Chapter 2
Section 3 Continued
Objectives
• Be able to define: accuracy, precision, percent error, significant
figures, scientific notation, conversion factor, hyperbola.
• Be able to distinguish between accuracy and precision.
• Be able to determine the number of significant figures in a
measurement.
• Be able to perform mathematical operations and express the
result in the proper number of significant figures.
• Be able to convert measurements into scientific notation.
• Be able to distinguish between inversely and directly
proportional.
• Be able to perform calculations involving measurements
(addition, subtraction, multiplication, division) and express the
result in the proper number of significant figures and the
proper units.
Significant Figures
***** This is IMPERATIVE: You MUST use
and recognize significant figures when
you work with measured quantities and
report your results, and when you
evaluate measurements reported by
others. So, LEARN THE RULES FOR
DETERMINING SIGNIFICANT
FIGURES.*****
A. Determining the number of
Significant Figures:
• All nonzero digits are significant.
• All zeros between two nonzero digits are significant
• Zeros at the end of a number and to the right of a
decimal point are significant.
• Zeros at the end of a number but to the left of a decimal
point may or may not be significant. If a zero has not
been measured or estimated but is a placeholder, it is
not significant. A decimal point placed after zeros
indicates that they are significant.
B. Rounding Significant
Figures:
• The answers given on a calculator can be
derived results with more digits than are
justified.
• Supposed you use a calculator to
divide 154g by 327g.
• Each of these numbers has three
significant figures but the calculator
will show an answer of 0.470948012 –
• Such an answer has to be rounded off.
Rounding Rules
Rounding Significant
Figures
When performing calculations
involving measurements certain
rules apply whether you are
adding, subtracting, multiplying,
or dividing. FOLLOW THOSE
RULES.
Addition and Subtraction
with Significant Figures:
When adding or subtracting decimals,
the answer must have the same
number of the digits to the RIGHT of
the decimal point as there are in the
measurement having the FEWEST
digits to the right of the decimal point.
Addition and Subtraction
Steps when adding or subtracting:
1. Place all measurements in a column.
2. Find the rightmost place where there is a
digit for each number.
3. Round all measurements to that place, then
add or subtract.
Addition and Subtraction
+
24.6 cm
120.003 cm
4.68 cm
0.006 cm
149.3 cm
Addition and Subtraction
25.1
+ 2.03
cm
cm
27.1
cm
Addition and Subtraction
•When working with WHOLE
numbers, the answer should be
rounded so that the FINAL sig fig
digit is in the SAME PLACE as the
leftmost uncertain digit.
5400 + 365 = 5800
Multiplication and
Division
For multiplication and division, the
answer can have NO MORE significant
figures than are in the measurement
with the FEWEST number of
significant figures
Multiplication and
Division
1.
Count the number of significant figures in each
measurement.
2.
Round your answer to the number of
significant figures in the measurement with the
LEAST number of significant figures.
3.05 g
= 0.360094451 g/mL = 0.360 g/mL
D=
8.47 mL
Scientific Notation
• The numbers we deal with in science can be
extremely small or extremely large.
• Converting to scientific notation or
exponential notation makes handling these
numbers much easier.
Scientific Notation
• In scientific notation, numbers are written in
the form of M x 10n, where the factor M is a
number greater than or equal to 1 but less
than 10 and n is a whole number.
Example: 6.02 x 1023 atoms/mole
Incorrect: 60.2 x 1022 or .602 x 1024
Steps of converting to
scientific notation:
1. Determine M by moving the decimal point in
the original number to the left or right so that
only one nonzero digit remains to the LEFT of
the decimal place.
2. Determine n by counting the number of places
that you move the decimal place.
a. If you move the decimal to the LEFT, n is
POSITIVE
b. If you move the decimal to the RIGHT, n is
NEGATIVE
Mathematical Operations
using Scientific Notation:
Addition and Subtraction:
• These operations can be performed ONLY if the values
have the same exponent (n factor).
• If they do not, adjustments must be made to the values
so that the exponents are EQUAL.
• Once the exponents are equal, the M factors can be
added or subtracted – the exponents can remain the
SAME or it may then require adjustment if the M
factor of the answer has MORE than ONE digit to the
left of the decimal.
Addition and Subtraction
• Convert all numbers to the same power of 10.
• Add or Subtract all the coefficients (M’s).
• Convert the sum to proper notation form.
4.2 x 104 kg
0.79 x 104 kg
4.99 x 104 kg  rounded to 5.0 x 104 kg
Multiplication
• Multiply coefficients (M’s)
• Add exponents
• Convert to proper notation form
(5.23 x 106 um) (7.1 x 10-2 um) = (5.23 x 7.1) (106 x 10-2)
= 37.133 x 104 um2
(adjust to two sig figs and one nonzero number to left
of decimal)
= 3.7 x 105 um2
Division
• Divide coefficients (M’s)
• Subtract exponents
• Convert to proper notation form
5.44 x 107 g
8.1 x 104 mol
=
5.44
x 107-4 g/mol
8.1 mol
= 0.6716049383 x 103 = 6.7 x 102 g/mol
Problem Solving
Analyzing and solving problems is an integral part of
Chemistry. You must follow a logical approach to solving
problems in Chemsitry.
1. ANALYZE
Analyze the Problem - read the problem at least twice to
analyze the information in the problem. If you don’t
understand it read it again.
Problem Solving
2.
PLAN
Develop a plan for solving the problem.
3. COMPUTE
Substitute the data and necessary conversion factors into
the plan you have developed.
4. EVALUATE
Examine your answer to determine whether it is
reasonable.
(Sample Prob. F, pg. 54)
Direct Proportions
• Two quantities are directly proportional to
each other if dividing one by the other gives a
constant value.
Inverse Proportions
• Two quantities are inversely proportional to
each other if their product is constant.
Inverse Proportion
• A graph of variables that are inversely
proportional produces a curve called a
hyperbola.