Chapter
3: Scientific Measurement
We often need
to explain to others the items that we study and observe.
We do this by classifying
the items around us.
In order to do this
we must have a system of measures also used by others.
There are two different
types of measures that chemists are concerned with in their studies.
A qualitative observation is one that can be made
without measurement.
Sucrose is composed
of the elements carbon, hydrogen, and oxygen. This is a qualitative
expression of composition.
A powerful way to
classify matter is by its composition. This is the broadest type of
classification.
Classification
by Composition
When you examine
an unknown piece of stuff, you first ask, “What is it made of?”
After a qualitative
analysis, the next question that you might ask is how much of each of
the elements is present.
A quantitative observation is one that uses measurement.
For sucrose, the
answer to that question is that 100 g of sucrose contains 42.1 g of
carbon, 51.4 g of oxygen, and 6.5 g of hydrogen. This is a quantitative
expression of composition.
Classification
by Composition
Scientific
notation expresses
numbers as a multiple of two factors: a number between 1 and10; and
ten raised to a power, or exponent.
Scientific
Notation
The exponent tells
you how many times the first factor must be multiplied by ten.
When numbers larger
than 1 are expressed in scientific notation, the power of ten is positive.
Scientific
Notation
When numbers larger
than 1 are expressed in scientific notation, the power of ten is positive.
When numbers smaller
than 1 are expressed in scientific notation, the power of ten is negative.
Change the following
data into scientific notation.
The diameter of
the Sun is 1 392 000 km.
Convert
Data into Scientific Notation
B. The density of the
Sun’s lower atmosphere
is 0.000 000 028 g/cm3.
Convert
Data into Scientific Notation
Move the decimal
point to produce a factor between 1 and 10. Count the number of places
the decimal point moved and the direction.
Remove the extra
zeros at the end or beginning of the factor.
Multiply the result
by 10n where n equals the number of places moved.
Convert
Data into Scientific Notation
Remember to add
units to the answers.
When adding or subtracting
numbers written in scientific notation, you must be sure that the exponents
are the same before doing the arithmetic.
Adding and
Subtracting Using Scientific Notation
Suppose you need
to add 7.35 x 102 m + 2.43 x 102
m.
Adding and
Subtracting Using Scientific Notation
You note that the
quantities are expressed to the same power of ten. You can add 7.35
and 2.43 to get 9.78 x 102 m.
If the quantities
are not expressed to the same power of ten, change one of the numbers
to match the power of ten of the other number.
Multiplying and
dividing also involve two steps, but in these cases the quantities being
multiplied or divided do not have to have the same exponent.
Multiplying
and Dividing Using Scientific Notation
For multiplication,
you multiply the first factors.
Then, you add the exponents.
For division, you
divide the first factors. Then, you subtract the exponent of the divisor
from the exponent of the dividend.
Suppose you are
asked to solve the following problems.
Multiplying
and Dividing Numbers in Scientific Notation
Accuracy
and Precision
When scientists
make measurements, they evaluate both the accuracy and the precision
of the measurements.
Accuracy refers to how close a measured value
is to an accepted value.
Precision
refers to how close a series of measurements are to one another.
Accuracy
and Precision
An archery target
illustrates the difference between accuracy and precision.
Accuracy
and Precision
An archery target
illustrates the difference between accuracy and precision.
Percent
error
To evaluate the
accuracy of experimental data, you can calculate the difference between
an experimental value and an accepted value.
The difference is
called an error.
Percent
error is the ratio
of an error to an accepted value.
Percent
error
Scientists want
to know what percent of the accepted value an error represents.
Percent
error
For this calculation,
it does not matter whether the experimental value is larger or smaller
than the accepted value. Only the size of the error matters.
When you calculate
percent error, you ignore plus and minus signs.
Calculating
Percent Error
Calculate the percent
errors. Report your answers to two places after the decimal point.
Calculating
Percent Error
Substitute each
error into the percent error equation. Ignore the plus and minus signs.
Note that the units for density cancel out.
Suppose you calculate your
semester grade in chemistry as 90.1, but you receive a grade of 89.4.
What is your percent error?
0.783%
Question
1
Answer
Assessment Questions
On a bathroom scale, a person
always weighs 2.5 pounds less than on the scale at the doctor’s office.
What is the percent error of the bathroom scale if the person’s actual
weight is 125 pounds?
Question
2
2.00%
Answer
Assessment Questions
Significant
Figures
Often, precision
is limited by the available tools.
Scientists indicate
the precision of measurements by the number of digits they report.
A value of 3.52
g is more precise than a value of 3.5 g.
Significant
Figures
Significant
figures
include all known digits plus one estimated digit.
The digits that
are reported are called significant figures.
Rules for
recognizing significant figures
Non-zero numbers
are always significant.
Zeros between
non-zero numbers are always significant.
All final zeros
to the right ofthe decimal place are significant.
72.3 g has three
60.5 g has three
6.20 g has three
Zeros that act
as placeholders are not significant. Convert quantities to scientific
notation to remove the placeholder zeros.
Counting numbers
and defined constants have an infinite number of significant
figures.
Rules for
recognizing significant figures
0.0253 g and 4320 g each have
three
6 molecules 60 s = 1
min
Applying
Significant Figure Rules
Determine the number
of significant figures in the following masses.
Applying
Significant Figure Rules
Count all non-zero
numbers (rule 1), zeros between non-zero numbers (rule 2), and final
zeros to the right of the decimal place (rule 3). Ignore zeros that
act as placeholders (rule 4).
has five significant figures.
has three significant figures.
Rounding
Off Numbers
Answers should have
no more significant figures than the data with the fewest significant
figures.
In the example for
each rule, there are three significant figures.
Rules for
Rounding Numbers
If the digit
to the immediate right of the last significant figure is less than five,
do not change the last significant figure.
If the digit
to the immediate right of the last significant figure is greater than
five, round up the last significant figure.
Rules for
Rounding Numbers
If the digit
to the immediate right of the last significant figure is equal to five
and is followed by a nonzero digit, round up the last significant figure.
Rules for
Rounding Numbers
If the digit
to the immediate right of the last significant figure is equal to five
and is not followed by a nonzero digit, look at the last significant
figure. If it is an odd digit, round it up. If it is an even digit,
do not round up.
Addition
and Subtraction
When you add or
subtract measurements, your answer must have the same number of digits
to the right of the decimal point as the value with the fewest digits
to the right of the decimal point.
The easiest way
to solve addition and subtraction problems is to arrange the values
so that the decimal points line up.
Addition
and Subtraction
Then do the sum
or subtraction. Identify the value with the fewest places after the
decimal point.
Round the answer
to the same number of places.
Applying
Rounding Rules to Addition
Add the following
measurements: 28.0 cm, 23.538 cm, and 25.68 cm.
Line up the measurements:
Because the digit
immediately to the right of the last significant digit is less than
5, rule 1 applies. The answer is 77.2 cm.
Multiplication
and Division
When you multiply
or divide numbers, your answer must have the same number of significant
figures as the measurement with the fewest significant figures.
Apply Rounding
Rules to Multiplication
Calculate the volume
of a rectangular object with the following dimensions:
length = 3.65 cm width = 3.20
cm height = 2.05 cm
Apply Rounding
Rules to Multiplication
To find the volume
of a rectangular object, multiply the length times the width times the
height.
Because the data
have only three significant figures, the answer can have only three
significant figures.
The answer is 23.9
cm3.
A graph
is a visual display of data.
Graphing
Using data to create
a graph can help to reveal a pattern if one exists.
Circle graphs
A circle graph is
sometimes called a pie chart because it is divided into wedges like
a pie or pizza.
A circle graph is
useful for showing parts of a fixed whole.
The parts are usually
labeled as percents with the circle as a whole representing 100%.
Circle graphs
Bar graph
A bar graph often
is used to show how a quantity varies with factors such as time, location,
or temperature.
In those cases,
the quantity being measured appears on the vertical axis (y-axis).
The independent
variable appears on the horizontal axis (x-axis).
The relative heights
of the bars show how the quantity varies.
Bar graph
Line Graphs
In chemistry, most
graphs that you create and interpret will be line graphs.
The points on a
line graph represent the intersection of data for two variables.
The dependent variable
is plotted on the y-axis.
Remember that the
independent variable is the variable that a scientist deliberately changes
during an experiment.
Line Graphs
Line Graphs
Sometimes points
are scattered, the line cannot pass through all the data points.
The line must be
drawn so that about as many points fall above the line as fall below
it.
This line is called
a best fit line.
Line Graphs
Line Graphs
If the best fit
line is straight, there is a linear relationship between the variables
and the variables are directly related.
This relationship
can be further described by the steepness, or slope, of the line.
If the line rises
to the right, the slope is positive.
Line Graphs
A positive slope
indicates that the dependent variable increases as the independent
variable increases.
If the line sinks
to the right, the slope is negative.
Line Graphs
A negative slope
indicates that the dependent variable decreases as the independent variable
increases.
Either way, the
slope of the graph is constant. You can use the data points to calculate
the slope of the line.
The slope is the
change in y divided by the change in x.
Interpreting
Graphs
An organized approach
can help you understand the information on a graph.
First, identify
the independent and dependent variables.
Look at the ranges
of the data and consider what measurements were taken.
Decide if the relationship
between the variables is linear or nonlinear.
Interpreting
Graphs
If the relationship
is linear, is the slope positive or negative?
If a graph has multiple
lines or regions, study one area at a time.
Interpreting
Graphs
When points on a
line graph are connected, the data is considered continuous.
You can read data
from a graph that falls between measured points.
This process is
called interpolation.
Interpreting
Graphs
You can extend the
line beyond the plotted points and estimate values for the variables.
This process is
called extrapolation.
Why might extrapolation
be less reliable than interpolation?
Assessment Questions
Mount Everest is 8847 m high.
How many centimeters high is the mountain?
884 700 cm
Question
3
Answer
Scientists need
to report data that can be reproduced by other scientists. They need
standard units of measurement.
SI Units
In 1795, French
scientists adopted a system of standard units called the metric system.
In 1960, an international
committee of scientists met to update the metric system.
The revised system
is called the Système Internationale d’Unités, which is abbreviated
SI.
There are seven
base units in SI.
Base Units
A base
unit is a defined unit
in a system of measurement that is based on an object or event in the
physical world.
A base unit is independent
of other units.
Some familiar quantities
that are expressed in base units are time, length, mass, and temperature.
Base Units
The SI base unit
for time is the second
(s).
Time
The frequency of
microwave radiation given off by a cesium-133 atom is the physical standard
used to establish the length of a second.
Time
To better describe
the range of possible measurements, scientists add prefixes to the base
units.
This task is made
easier because the metric system is a decimal system.
The SI base unit
for length is the meter (m).
Length
A meter is the distance
that light travels through a vacuum in 1/299 792 458 of a second.
A vacuum is a space
containing no matter.
A meter, which is
close in length to a yard, is useful for measuring the length and width
of a room.
The SI base unit
for mass is the kilogram
(kg).
Mass
Recall that mass
is a measure of the amount of matter.
A kilogram is about
2.2 pounds. The kilogram is defined by a platinum-iridium metal cylinder.
Not all quantities
can be measured with base units.
Derived
Units
For example, the
SI unit for speed is meters per second (m/s).
Notice that meters
per second includes two SI base units—the meter and the second. A
unit that is defined by a combination of base units is called a derived unit.
Derived
Units
Two other quantities
that are measured in derived units are volume and density.
Volume is the space
occupied by an object.
Volume
The derived unit
for volume is the cubic meter, which is represented by a cube whose
sides are all one meter in length.
For measurements
that you are likely to make, the more useful derived unit for volume
is the cubic centimeter (cm3).
Volume
The cubic centimeter
works well for solid objects with regular dimensions, but not as well
for liquids or for solids with irregular shapes.
The metric unit
for volume equal to one cubic decimeter is a liter
(L).
Volume
Liters are used
to measure the amount of liquid in a container of bottled water or a
carbonated beverage.
One liter has about
the same volume as one quart.
Density is a ratio that compares the mass
of an object to its volume.
Density
The units for density
are often grams per cubic centimeter (g/cm3).
You can calculate
density using this equation:
If a sample of aluminum
has a mass of 13.5 g and a volume of 5.0 cm3, what is its
density?
Density
Insert the known
quantities for mass and volume into the density equation.
Density is a property
that can be used to identify an unknown sample of matter. Every sample
of pure aluminum has the same density.
Scientists use two
temperature scales.
Temperature
Scales
The Celsius scale
was devised by Anders Celsius, a Swedish astronomer.
He used the temperatures
at which water freezes and boils to establish his scale because these
temperatures are easy to reproduce.
Temperature
Scales
He defined the freezing
point as 0 and the boiling point as 100.
Then he divided
the distance between these points into 100 equal units, or degrees Celsius.
Temperature
Scales
The Kelvin scale
was devised by a Scottish physicist and mathematician, William Thomson,
who was known as Lord Kelvin.
A kelvin
(K) is the SI base
unit of temperature.
On the Kelvin scale,
water freezes at about 273 K and boils at about
373 K.
Temperature
Scales
It is easy to convert
from the Celsius scale to the Kelvin scale.
For example, the
element mercury melts at -39oC and boils at 357oC.
To convert temperatures
reported in degrees Celsius into kelvins, you just add 273.
It is equally easy
to convert from the Kelvin scale to the Celsius scale.
For example, the
element bromine melts at 266 K and boils at 332 K.
To convert temperatures
reported in kelvins into degrees Celsius, you subtract 273.
Temperature
Scales
Assessment Questions
What is the volume of chemical
sample that has a mass of 24 g and a density of 6 g/mL?
4 mL
Question
4
Answer
Assessment Questions
Convert the following Celsius
temperatures to Kelvin.
A. 42oC
Question
5
B. 100oC
C. 68oC
Assessment Questions
A. 42oC
B. 100oC
C. 68oC
315 K
373 K
341 K
Answers
Assessment Questions
Convert the following Kelvin
temperatures to Celsius.
A. 345 K
Question
6
B. 240 K
C. 510 K
Assessment Questions
A. 345 K
B. 240 K
C. 510 K
Answers
72oC
-33oC
237oC
Assessment Questions
Simplify the following scientific
notation problems.
Question
7
B.
C.
D.
A.
Assessment Questions
Answers
B.
C.
D.
A.