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You Try It!
How many significant digits are present in each of the following?
________ 1. 6.908 g ________ 2. 81801000 g ________ 3. 893.760 g ________ 4. 56890 cm ________ 5. 8970000 cm ________ 6. 0.000136 cm ________ 7. 345000 cm ________ 8. 0.008710 L ________ 9. 7890.0 L Answers:
1. 4 (rule 2)
2. 5 (rule 5—zeros at the end are place holders) 3. 6 (rule 3)
4. 4 (rule 5) 5. 3 (rule 5)
6. 3 (rule 4—zeros are place holders) 7. 3 (rule 5)
8. 4 (rule 4—first zeros are not significant; rule 3—last zero is significant)
9. 5 (rule 3— last zero is significant; rule 2—middle zero is significant because it is between two significant numbers)
With a basic understanding of significant numbers, you can now understand how to determine significant numbers in calculations. Assume that you measure the density of an unknown substance. Remember the equation for density: D 5 mv. The object had a mass of 25.35 g and a volume of 4.2 cm3. Each of these measurements was taken by some measuring device and was measured as accurately as possible. If you substitute these numbers into the density equation, you get an answer of 6.03571429 g/cm3. But just because the calculator gives all of those digits doesn’t mean that your measurements are accurate. So, where do you draw the line in rounding those long numbers? When measurements are used in calculations, it is important to keep track of the number of significant figures throughout the problem. This assures you that your answer will be no more accurate than the least accurate measurement.
One important consideration for all problems is that the calculations should be completed before you round. Wait until you have an answer, and then round it to the proper place or number of significant figures.
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NOTE
The rules for significant figures are slightly different for addition, subtraction, multiplication, and division.
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Addition and Subtraction
The answer can only be as significant as the least accurate number (least place value).
Examples: 3.245 m 1 3.98765 m 1 5.98 m 1 9 m 5 22 m 6.234 g 2 4.0 g 5 2.2 g
You Try It!
1. 14.2 1 23.89 1 37.891 5 2. 345.178 2 4.58 5 3. 892.5 1 234 1 27.88 5
4. 94.234 2 2.7 5
Answers:
1. 76.0 2. 340.60 3. 1154 4. 91.5
Multiplication and Division
The answer can be only as significant as the least significant number involved (number of significant figures).
Samples: 78.35 m 3 3400 m 5 270 000 m2 56.78 g 4 6.7 ml 5 8.5 g/ml
Numbers that are written using scientific notation are treated the same way. The root portion of each number is what is counted.
Sample: 6.02 3 1023 atoms/mol 3 1.4 mol 5 8.4 3 1023 atoms
You Try It!
1. 3.08 J 3 5.2 s 5
2. 0.075 kg 4 0.030 m 5
3. 4.50 3 1027m 3 6.67 3 1014s21 5 4. 3.00 3 108m s214 6.8 3 1027m 5 Answers:
1. 16 J s
2. 0.0023 kg m, or 2.3 3 1023kg m 3. 3.00 3 108m s21
4. 4.4 3 1014s21
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REVIEWING SCIENTIFIC NOTATION
Many of the numbers you will deal with will either be very large (e.g., Avogadro’s number—
6.02 3 1023) or very small (e.g., Planck’s constant—6.63 3 10234J s). Rather than write these numbers with all of the zeros, it is much easier to use scientific (or exponential) notation:
M 3 10n
Where M is a number equal to or greater than 1 and less than 10, M must have one significant digit to the left of the decimal point. n is any positive or negative integer.
To change a number into scientific notation you must do two things:
Determine M by moving the decimal point so that you leave only one nonzero digit to the left of the decimal.
Determine n by counting the number of places that you moved the decimal point. If you move it to the left, the value of n is positive. If you move the decimal to the right, the value of n is negative.
Sample: Write the following numbers in scientific notation:
105,000,000,000 5 1.05 3 1011 0.00000587 5 5.87 3 1026
You Try It!
Write the following numbers in scientific notation:
1. 400,780,000,000 5 2. 0.00052 5
Answers:
1. 4.0078 3 1011 2. 5.2 3 1024
USING DIMENSIONAL ANALYSIS TO ORGANIZE YOUR WORK
There are a variety of problem-solving strategies that you will use as you prepare for and take the AP test. Dimensional analysis, sometimes known as the factor label method, is one of the most important of the techniques for you to master. Dimensional analysis is a problem-solving technique that relies on the use of conversion factors to change measurements from one unit to another. It is a very powerful technique but requires careful attention during setup. The conversion factors that are used are equalities between one unit and an equivalent amount of some other unit. In financial terms, we can say that 100 pennies is equal to 1 dollar. While the units of measure are different (pennies and dollars) and the numbers are different (100 and 1), each represents the same amount of money. Therefore, the two are equal. Let’s use an example that is more aligned with science. We also know that 100 centimeters are equal to 1 meter. If we express this as an equation, we would write:
100 cm 5 1 m
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Since these represent the same distance, the values can be thought of as equivalent.
Therefore, we can say that:
100 cm 1 m 5 1
And, because each of these values is equal to the same thing, we can also say that:
1 m 100 cm5 1
Since these two values are interchangeable, we now have a conversion factor that can be used to convert between meters and centimeters.
Sample: Convert 455 centimeters to meters.
Answer: First, set up the conversion factor to eliminate the centimeters unit and change it to meters. When you set up a problem like this, always begin by writing down your given (or starting) information. The next step is to set up the conversion factor so that the units in the denominator will cancel the units of the given. This can be accomplished by
455 cm 3 1 m
100 cm, which allows you to cancel units and solve the following problem:
455 cm 3 1 m
100 cm=455 m
100 = 4.55 m
Sometimes, students prefer to use a slightly different setup, which can be especially helpful for longer stoichiometric conversions. Some call this technique the “egg carton” approach because the problem is set up in a grid that has slots to fill in quantities, much like the slots that hold eggs in an egg carton. The same problem above, written in the “egg carton” format, would look like this:
455 cm 1 m
5 4.55 m 100 cm
Not all problems lend themselves to such a linear method, but you can still use the most important elements of this technique if you include units of measure with every quantity and carefully check to be sure that the units are canceling appropriately. To see a more complex example, let’s look at a gas law problem.
Sample: What is the pressure of 2.0 moles of nitrogen gas (N2) that occupies a volume of 1.5 L and is at 298 K?
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Answer: Using the ideal gas equation, we know that PV 5 nRT. We also know that when we solve for P, our answer should come out to be in units of atmospheres (atm). When we set up and solve the problem, all units should cancel to give us atm.
PV 5 nRT, rearrange the equation to solve for P
P =nRT V =
~2.0 mol!
S
0.0821L atmmol KD
~298 K!1.5 L ; canceling units, we see
~2.0 mol!
S
0.0821L atmmol KD
~298 K!1.5 L =48.9316 L atm
1.5 L = 33 atm
LABORATORY COMPONENT
Table 3.1 provides you with a summary of the laboratory experiences and equipment the College Board recommends that you be familiar with. Further discussions of many of the labs will take place within the appropriate chapters. Emphasis will be placed on those labs that are most frequently referred to on the AP test. Table 3.2 provides you with a list of the recommended equipment to help you remember the specific names for each piece. Words like “thingy” and “whatchamacallit” are not highly looked upon by the AP graders and earn no credit.
TABLE 3.1 LABORATORY EXERCISES THAT ARE RECOMMENDED BY THE COLLEGE BOARD
Lab Equipment Used
Determination of the formula of a compound
crucible and cover, tongs, analytical balance, support stand, triangle crucible support, burner
Determination of the percentage of water in a hydrate
crucible and cover, tongs, test tube, analytical balance, support stand, triangle crucible support, wire gauze, burner
Determination of molar mass by vapor density
barometer, beaker, Erlenmeyer flask, graduated cylinder, clamp, analytical balance, support stand
Determination of molar mass by freezing-point depression
test tube, thermometer, pipet, beaker, stirrer, stopwatch, ice
Determination of the molar volume of a gas
barometer, beaker, Erlenmeyer flask, test tubes, graduated cylinder, clamp, analytical balance, thermometer, rubber tubing
Standardization of a solution using a primary standard
pipet, buret, Erlenmeyer flasks, volumetric flask, wash bottle, analytical balance, drying oven, desiccator, support stand, pH meter
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TABLE 3.1 (continued)
Lab Equipment Used
Determination of concentration by acid-base titration, including a weak acid or weak base
pipet, buret, Erlenmeyer flasks, wash bottle, analytical balance, drying oven, desiccator, support stand and clamp, pH meter
Determination of concen-tration by oxidation-reduction titration
pipet, buret, Erlenmeyer flasks, wash bottle, analytical balance, drying oven, desiccator, support stand and clamp, pH meter as millivoltmeter
Determination of mass and mole relationship in a chemical reaction
beaker, Erlenmeyer flask, graduated cylinder, hot plate, desiccator, analytical balance
Determination of the equilibrium constant for a chemical reaction
pipet, test tubes and/or cuvettes, volumetric flask, analytical balance, spectrophotometer (Spec 20 or 21)
Determination of appro-priate indicators for vari-ous acid-base titrations;
pH determination
pipet, Erlenmeyer flasks, graduated cylinder, volumetric flask, analytical balance, pH meter
Determination of the rate of a reaction and its order
pipet, buret, Erlenmeyer flasks, graduated cylinder or gas measuring tubes, stopwatch, thermometer, analytical balance, support stand and clamp
Determination of enthalpy change associated with a reaction
calorimeter (can be polystyrene cup), graduated cylinder, thermometer, analytical balance
Separation and qualitative analysis of cations and anions
test tubes, beaker, evaporating dish, funnel, watch glass, mortar and pestle, centrifuge, Pt or Ni test wire
Synthesis of a coordination compound and its chemical analysis
beaker, Erlenmeyer flask, evaporating dish, volumetric flask, pipet, analytical balance, test tubes/cuvettes,
spectrophotometer Analytical gravimetric
determination
beakers, crucible and cover, funnel, desiccator, drying oven, Meker burner, analytical balance, support stand and crucible support triangle
Colorimetric or spectrophotometric analysis
pipet, buret, test tubes and/or cuvettes, spectrophotometer, buret support stand
Separation by chromatography
test tubes, pipet, beaker, capillary tubes or open tubes or burets, ion exchange resin or silica gel (or filter paper strips, with heat lamp or blow dryer)
Preparation and properties of buffer solutions
pipet, beaker, volumetric flask, pH meter
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TABLE 3.1 (continued)
Lab Equipment Used
Determination of electrochemical series
test tubes and holder rack, beakers, graduated cylinder, forceps
Measurements using electrochemical cells and electroplating
test tubes, beaker, filter flasks, filter crucibles and adapters, electrodes, voltmeter, power supply (battery)
Synthesis, purification, and analysis of an organic compound
Erlenmeyer flask, water bath, thermometer, burner, filter flasks, evaporating dish (drying oven), analytical balance, burets, support stand, capillary tubes
TABLE 3.2 LIST OF EQUIPMENT THE COLLEGE BOARD RECOMMENDS YOU KNOW Analytical Balance
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SUMMING IT UP
• Accuracy is about how close your measurements are to the actual, or true, value.
Precision means that you are consistent in your measurements.
• There are five basic rules for determining whether or not digits are significant. These rules are important to know to earn all possible points during the free-response section of the test. Significant figures do not appear in the multiple-choice portion.
• Scientific notation provides a convenient way to write very large or very small numbers using powers of 10. You should be able to write, interpret, and perform calculations with numbers written using scientific notation.
• Dimensional analysis is a useful technique for organizing information in computations.
By using this technique of canceling units and unit conversion, you can decrease your chances of making careless errors and can improve your score.
• Laboratory-based questions are becoming more common on the AP test. You should review your old labs and look over the examples of the labs in this book. You should also know the names and uses for the equipment listed in this chapter.
While these are some of the tools you will need to continue with the book, they are by no means a comprehensive list. The more techniques you can learn and problem-solving strategies you can use, the more likely you are to experience success on the test problems.
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Atomic Structure
OVERVIEW
• The historical development of modern atomic theory
• Modern atomic theory
• The Periodic Table of the Elements
• The quantum model of the atom
• Quantum numbers
• Electron configurations
• Periodic trends
• Summing it up
Just as the atom is the building block from which all materials are made, this chapter must be the building block upon which the rest of the book is made.
The remaining chapters of this book are devoted to topics directly related to the behavior that can only be understood by first learning atomic structure.
The information in this chapter will provide you with a solid foundation of knowledge upon which to build the remainder of your review. The AP test has a few multiple-choice questions on the topics covered in this chapter and will periodically have one essay question as well. You will have a much easier time on the other sections of this book and the AP test if you can develop a firm understanding of atomic structure and its impact.
THE HISTORICAL DEVELOPMENT OF MODERN ATOMIC THEORY
The earliest known descriptions of atoms date back to between 460 and 370 B.C.E., when the Greek philosopher Democritus first proposed the idea that matter was composed of indivisible particles. He used the term atomos (indivisible) to describe these. Other philosophers, including Plato and Aristotle, opposed Democritus’s ideas, believing that matter was infinitely
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divisible—views that were widely held until the nineteenth century. Table 4.1 summarizes the major contributions to modern atomic theory.
Dalton’s Solid Sphere Model of the Atom
During the period between 1803 and 1807, John Dalton proposed a theory of the atomic nature of matter. Dalton’s four basic postulates stated the following:
All matter is composed of extremely small particles called atoms.
Atoms of like elements are identical, while elements of different elements are different.
Atoms are neither created nor destroyed in chemical reactions, nor do they change forms.
Compounds are formed when atoms of different elements combine. Atoms will combine in certain fixed ratios with other atoms.
Thomson’s “Plum Pudding” Model of Atomic Structure
These ideas remained relatively unchallenged until late in the nineteenth century when work with electricity uncovered additional aspects of atomic structure. The first discovery was that the atom, rather than being indivisible, actually consisted of small subatomic particles. The study of cathode rays uncovered the first subatomic particle, the negatively charged electron.
While studying cathode rays, scientists also observed a second type of ray, the canal ray, which led to the discovery of a positively charged particle (later shown to be a proton). J. J.
Thomson, a leading researcher in this area, proposed the second major model of atomic structure, the “plum pudding” model of the atom (named for a popular English dessert). The atom, he proposed, consisted of a positively charged, spherical mass (the pudding) with negatively charged electrons (raisins) scattered throughout. Another of Thomson’s significant contributions was the determination of the charge-to-mass ratio of the electron. This bit of evidence aided physicist Robert Millikan, in 1909, to determine the mass of the electron in his famous oil-drop experiment.
While Thomson and others were busy studying electrical phenomena, Henri Becquerel discovered a new phenomenon—radiation. (We will discuss radiation in more detail in Chapter 5.) The study of this new type of high-energy emission from materials was the principal focus of Ernest Rutherford. Rutherford’s initial work discovered two new types of particles associated with the high-energy emissions, the alpha (a) particle and the beta (b).
These are now known to be a helium nucleus and an electron, respectively (more on this in Chapter 5).
Rutherford’s Discovery of the Nucleus
In 1910, Rutherford performed his famous gold-foil experiment, the results of which ended the brief reign of Thomson’s plum pudding model. In the experiment, he devised an ingenious apparatus to confirm current ideas about atomic structure. His design involved shooting a beam of alpha particles at a very thin sheet of gold foil. His hypothesis suggested that the distribution of charge and mass throughout a plum pudding atom should allow the positively
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charged alpha particle to blast right through the foil with little or no consequence. To his amazement, there were a significant number of particles that did not pass directly through the foil, but instead were deflected at various angles—some even straight backward! He ultimately concluded that the atoms in the foil must contain an extremely dense, positively charged core, sufficient to deflect the positively charged alpha particles. Rutherford’s model, the third major atomic model, consisted of a dense positively charged nucleus, surrounded by tiny, negatively charged electrons in a large amount of empty space (see Figure 4.1).
Several years later, in 1919, the positively charged particle, the proton, was discovered that makes up the dense nucleus of an atom. It would be another thirteen years before James Chadwick discovered the neutrally charged neutron, the second component of the nucleus.
At this point, scientists knew of three fundamental components of atoms (and hence matter):
the electron, the neutron, and the proton. They knew that protons were located in a dense region in the center of the atom and that they were positively charged. They also knew that protons had a mass nearly 2,000 times greater than that of an electron. In addition, they knew that the charge on an electron was equal in magnitude, but opposite in sign, to a proton (despite its much smaller size). The region outside the dense nucleus was mostly empty space;
however, the electrons were believed to be scattered throughout the empty space. The exact positions and behavior of the electron were still uncertain.
Figure 4.1 Schematic diagram of the setup in Rutherford’s gold-foil experiment
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MODERN ATOMIC THEORY
Planck’s Quantized Energies and Einstein’s Photoelectric Effect
In order to better understand how our existing model of the atom evolved, we must step back to about 1900, when physicist Max Planck discovered an unusual property of atoms. While studying the spectra emitted from glowing objects, Planck concluded that energy could be emitted or absorbed from atoms only in fixed amounts, or quanta. He proposed that this amount of energy (E) was directly proportional to the frequency (n) of the electromagnetic wave. Mathematically, this is expressed in the formula:
E 5 hn, where h, known as Planck’s constant, is 6.63 3 10234J•s
Planck’s work provided Albert Einstein with valuable information that helped him propose the photoelectric effect in 1905. Einstein provided the explanation that when light with certain frequencies struck a metal plate, it could emit electrons from the metal. He explained this by describing radiant energy (such as light) as a stream of tiny packets of energy. These tiny packets of energy behave like a tiny particle containing a fixed amount of energy. These
“particles” of light became known as photons. This discovery created new problems for physicists whose existing models viewed light as a wave. Einstein’s work suggested that this
“wave” also behaved like a particle. This dual nature of light has yet to be completely understood.
Bohr’s Planetary Model of Atomic Structure
The next major modification to Rutherford’s nuclear model of the atom came from Danish physicist Niels Bohr. Bohr was attempting to explain the emission spectrum of hydrogen gas.
At this time, the spectral lines research, coupled with Rutherford’s work, led scientists to believe that electrons may orbit the nucleus much like planets in the solar system orbit the
At this time, the spectral lines research, coupled with Rutherford’s work, led scientists to believe that electrons may orbit the nucleus much like planets in the solar system orbit the