Download Example 1-22

Unit 2: Math of Chemistry
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I. Algebraic Manipulation of equations
Ex. 2-1 Rearrange the following equations to solve for the variable that is in bold/italics.
PV = nRT
D=
Mass
V
E = ½ mv2
v1
v2
=
√ M2
√ M1
II. Rules for Scientific Notation
Scientific Notation – shorthand way of writing very small or very large numbers. May be used for any number EXCEPT (1)
temperature or (2) money.
A. Writing and Converting to scientific notation:
1. Coefficient (mantissa) must be a number: >-10 but <10
2. Exponent (characteristic) is positive if original number is <-10 or >10
3. Exponent is negative if original number is >-1 but <1 (i.e. a decimal)
B. Converting from scientific notation to decimal form:
1. If exponent is +, move decimal right (add zeros if needed)
2. If exponent is -, move decimal left (add zeros to left of coefficient)
C. Calculations in scientific notation: (CALCULATOR WILL DO THIS FOR YOU!!)
1. Addition or Subtraction: Exponent must be the same
2. Multiplication: Multiply the coefficient and add the exponents
3. Division: Divide the coefficient and subtract the exponents
Ex. 2-2: Convert to correct scientific notation.
a) 75100000 ______________________
d)
0.00000231_______________________
b)
-234900 ______________________
e)
0.95000 _______________________
c)
9260 ______________________
f)
-0.00003549 ______________________
Ex. 2-3: Convert to decimal form.
a) 5.39 x 107 ______________________
d) 5.39 x 10-7 _______________________
b) 1.12 x 103 ______________________
e)
c) -2.35 x 105 ______________________
1.12 x 10-3 _______________________
f) -2.35 x 10-5 _______________________
Ex. 2-4: Correct the following incorrect scientific notation.
36.7 x 101 ______________________
d)
97723 x 10-2 ______________________
b) -0.015 x 10-3 ______________________
e)
851.6 x 10-3 ______________________
0.123 x 104 ______________________
f)
966 x 10-1 ______________________
a)
c)
III. USING YOUR CALCULATOR
A. The "EE" or "EXP" key on your calculator allows scientific notation.
B. To enter 1.2 x 10-5, press 1.2 EE -5 (don't press “x10” because your calculator automatically puts
it in when you press EE!)
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Unit 2: Math of Chemistry
C. To enter 2.5 x 102 x 3 x 10-4 press 2.5 EE 2 x 3 EE -4 (answer 7.5 x 10-2)
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D. If 10x key is used parenthesis should be put around EVERY number entered. I suggest you use
the EE key in lieu of the 10x key to avoid errors.
Ex. 2-5: Perform the following mathematical functions and express the answer in correct scientific notation.
a) 3.20 x 103 + 9.77 x 102 = ______________________
b) 3.20 x 103 - 9.77 x 102 = ______________________
c) 3.20 x 103 x 9.77 x 102 = ______________________
d) 3.20 x 103  9.77 x 102 = ______________________
IV. SIGNIFICANT FIGURES – what is significant in a measurement
A. Guidelines for determining the number of Significant Digits (Figures)
1.
2.
3.
4.
5.
Any digit that is not zero is significant.
Zeros between nonzero digits are significant.
Zeros to the left of the first nonzero digit are not significant. These zeros are used to indicate the placement of
the decimal point.
If the number is greater than one, all zeros written to the right of the decimal point count as significant.
Numbers that do not contain decimal points, the trailing zeros (i.e. zeros after the last nonzero digit) may or may
not be significant. We will assume they are not significant.
Pacific Ocean
Present Decimal
Start from left, begin counting
at first non-zero. Count all
numbers – including zeros –
until the end of number
Atlantic Ocean
Absent Decimal
Start from right, begin counting
at first non-zero. Count all
numbers – including zeros –
until the end of number
Ex. 2-6: Give the number of sig figs in each of the following numbers.
a)
250
______________________________
f) 0.000090
_____________________________
b) 3.990
______________________________
g) 9.8801
_____________________________
c)
______________________________
h) 100.
_____________________________
0.0045
d) 4000
______________________________
e)
______________________________
100.0
I) 4 x 10
-3
_____________________________
A. Using Significant Digits with multiplication and/or division.
1. In multiplication and division, the number of significant digits in the final answer is determined by looking that
the original numbers used in the problem. The original number that has the smallest number of significant
figures determines the number of sig figs in the answer.
2. Exact numbers obtained from definitions or by counting number of objects are not used in determining the
number of sig figs in the answer.
Example: If an object has a mass of 0.2553 then 8 of those objects has a mass of 2.042.
Ex. 2-7:
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Divide 0.006750 by 32 and round your answer correctly
Unit 2: Math of Chemistry
Ex. 2-8:
Multiply 278.4 by 25.2 and round your answer correctly.
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The calculator shows an answer of 7015.680 which is too many sig figs. However, when you decide to cut it to 3 sig figs, it will
mean making a cut BEFORE the decimal point and YOU CANNOT DO THIS. The number must first be put in scientific notation
and then a cut is made. 7.015680 x 10 3 = 7.02 x 103.
Would it be OK to make the answer 7020?
Hint: If the cut occurs AT or WITHIN a decimal, you may simply make the cut, round your last digit and throw the rest away.
If the cut occurs BEFORE the decimal, you must first put your answer in scientific notation and then make the cut and round the
last digit retained correctly.
Ex. 2-9:
Multiply 89.5540 by 43.1 and round your answer correctly.
Ex. 2-10:
Divide 3388.01 by the cube root of 457.9 and round your answer correctly.
NOTE: Exact numbers (like counts) do not count in determining the number of sig figs. If I have 5 toes, I
have 5.0 toes, 5.00 toes, etc. Sig Figs are used to indicate the limits of a measurement.
B. Determining Significant Figures using addition/subtraction:
Addition/subtraction problems are rounded to the LEAST NUMBER OF DECIMAL PLACES given in the data.
You do not count significant at all—count decimal places only.
Example 2-11: Solve the following problem on your calculator and round your answer correct (based on decimal places, not
significant figures). It helps to line up the numbers vertically instead of horizontally.
+
67.4432
3.07
245.21
2.9
(b)
318.6232
???
-861.2
+
78.554
-
91.0045
+
0.0203
= ______________________
318.6
Once you have decided where to make the cut in your answer, be sure to look at the number to the immediate right of the last digit
you plan to retain. If this number is 5 or greater, round the last number you retain up 1. If this number is less than 5, retain the last
digit just as it is.
Example 2-12: Perform the following calculation on your calculator and round your answer correctly: Do the addition or
subtraction first because this step may alter the final number of sig figs.
a)
(78.4)(0.00513)
3.2 + 1.4
=
b)
(34.990) (889.345
c)
(3.102x10-4) (4.55 x 10-4) = ______________________
(6 x 103)
d)
(4.78 x 10-2) (-5.9 x 106) = ______________________
(3.02 x 102)
e)
(0.136)2 (0.019) = ______________________
(1.40 x 103)
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-
______________________
766.1) = ______________________
Unit 2: Math of Chemistry
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V. METRIC
Metric Prefixes:
Prefix
The prefixes which you need to memorize NOW are the following
Abbreviation
S.N. equivalent
Giga
G
1 x 109
Mega
M
1 x 106
Kilo
K
1 x 103
Deci
d
1 x 10-1
Centi
c
1 x 10-2
Milli
m
1 x 10-3
Micro

1 x 10-6
Nano
n
1 x 10-9
Pico
p
1 x 10-12
If you memorize the conversions this
way it will help with conversions
because the “1” always goes with the
prefix and the scientific notation goes
with the base unit. For example:
1 Gm = 1 x 109 m
1 nL = 1 x 10-9 L
1 g = 1 x 10-6 g
NO PREFIX IS EVER USED ALONE; IT IS ALWAYS ATTACHED TO A UNIT OF SOME KIND.
Example 2-13:
What is the metric abbreviation for each of the following?
______ (a) one-millionth of a gram
_______ (d) one thousand grams
______ (b) one billion liters
_______ (e) one-thousandth of a meter
______ (c) one-hundredth of a liter
_______ (f) one million grams
Example 2-14:
What is the metric abbreviation for each of the following?
______ (a) 1 x 103 grams
_______ (d) 1 x 10-12 meters
______ (b) 1 x 106 liters
_______ (e) 1/10 of a gram
______ (c) 1000 meters
_______ (f) 1/1,000,000,000 of a liter
VI. DIMENSIONAL ANALYSIS (Factor Label Method of Conversions)
Solving problems in this way of arranging labels so that they can cancel out is known as DIMENSIONAL ANALYSIS. Instead of
writing fractions in parentheses beside each other and multiplying them, we are going to place them on a straight-line dimensional
analysis grid. ALL WORK MUST BE SHOWN TO RECEIVE CREDIT. YOU MUST SET THESE UP AS TAUGHT! We are
teaching a problem solving technique – we care more about the technique than your final answer.
Example 2-15: Convert 3.179 hours to minutes. It is helpful to write the conversion factor as an equality. We can flip the
conversion factor at our convenience in order to cancel units. In this case: 60 min = 1 hr is the same as 1 hr = 60 min.
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Unit 2: Math of Chemistry
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A.
(1)
(2)
(3)
Instructions to set up your dimensional analysis problems:
Draw the dimensional analysis grid like this: ---------|--------Put what is given in the UPPER LEFT corner of the grid (with label)
Whatever label is in the upper left corner must be copied diagonally to the lower right corner.
This is arranging your labels correctly for canceling later.
(4) Put the label of what you are converting TO in the UPPER RIGHT corner.
(5) Fill in a correct numerical relationship which exists between the 2 labels.
Hint: a good rule of thumb is to always put the “1” in front of the prefix and then simply define that
prefix in the other part of the dimension. Don’t make rules about always putting the “1” on the top of the
dimension or on the bottom because this will not work.
(6) Cancel any labels which appear both in the numerator and the denominator of the grid
(7) Multiply everything together that is above the grid line; divide by everything that is below
(8) Express your answer in the same number of significant figures as were given in the ORIGINAL problem
(9) BE SURE YOUR ANSWER HAS A LABEL
Example 2-16: How many eggs are there in 10.25 dozen?
B. Dimensional analysis to convert the metric system units - SINGLE STEP
Example 2-17: Convert 378.4 cm to meters
Example 2-18: Convert 4.32 x 10-4 g to micrograms
Example 2-19: How many picoliters are there in 4.56 x 10-7 liters?
Example 2-20: Convert 88.1 km to meters.
C. Metric — TWO STEP: ALWAYS GO THROUGH THE BASE UNIT!
Example 2-21: Convert 231 mm to nm.
Example 2-22: Convert 5.43 GL to dL.
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Unit 2: Math of Chemistry
Example 2-23: Convert 6.99 x 108 pg to cg.
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Example 2-24: How many micrometers are there in 45.2 kilometers?
D. Metric system to the American system and vice versa.
1 ton =
1 pound =
1 mile =
1 foot =
1 yard =
1 pint =
1 quart =
1 gallon =
1 day =
1 hour =
1 minute =
1 year =
2000
16
5280
12
3
2
2
4
24
60
60
365
pounds
ounces
feet
inches
feet or
cups
pints
quarts
hours
minutes
seconds
days
36
inches
These will be provided
for you on quizzes and
exams BUT you will
speed up your
problem solving time
if you MEMORIZE
them!
THERE ARE ONLY 4 METRIC TO AMERICAN CONVERSIONS TO MEMORIZE and you will be able to
convert any units of length, volume, and mass from one system to the other. They are:
2.54 cm
=
1 inch (length)
1.61 Km
=
1 mile (length)
0.943 L
=
1 quart (volume)
454 grams
=
1 pound (mass)
Example 2-25: Convert 3.92 yards to picometers.
Example 2-26: Convert 1.75 x 106 grams to ounces
Example 2-27: Convert 25.6 tons to mg.
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Unit 2: Math of Chemistry
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E. “DOUBLE” UNIT Conversions: (such as miles/hour being converted to km/sec)
(1)
(2)
(3)
(4)
Put the number and the TOP (numerator) label in the upper left corner of the grid.
Put the bottom (denominator) label in the lower left corner of the grid.
You may begin working with either one—just be sure to arrange labels diagonally so they will cancel.
Work until you get one of the labels to the unit that you want. NOW YOU ARE THROUGH WITH THAT LABEL,
SO JUST LEAVE IT ALONE AND BEGIN WORK ON THE OTHER ONE by arranging labels diagonally and
filling in correct numerical equivalents in each dimension.
Example 2-28: Convert 35 miles/hour to km/sec.
Example 2-29: Convert 45.8 pounds/gallon to ng/microliter.
Example 2-30: Convert 1.023 x 107 pm/day to inches/sec.
F. Converting CUBIC UNITS: (such as cm3 to in3) Squared units are handled the same way.
**You must be VERY CAREFUL about canceling units.
Example 2-31: Convert 70.6 cm3 to m3.
(1) Write what is given in the problem in the upper left part of the grid, with label.
(2) Before you simply copy the cm3 label, stop and think! You do not know one single conversion, which utilizes cubic
units, so transfer just the “cm” label in the lower right corner and write just “m” in the upper right. Go ahead and fill
in the relationship which exists between centimeters and inches, BUT SINCE THE LABLE THAT YOU STARTED
WITH WAS CUBIC, YOU MUST CUB THE ENTIRE DIMENSION (both number and label), TOP AND
BOTTOM, in order to be able to cancel units.
(3) Sometimes it is necessary to cube MORE THAN ONE OF THE DIMENSIONS to get to the label you want, so don’t
panic.
Example 2-32: Convert 35 km3 to cubic feet.
Example 2-33: Convert 231.4 cubic yards to cubic millimeters.
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Unit 2: Math of Chemistry
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G. Challenging problems: (Let’s see what you are made of!)
To solve problems with dimensional analysis:
(1) Read the problem and write down ALL “double label” conversions given or implied.
(2) Begin your dimensional analysis with anything in the problem which is NOT a double label conversion factor
(3) Use each double label conversion factor you have written down only ONCE and arrange the conversion factors on
the dimensional analysis grid so that the labels cancel out.
Example 2-34: Apples sell for $.79 per pound. There are (on the average) 15 apples per 5 pound bag. How much would 20
apples cost?
Example 2-35: What is the cost of 60 onions if 3 onions weigh 1.50 pounds and the price of onions is $0.15 per pound? (give
your answer in dollars and cents and don’t worry about significant figures in this problem.)
Example 2-36: The price of a ream of paper is $2.50. There are 500 sheets of paper in a ream. If a sheet of paper weighs 0.219
oz, what is the price per pound of paper? (give your answer in dollars and cents and don’t worry about significant figures.)
Example 2-37: In a primitive barter society the following exchange rates of exchange exist: 1 fot = 5 vum, 2 sop = 3 tuz, 4 bef =
3 tuz and 9 fot = 2 bef. A man has 4 sop and wants to convert all of his possessions into vum. How many vum can he trade for?
Example 2-38: The price of a dozen bars of soap at Neiman-Marcus is $38. If each bar of soap weighs 6.5 ounces, what is the
price of the soap per pound?
Example 2-29: If pecans sell for $3.00 per pound and there are 27 pecans per pound, how many pecans can you buy for $14.12?
VII. Density: is the mass of a substance per unit volume or how much it weighs per given volume; it is an
intensive property and therefore can be used for identification.
D =
Mass
V
Mass: grams
Remember: 1 cm3 = 1 mL and 1 dm3 = 1 L
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Volume: mL (liquids and solids), L (gases)
Unit 2: Math of Chemistry
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Problem solving rules:
 List givens
 Check units, convert if necessary
 Write down formula – it is a good idea to solve for the unknown before substituting numbers.
 Substitute and solve (i.e. plug and chug!)
There are 3 variables in the equation, and you will be asked to solve for any one of these when the
other two are given.
Example 2-30
A certain solid has a volume of 35.7 cm3 and weighs 85 grams. What is its density?
Example 2-31
If the density of gold is 19.3 g/cm3, what is the volume of 200 g of gold? (10 cm3)
Example 2-32
The density of liquid mercury is 13.6 g/mL. How much would 35.0 mL of mercury weigh? (476 g)
Example 2-33 A liquid fills a graduated cylinder to the 45.8 mL mark. The empty graduated cylinder weighs 26.5 grams and
when it is filled with an unknown liquid to the 45.8 mL mark, the cylinder and the liquid together weigh 70.0 grams.
What is the density of the unknown liquid? (0.950 g/mL)
Example 2-34 Find the density of a rectangular solid whose dimensions are 3.4” by 1.2” by 1.7” if it is known to weigh
500. grams. (4.40 g/cm3)
You must memorize the density of water!



Example 2-35
D(H2O) = 1.0 g/mL
This means that 1.0 gram of water has a volume of 1.0 mL and that 1.0 mL of water will weigh 1 gram.
This is only true at ONE temperature (3.98C); at all other temperatures the density of water is slightly less
than 1.0 g/mL, but it will round to 1.0 g/mL.
How much would 500. mL of water weigh in grams? 1.0 liters of water in grams?
Substances which are LESS DENSE than WATER WILL FLOAT assuming that they do not
dissolve in the water nor react with the water.
Example 2.36 A certain substance has a mass of 3.2 kg and a volume of 8.83 x 10 -3 cubic feet. Would you expect this
substance to float in water, assuming that it did not dissolve in the water nor react with it?
(b)
Would you expect this substance to float in mercury (D = 13.6 g/mL), assuming that it did not dissolve in the mercury
nor react with it?
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