mass of the molecule and with an increase in the complexity of the gas molecule (i.e. volume and number of atoms)
a (L2 atm/mol2)
He 0.0341 0.0237
H2 0.244 0.0266
O2 1.36 0.0318
H2O 5.46 0.0305
CCl4 20.4 0.1383

Use the van der Waals equation to calculate the pressure exerted by 100.0 mol of oxygen gas in 22.41 L at 0.0°C
V = 22.41 L
T = (0.0 + 273) = 273°K
a (O2) = 1.36 L2 atm/mol2
b (O2) = 0.0318 L /mol
P = 117atm - 27.1atm

P = 90atm

  • The pressure will be 90 atm, whereas if it was an ideal gas, the pressure would be 100 atm
  • The 90 atm represents the pressure correction due to the molecular volume. In other words the volume is somewhat less than 22.41 L due to the molecular volume. Therefore the molecules must collide a bit more frequently with the walls of the container, thus the pressure must be slightly higher. The -27.1 atm represents the effects of the molecular attraction. The pressure is reduced due to this attraction.


  • Chemical kinetics- the area of chemistry that studies the rates of reactions
  • Reaction mechanism- the steps involved in a chemical reaction.
With the understanding of the speed and the process of a reaction, chemists can facilitate reactions to their needs.
Reaction rate is simply the change in the amount of product or reactant over time. The amount is measured in molarity while the change in time is in seconds. Therefore the reaction rate is:
The triangle in front of the letters is called "delta" and represents "change of."
A is the reactant or product, and the enclosed brackets stands for the molarity. Think of the reaction rate as the slope of the concentration vs. time graph.
  • The reaction rate of the reactant is always negative because the reactant is decreasing to form the product.
  • The reaction rate of the product is always positive because the product is being formed.
Ex. N2(g)+ 3H2(g) Þ 2NH3(g)
The reaction rate is: -D [N2 ]/D T. The rate of consumption of hydrogen is 1.5 times as fast as the formation of ammonia because 3 moles of hydrogen will produce 2 moles of ammonia.
A reaction can proceed forward as well as backwards. For now the reverse reaction is considered negligible because the reactions in this tutorial are studied under conditions where the reverse reaction is insignificant.

The rate law, another form of is: Rate = k[A]n
Where k is the rate constant and n is the order, both values must be determined through experiments.
For the rate law to hold true:
  • We assume that the reverse reaction is insignificant.
  • The order, n, is an experimental value.
The rate law of N2(g)+ 3H2(g) Þ 2NH3(g)
Rate = k[N2]n[3H2]m
The overall reaction order is the total order in the equation: m + n
Again, n and m are experimental figures.

There are two types of rate laws.
Differential rate law- relates the rate law with the concentration of reactant. Ex. Rate = k[A]n
Integrated rate law- relates the rate law with time. For those of you who have studied calculus, this law is the integration of the differential rate law. Ex. ln[A]=-kt+[A]0
Three types of rate orders.
0 order
the rate of the reaction is independent of the concentration. The rate is constant.
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Half-life is the amount of time it takes for something to decompose to half of its original amount. With a half-life of 5 days, 1kg of road kill would become 0.5 kg in 5 days.
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First order
the concentration of reaction is directly proportional to the rate. If you triple the concentration, the reaction rate will triple.
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Second order
the reaction rate grows exponentially with concentration. If you double the concentration, the reaction rate quadruples.
Since a reaction occurs in a series of steps, names are given to each step along with the substances involved in the reaction.
O3(g) Þ O2(g) + O(g)
O3(g) + O(g) Þ 2 O2(g)
O(g) in called an intermediate.
  • An intermediate does not qualify as a reactant or product, but its consumed or formed.
  • Each step of reaction is called an elementary step.
  • Molecularity- minimum number of molecules that must collide for the reaction in that step to take place. Here is how you name them:
Number of reactants Name Rate Law
1;[A]Þ products Unimolecular k[A]
2;2[A]Þ products Bimolecular k[A]2
2;[A]+[B]Þ products Bimolecular k[A][B]
3;2[A]+[B]Þ products Termolecular k[A] 2 [B]
3;[A]+[B]+[C]Þ products Termolecular k[A][B][C]
So what does temperature have on the reaction rate? For a reaction to occur, molecules must collide, and raising the temperature increases the movement of molecules. Therefore, as temperature increases, the reaction rate also increases.
But the for a reaction to proceed, a specific amount of energy must first be achieved. This energy is called activation energy. A rise in temperature also lowers the activation energy.
Pg. 572 fig. 12.9
Pg. 573 fig. 12.10
For a reaction to occur:
  • Molecules must produce enough energy to overcome the activation energy.
  • The molecules in the reactants must be orientated in such a manner that old bonds are broken so new bonds in the product can form.
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The most common equation for activation energy is:
Learn this equation, it’s very useful. After all, anybody can plug in numbers and use a calculator.
A catalyst is a substance that speeds up chemical reaction without being consumed in the process. A prime example is the enzymes in the human body. Chemists use catalysts to speed up slow reactions. An example of this is the production of ammonia, know as the Haber Process.
A catalyst helps a reaction by lowering its activation energy; therefor, speeding up the reaction. Since the reaction is very slow on its own, chemists add a catalyst to speed it up.
Two types of catalyst:
Homogeneous-the catalyst exists in the same phase during the reaction.
O3(g) Þ O2(g) + O(g)
O3(g) + O(g) Þ 2 O2(g)
For this reaction, the O(g) is the catalyst.
Heterogeneous-the catalyst enters and exists in different phases of the reaction. The catalyst usually provided the area where gaseous molecules can adsorb on its surface, then it forms the product.
The term adsorption is usually related with heterogeneous catalysts. Adsorption is the collection of a substance on the surface of another.