Name: ___________________
Period: _____
Honors Chemistry Cram Session
Honors PS Packet
Chapter 1
Mathematical Concepts
Part I Expressing Numbers that are Very large or Very Small
1. Scientific Notation
In the study of chemistry we often encounter numbers that are very large or very
small. It is more convenient to work with these numbers if we express them in scientific
notation. This means that we move the decimal point to a position immediately to the right of
the first non-zero digit and use a power of ten to indicate how many places the decimal point
has been moved. Thus any number expressed in scientific notation has the following format:
Remember that the decimal point must always end up immediately to the right
of the first digit in scientific notation. Let's look at some numbers and see how they
are converted to scientific notation,
Notice that... In scientific notation the first part is
always a number between one and ten.
Notice that... In scientific notation the exponent of the
power of ten indicates how many places the decimal point has
been moved.
Notice that... If the original number was large (greater than one), the exponent is positive. If the
original number was small (less than one), the exponent is negative.
Problems
Copy each problem into your spiral notebook and convert each number into scientific
notation.
1.
2.
3.
23,000
0.0045
241
4.
5.
6.
14.9
0.082
0.92
7.
8.
9.
121.1
40,000
0.00016
Copy each problem into your notebook. Then get rid of the scientific notation and
express each as a regular number.
1.6 x 10-4
9.74 x 107
10.
11.
12.
13.
4 .0 8 8 x 1 0 5
6.05 x 10-9
2.
Multiplication
When we multiply numbers in scientific notation, we multiply the first part of each number as
usual and add the exponents of the power of ten. Look over the following sample problems to see
how this is done.
Sample Problem 1
Solution
Multiply:
1. Multiply 2 x 3 = 6
2. Add 4 + 5 = 9
Sample Problem 2
Solution
(2 x l04) (3 x 105)
the answer... 6 x 109
Multiply:
1. Multiply 4x2 = 8
2. Add (-5) + (+7) = +2
(4 x 10-5) (2 x 107)
the answer... 8 x 102
3.
Division
When we divide numbers in scientific notation, we divide the first part of each number as
usual and subtract the exponents of the power of ten (the top exponent minus the bottom
exponent). Look over the next sample problems to see how this is done.
Sample Problem 3
Solution
1. Divide 6  3 = 2
2. Subtract 9 – 5 = 4
Divide:
6 x 109
3 x 10 5
the answer... 2 x 104
Divide: 2 x 103
Sample Problem 4
4 x 10-9
Solution
1. Divide 2 + 4 = 0.5
2. Subtract (+3) - (-9) = +12
3. Move the decimal point to the standard position and adjust the power of ten
accordingly.
0.5 x 1012
the answer... 5 x 1011
Remember... the decimal point must end up in the standard position, immediately to the right
of the first digit. If it does not, it must be moved to the standard position. Then adjust the
power of ten to compensate as follows:
Sample Problem 5
Solution
Divide:
1. Divide 2  8 = 0.25
2. Subtract (+3) - (+5) = -2
3. Move the decimal point to the
standard position and adjust
the power of ten accordingly.
2 x 103
8 x 105
0.25 x 10-2
the answer... 2.5x 10
-3
Problems
Move the decimal point to its standard position and adjust the power of ten accordingly.
14.
15.
16.
0.85 x 105
42 x 108
16x104
17.
18.
19.
0.55 x 10 -23
0. 7 x 1 0 4
0.036 x 108
20. 248 x 10 -6
21. 0.6x102
22. 95x10-1
Problems
Copy each problem into your notebook and do each calculation as indicated. Show any
intermediate steps. Make sure your answers are expressed in correct scientific notation with
the decimal point immediately to right of first digit.
Problems
Copy each problem into your notebook and do each calculation as indicated. Show
any intermediate steps. Make sure your answers are expressed in correct scientific
notation.
Part II Expressing Accuracy in Measurements and Calculations
4.
Accuracy
Most of the numbers we work with in chemistry are measurements. In a mathematics class,
the numbers which you have worked with are usually assumed to be exact and therefore you were only
concerned with their value.
But in science, since our numbers are measurements, each one has not only a particular value
but also a particular accuracy. The accuracy of a measurement depends upon two things: the size
of the measurement and its precision. Let's first look at the precision of a measurement, how it is
determined, how it is expressed, and how it effects calculations.
5.
Precision
The precision of measurement depends upon the precision of the instrument used to make
the measurement. In turn the precision of the instrument is determined by the smallest division
readable on its scale. For example, if you measure several objects with the same meter stick
which can be read to the nearest tenth of a centimeter, you might get measurements like the
following: 2.5 cm, 5.8 cm, and 0.7 cm ---- all with the same precision )(i.e. accurate to the
nearest tenth of a centimeter).
Everyone has used a graduated cylinder to measure the volume of a liquid. Let's look at the
difference in precision in the readings obtained using two different sizes of graduated cylinders.
Size 1
The smallest division readable on this size
graduated cylinder is tenths of a milliliter. Thus all of
the readings on this instrument will be accurate to
the nearest tenth of a milliliter.
When the smallest division on the instrument is a tenth of a
milliliter, ALL the measurements on that instrument must be made
to a tenth of a milliliter!! If the reading lines up on a milliliter
mark, like the third one above, a zero must be supplied for the
tenths column to record the measurement to the correct
precision.
Size II
The smallest division readable on this size graduated cylinder is milliliters. All of the readings on this
instrument will be accurate to the nearest one milliliter.
Precision Defined
Thus we express the precision of a measurement by
writing it out to the smallest division readable on the instrument
used to make that measurement. This means expressing the
measurement to the proper column (e.g. tens, ones, tenths,
hundredths, etc.).
You might notice that the third graduated cylinder above reads 10. mL rather than 10 mL.
The decimal point in this reading tells us that the ones column has been measured even though it
came out to zero. Without the decimal point, the zero would be just a place holder and would not
indicate the precision to which the measurement was made.
Trailing Zeros Indicate Precision Only if a Decimal Point is Shown
40
40.
300
300.
35,000
is measured to the nearest
is measured to the nearest
is measured to the nearest
is measured to the nearest
is measured to the nearest
ten
one
hundred
one
thousand
Problems
Copy each problem into your notebook and indicate the precision of each number (to
the nearest: hundred, ten, one, tenth, hundredth, etc.).
42.
42.796
46.
19
50.
6,000.
43.
0.72
47.
19.0
51.
200
44.
1.000
48.
400.0
52.
460
45.
145.8
49.
3,000
53.
30.0
When doing calculations involving measurements, the answer can only be as accurate as the
least accurate measurement made. For addition and subtraction this can be expressed in the
following rule:
Sample Problem 7
Solution
Sample Problem 8
Solution
Problems
Subtract:
246.58
- 87.3
Calculator answer ------------------>
159.28
Rounded to the correct precision -->
(to the nearest tenth)
159.3
Add:
4.300.
+928.6
Calculator answer ------------------>
Rounded to the correct precision -->
(to the nearest hundred)
5,228.6
5,200
Copy each problem into your notebook, do the addition or subtraction indicated, and round
your answer to the correct precision.
7. Significant Figures
The significant figures are the digits in a number which represent the accuracy of that
number. All non-zero digits in a number are significant. But zeros may be just "place
holders". The following two examples show the use of place holders in numbers.
0.085 This number has an accuracy of two significant figures.
In this number the "8" and "5° are measured digits and are therefore significant. The zero
is just a place holder that shows the position of the decimal point; it is not a significant
figure.
400 This number has an accuracy of one significant figure.
Trailing zeros are often only place holders. In this number the zeros are there to show that
the "4" is in the hundreds column. Since no decimal point is shown, the zeros have not been
measured and are not significant.
For the beginner, it is often difficult to decide which digits are significant and which
are not. For this reason, it is best to follow strictly a set of rules for determining the number
of significant figures in any number.
Rules for Determining Significant Figures
1. All non-zero digits are significant.
2. Zeros to the left of non-zero digits are NEVER significant.
3. Zeros between non-zero digits are ALWAYS significant.
4. Zeros to the right of non-zero digits are significant ONLY if a decimal
point is shown.
'Notice that the terms left, between and right refer to the placement of the zeros in
relationship with non-zero numbers NOT in relationship with the decimal point.
All non-zero digits are always significant. The following examples illustrate the
rules shown above as they apply to zeros:
rule 2
rule 3
rule 4
number
sig figs
number
sig figs
007
1
408
3
0.025
2
7.002
4
0.09
1
30.7
3
0.0081
2
50,009
5
number
sig figs
600
8,500
1
2
30.0
46,000.
3
5
Problems
Copy each problem into your notebook. Indicate the number of
significant figures and list the rules (by number) that apply to each.
66.
247
71.
0.3
76.
200
67.
68.
69.
70.
2.47
4,105
0.1002
250
72.
73.
74.
75.
0.0074
8.00
62.000
0.030
77.
78.
79.
80.
0.04030
0.00007
3,000.
1,200
8. Multiplication and Division
When multiplying of dividing numbers you must count the number of significant figures in each
number and round off the answer to the same number of significant figures as the least accurate
number.
Sample Problem 9
Solution
Multiply:
Count significant figures ---------------------->
(34.0) (.0921084)
3 S.F. 6 S.F.
Calculator answer --------------------------->
3.1316856
Rounded to the correct accuracy ----------->
(to 3 significant figures)
Sample Problem 10
Solution
3.13
534.168
0.07
Divide:
Count significant figures ----------------------->
6 sig figs
1 sig fig
Calculator answer ----------------------------->
76.30971
Rounded to the correct accuracy --------->
80
(to 1 significant figure)
Problems
Copy each problem into your notebook. Label each number as to how many significant
figures it contains. Write down your calculator answer and then the answer rounded to the
correct accuracy.
81.
(16.00) (.617289)
85.
(560) (0.0031842)
82.
65.431
0.003
86.
(0.050002) (406)
83.
(.030040) (78.00000)
87.
30.5
0.050817
84.
0.8000
0.20
88.
128
16
9. Summary of Methods
The most common mistake is to mix up the two methods we have learned for rounding.
Here are the two rules we have learned:
Mixed Problems
Copy each problem into your notebook and show both your calculator answer and
the final answer (rounded correctly). Be careful to use the right method for each problem.
10.
Rounding Off
When rounding off numbers, if what you drop off is greater than five, round up; if
what you drop off is less than five, leave what remains alone. This rule will take care of 99%
of all situations. But on the rare occasion in which the part you drop off is exactly five, round
what remains to an even number.
These three situations are illustrated below as each number is rounded to two
significant figures.
The special "round even" rule is used rarely since even a number such as 28.500001 will be
rounded up to 29 (if rounded to two significant figures).
This ruler illustrates how numbers that are exactly halfway round off to an even number.
11.
Counting Numbers
All of our discussion of accuracy and rounding has been directed to numbers that are
measurements, as most of the numbers we deal with are. But occasionally we work with counting
numbers. Numbers that have been arrived at by counting (rather than by measuring) are exact
and have unlimited significant figures. To decide if a number is a counting number you must
relate it to the context of the problem as illustrated in the next sample problem.
Sample Problem 11 A new penny weighs 3.8 grams. How much do 7 new pennies weigh?
1.
Multiply --------------------------------------->
2.
Number of Significant figures ------------->
3. Calculator answer --------------------------->
3. Rounded off to the correct accuracy ----->
(to 2 significant figures)
(3.8) (7)
2 unlimited
26.6
27
12. Using Scientific Notation to Express the Correct Number of Significant Figures
Once in a while we come across a number whose accuracy cannot be expressed
correctly as a regular number. ---- For example: the number two hundred, if measured on an
instrument which measures to the nearest ten, must be expressed as two significant figures.
105. (6.50) (330.)
106.
6650
700
107. (25) (32)
108.
2048
.256
109. A toothpick weighs 1.45 grams. How much do 70 toothpicks weigh?
110. A sample of metal is measured and found to be 37.42 inches in length. What. is its
length in feet? (Twelve inches are "defined" to equal one foot. This is an example of a
counting number, not a measurement.)