Transcript Chapter 2

Introductory Chemistry:
Concepts & Connections
4th Edition by Charles H. Corwin
Chapter 2
Scientific
Measurements
Christopher G. Hamaker, Illinois State University, Normal IL
© 2005, Prentice Hall
Uncertainty in Measurements
• A measurement is a number with a unit attached.
• It is not possible to make exact measurements,
and all measurements have uncertainty.
• We will generally use metric system units, these
include.
– the meter, m, for length measurements
– the gram, g, for mass measurements
– the liter, L, for volume measurements
Chapter 2
2
Length Measurements
• Lets measure the length of a candy cane.
• Ruler A has 1 cm divisions, so we can estimate the
length to ±0.1 cm. The length is 4.2±0.1 cm.
• Ruler B has 0.1 cm divisions, so we can estimate
the length to ±0.05 cm. The length is 4.25±0.05
cm.
Chapter 2
3
Uncertainty in Length
• Ruler A: 4.2 ±0.1 cm; Ruler B: 4.25 ±0.05 cm.
• Ruler A has more uncertainty than Ruler B.
• Ruler B gives a more precise measurement.
Chapter 2
4
Mass Measurements
• The mass of an object
is a measure of the
amount of matter it
posses.
• Mass is measured with
a balance and is not
affected by gravity.
• Mass and weight are
not interchangeable.
Chapter 2
5
Volume Measurements
• Volume is the amount of space occupied by
a solid, liquid, or gas.
• There are several instruments for measuring
volume, including:
– graduated cylinder
– syringe
– buret
– pipet
– volumetric flask
Chapter 2
6
Significant Digits
• Each number in a properly recorded measurement
is a significant digit (or significant figure).
• The significant digits express the uncertainty in
the measurement.
• When you count significant digits, start counting
with the first non-zero number.
• Lets look at a reaction measured by three
stopwatches.
Chapter 2
7
Significant Digits Cont.
• Stopwatch A is calibrated to seconds (±1 s),
Stopwatch B to tenths of a second (±0.1 s), and
Stopwatch C to hundredths of a second (±0.01 s).
• Stopwatch A reads
35 s, B reads 35.1 s,
and C reads 35.08 s.
– 35 s has 1 sig fig
– 35.1 s has 2 sig figs
– 35.08 has 3 sig figs
Chapter 2
8
Significant Digits and Placeholders
• If a number is less than one, a placeholder zero is
never significant.
• Therefore, 0.5 cm, 0.05 cm, and 0.005 cm all have
one significant digit.
• If a number is greater than one, a placeholder zero
is usually not significant.
• Therefore, 50 cm, 500 cm, and 5000 cm all have
one significant digit.
Chapter 2
9
Exact Numbers
• When we count something, it is an
exact number.
• Significant digit rules do not apply to
exact numbers.
• An example of an exact number:
there are 3 coins on this slide.
Chapter 2
10
Rounding Numbers
• All numbers from a measurement are significant.
However, we often generate nonsignificant digits
when performing calculations.
• We get rid of nonsignificant digits by rounding
off numbers.
• There are three rules for rounding off numbers.
Chapter 2
11
Rules for Rounding Numbers
1. If the first nonsignificant digit is less than 5, drop
all nonsignificant digits.
2. If the first nonsignificant digit is greater than or
equal to 5, increase the last significant digit by 1
and drop all nonsignificant digits.
3. If a calculation has two or more operations,
retain all nonsignificant digits until the final
operation and then round off the answer.
Chapter 2
12
Rounding Numbers
• A calculator displays 12.846239 and 3 significant
digits are justified.
• The first nonsignificant digit is a 4, so we drop all
nonsignificant digits and get 12.8 as the answer.
• A calculator display 12.856239 and 3 significant
digits are justified.
• The first nonsignificant digit is a 5, so the last
significant digit is increased by one to 9, all the
nonsignificant digits are dropped, and we get 12.9
as the answer.
Chapter 2
13
Adding & Subtracting Measurements
• When adding or subtracting measurements, the
answer is limited by the value with the most
uncertainty.
• Lets add three mass measurements.
5
g
• The measurement 5 g has the
greatest uncertainty (±1 g).
• The correct answer is 15 g.
Chapter 2
5.0
g
+ 5.00 g
15.00 g
14
Multiplying & Dividing Measurements
• When multiplying or dividing measurements, the
answer is limited by the value with the fewest
significant figures.
• Lets multiply two length measurements.
 5.15 cm × 2.3 cm = 11.845 cm2
• The measurement 2.3 cm has the fewest
significant digits, two.
• The correct answer is 12 cm2.
Chapter 2
15
Exponential Numbers
• Exponents are used to indicate that a number has
been multiplied by itself.
• Exponents are written using a superscript; thus,
2×2×2×2 = 24.
• The number 4 is an exponent and indicates that
the number 2 is multiplied by itself 4 times. It is
read “2 to the fourth power”.
Chapter 2
16
Powers of Ten
• A power of 10 is a number that results when 10 is
raised to an exponential power.
• The power can be positive (number greater than 1)
or negative (number less than 1).
Chapter 2
17
Scientific Notation
• Numbers in science are often very large or very
small. To avoid confusion, we use scientific
notation.
• Scientific notation utilizes the significant digits in
a measurement followed by a power of ten. The
significant digits are expressed as a number
between 1 and 10.
Chapter 2
18
Applying Scientific Notation
• To use scientific notation, first place a decimal
after the first nonzero digit in the number
followed by the remaining significant digits.
• Indicate how many places the decimal is moved
by the power of 10.
– A positive power of 10 indicates that the decimal
moves to the left.
– A negative power of 10 indicates that the decimal
moves to the right.
Chapter 2
19
Scientific Notation Continued
• There are 26,800,000,000,000,000,000,000
helium atoms in 1.00 L of helium gas. Express
the number in scientific notation.
• Place the decimal after the 2, followed by the
other significant digits.
2.68 × 1022 atoms
• Count the number of places the decimal has
moved to the left (22). Add the power of 10 to
complete the scientific notation.
Chapter 2
20
Another Example
• The typical length between two carbon atoms in a
molecule of benzene is 0.000000140 m. What is
the length expressed in scientific notation?
• Place the decimal after the 1, followed by the
other significant digits.
1.40 × 10-7 m
• Count the number of places the decimal has
moved to the right (7). Add the power of 10 to
complete the scientific notation.
Chapter 2
21
Unit Equations
• A unit equation is a simple statement of two
equivalent quantities.
• For example:
– 1 hour = 60 minutes
– 1 minute = 60 seconds
• Also, we can write:
– 1 minute = 1/60 of an hour
– 1 second = 1/60 of a minute
Chapter 2
22
Unit Conversions
• A unit conversion factor, or unit factor, is a ratio
of two equivalent options.
• For the unit equation 1 hour = 60 minutes, we can
write two unit factors:
1 hour
60 minutes
or
Chapter 2
60 minutes
1 hour
23
Unit Analysis Problem Solving
• An effective method for solving problems in
science is the unit analysis method.
• It is also often called dimensional analysis or the
factor label method.
• There are three steps to solving problems using
the unit analysis method.
Chapter 2
24
Steps in the Unit Analysis Method
1. Write down the unit asked for in the answer
2. Write down the given value related to the answer.
3. Apply a unit factor to convert the unit in the
given value to the unit in the answer.
Chapter 2
25
Unit Analysis Problem
• How many days are in 2.5 years?
• Step 1: We want days.
• Step 2: We write down the given: 2.5 years.
• Step 3: We apply a unit factor (1 year = 365 days)
and round to two significant figures.
365 days
2.5 years 
 910 days
1 year
Chapter 2
26
Another Unit Analysis Problem
• A can of Coca-Cola contains 12 fluid ounces.
What is the volume in quarts (1 qt = 32 fl oz)?
• Step 1: We want quarts.
• Step 2: We write down the given: 12 fl oz.
• Step 3: We apply a unit factor (1 qt = 32 fl oz) and
round to two significant figures.
1 qt
12 fl oz. 
 0.38 qt
32 fl oz.
Chapter 2
27
Another Unit Analysis Problem
• A marathon is 26.2 miles. What is the distance in
yards (1 mi = 1760 yards)?
• Step 1: We want yards.
• Step 2: We write down the given: 26.2 miles.
• Step 3: We apply a unit factor (1 mi = 1760 yards)
and round to three significant figures.
1760 yd
26.2 mi 
 46,100 yd
1mi
Chapter 2
28
The Percent Concept
• A percent, %, expresses the amount of a single
quantity compared to an entire sample.
• A percent is a ratio of parts per 100 parts.
• The formula for calculating percent is shown
below:
quantity of interest
%
 100%
total sample
Chapter 2
29
Calculating Percentages
• Sterling silver contains silver and copper. If a
sterling silver chain contains 18.5 g of silver and
1.5 g of copper, what is the percent silver in
sterling silver?
18.5 g silver
 100%  92.5% silver
(18.5  1.5) g
Chapter 2
30
Percent Unit Factors
• A percent can be expressed as parts per 100 parts.
• 25% can be expressed as 25/100 and 10% can be
expressed as 10/100.
• We can use a percent expressed as a ratio as a
unit factor.
4.70 g iron
– A rock is 4.70% iron, so
100 g of sample
Chapter 2
31
Percent Unit Factor Calculation
• The earth and moon have a similar composition;
each contains 4.70% iron. What is the mass of
iron in a lunar sample that weighs 235 g?
• Step 1: We want g iron.
• Step 2: We write down the given: 235 g sample.
• Step 3: We apply a unit factor (4.70 g iron = 100 g
sample) and round to three significant figures.
4.70 g iron
235 g sample 
 11.0 g iron
100 g sample
Chapter 2
32
Summary
• A measurement is a number with an attached unit.
• All measurements have uncertainty.
• The uncertainty in a measurement is dictated by
the calibration of the instrument used to make the
measurement.
• Every number in a recorded measurement is a
significant digit.
Chapter 2
33
Summary Continued
• Place holding zeros are not significant digits.
• If a number does not have a decimal point, all
nonzero numbers and all zeros between nonzero
numbers are significant
• If a number has a decimal place, significant digits
start with the first nonzero number and all digits
to the right are also significant.
Chapter 2
34
Summary Continued
• When adding and subtracting numbers, the answer
is limited by the value with the most uncertainty.
• When multiplying and dividing numbers, the
answer is limited by the number with the fewest
significant figures.
• When rounding numbers, if the first
nonsignificant digit is less than 5, drop the
nonsignificant figures…If the number is 5 or
more, raise the first significant number by one and
drop all of the nonsignificant digits.
Chapter 2
35
Summary Continued
• Exponents are used to indicate that a number is
multiplied by itself n times.
• Scientific notation is used to express very large or
very small numbers in a more convenient fashion.
• Scientific notation has the form D.DD × 10n,
where D.DD are the significant figures (and is
between 1 and 10) and n is the power of ten.
Chapter 2
36
Summary Continued
• A unit equation is a statement of two equivalent
quantities.
• A unit factor is a ratio of two equivalent
quantities.
• Unit factors can be used to convert measurements
between different units.
• A percent is the ratio of parts per 100 parts.
Chapter 2
37