Chapter 2

Transcription

Chapter 2

INTRODUCTORY CHEMISTRY
Concepts and Critical Thinking
Sixth Edition by Charles H. Corwin
Chapter 2
Scientific
Measurements
by Christopher Hamaker
© 2011 Pearson Education, Inc.
Chapter 2
1
Uncertainty in Measurements
• A measurement is a number with a unit attached.
• It is not possible to make exact measurements,
thus 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
• Let’s 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
possesses.
• Mass is measured
with a balance and is
not affected by
gravity.
• Mass and weight are
not interchangeable.
Chapter 2
5
Mass Versus Weight
• Mass and weight are not the same.
– Weight is the force exerted by gravity on an object.
Chapter 2
6
Volume Measurements
• Volume is the amount of space occupied by a
solid, a liquid, or a gas.
• There are several instruments for measuring
volume, including:
–
–
–
–
–
Graduated cylinder
Syringe
Buret
Pipet
Volumetric flask
Chapter 2
7
Significant Digits
• Each number in a properly recorded measurement
is a significant digit (or significant figure).
• Significant digits express the uncertainty in the
measurement.
• When you count significant digits, start counting
with the first nonzero number.
• Let’s look at a reaction measured by three
stopwatches.
Chapter 2
8
Significant Digits, Continued
• 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 one significant figure.
– 35.1 s has two significant figures.
– 35.08 has three significant figures.
Chapter 2
9
Significant Digits and Placeholders
• If a number is less than 1, 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 1, a placeholder zero is
usually not significant.
• Therefore, 50 cm, 500 cm, and 5000 cm all have
one significant digit.
Chapter 2
10
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 seven
coins on this slide.
Chapter 2
11
Rounding Off Nonsignificant Digits
• 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
12
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 the nonsignificant digits until the final
operation and then round off the answer.
Chapter 2
13
Rounding Examples
• 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 displays 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
14
Rounding Off and Placeholder Zeros
• Round the measurement 151 mL to two significant
digits.
– If we keep two digits, we have 15 mL, which is only
about 10% of the original measurement.
– Therefore, we must use a placeholder zero: 150 mL
• Recall that placeholder zeros are not significant.
• Round the measurement 2788 g to two significant
digits.
– We get 2800 g.
• Remember, the placeholder zeros are not significant, and 28 grams is
significantly less than 2800 grams.
Chapter 2
15
Adding and Subtracting Measurements
• When adding or subtracting measurements, the
answer is limited by the value with the most
uncertainty.
• Let’s add three mass
measurements.
• The measurement 106.7 g has
the greatest uncertainty
(± 0.1 g).
106.7
0.25
+ 0.195
107.145
g
g
g
g
• The correct answer is 107.1 g.
Chapter 2
16
Multiplying and Dividing Measurements
• When multiplying or dividing measurements, the
answer is limited by the value with the fewest
significant figures.
• Let’s 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
17
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) = 23.
• The number 3 is an exponent and indicates that the
number 2 is multiplied by itself 3 times. It is read
“2 to the third power” or “2 cubed”.
• (2)(2)(2) = 23 = 8
Chapter 2
18
Powers of 10
• 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
19
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 10.
The significant digits are expressed as a number
between 1 and 10.
power of 10
D.DD
n
x 10
significant digits
Chapter 2
20
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
21
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.
• Count the number of places the decimal has
moved to the left (22). Add the power of 10 to
complete the scientific notation.
2.68 x 1022 atoms
Chapter 2
22
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.
• Count the number of places the decimal has
moved to the right (7). Add the power of 10 to
complete the scientific notation.
1.40 x 10-7 m
Chapter 2
23
Scientific Calculators
• A scientific calculator has an exponent key (often
“EXP” or “EE”) for expressing powers of 10.
• If your calculator reads 7.45 E-17,
the proper way to write the answer
in scientific notation is 7.45 x 10-17.
• To enter the number in your
calculator, type 7.45, then press
the exponent button (“EXP” or
“EE”), and type in the exponent
(17 followed by the +/– key).
Chapter 2
24
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
25
Unit Factors
• A unit conversion factor, or unit factor, is a ratio
of two equivalent quantities.
• 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
26
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
27
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
28
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 x
 910 days
1 year
Chapter 2
29
Another Unit Analysis Problem
A can of soda 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 = 12 fl oz) and round to two
1 qt
12 fl oz. x
 0.38 qt
32 fl oz.
Chapter 2
30
Another Unit Analysis Problem,
Continued
A marathon is 26.2 miles. What is the distance in
kilometers (1 km = 0.62 mi)?
• Step 1: We want km.
• Step 2: We write down the given:
26.2 mi.
• Step 3: We apply a unit factor
(1 km = 0.62 mi) and round to
three significant figures.
1 km
26.2 mi x
 42 km
0.62 mi
Chapter 2
31
Critical Thinking: Units
• When discussing measurements, it is critical that
we use the proper units.
• NASA engineers mixed metric and English units
when designing software for the Mars Climate
Orbiter.
– The engineers used kilometers rather than miles.
– 1 kilometer is 0.62 mile.
– The spacecraft approached too close to the Martian
surface and burned up in the atmosphere.
Chapter 2
32
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
% 
x 100%
total sample
Chapter 2
33
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 of silver in
sterling silver?
18.5 g silver
x 100 %  92.5% silver
(18.5  1.5) g
Chapter 2
34
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
35
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 x
 11.0 g iron
100 g sample
Chapter 2
36
Chemistry Connection: Coins
• A nickel coin contains 75.0 % copper metal and
25.0 % nickel metal, and has a mass of 5.00
grams.
• What is the mass of nickel metal in a nickel coin?
25.0 g nickel
5.00 g coin x
 12.5 g nickel
100 g coin
Chapter 2
37
Chapter 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
38
Chapter Summary, Continued
• Placeholding 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
39
• 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
• 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 1, and drop all of the
nonsignificant digits.
Chapter 2
40
• 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 x 10n,
where D.DD are the significant figures (and is
between 1 and 10) and n is the power of ten.
Chapter 2
41
• 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
42

Chapter 2

Transcription

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