General Chemistry Unit 1: Measurement and Atomic Theory
Measurement
Measurement is the process of gaining quantitative information about the
physical world. The science of measurement is closely linked to the
development of technology because more advanced technologies demand
better systems of measurement. A farmer may be satisfied to know the
length of a furrow to within one meter, but an electronic engineer may
require a tolerance of one billionth of a meter when designing an
integrated circuit.
All measurements are approximations. Suppose we use the two metric
rulers below to measure the length of a metal rod. The bottom ruler is
graduated in centimeters while the top is graduated in millimeters. Using
the cm-ruler we can see that the length of the rod is between 2 and 3
centimeters and the best we can do is estimate that the length of the
rod is about 50% of the distance from the 2-centimeter mark to the 3centimeter mark. Our best guess is 2.5 cm. The first digit of our answer
is certain (it is clearly more than 2 and less than 3), but the second digit,
an estimate, is uncertain. Using the centimeter ruler we measure the
length of the rod to two significant digits (one measured plus one
estimated).
Using the millimeter ruler, we can see that the end of the rod is about
60% of the distance between the 2.5 and 2.6 marks. Using this ruler we
estimate the length of the rod to be 2.56 cm. The first two digits of the
answer are certain (it is clearly more than 2.5 and less than 2.6) but the
last digit is an estimate and is uncertain. Using the millimeter ruler we
measure the length of the rod to three significant figures (two measured
plus one estimated). Whenever you make measurements, make sure that
you include only significant digits: all measureable plus one estimated.
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General Chemistry Unit 1: Measurement and Atomic Theory
Activity 1: Estimating length
To think in metric units, it is helpful to have some common benchmarks.
Use a meter stick and a ruler to find common objects that are
approximately 1mm, 1cm and 1m in length. For example , the tip of a
sharpened pencil may be approximately 1mm in diameter and the width of
your fingernail may be about 1cm.
Table 1: Benchmarks for “length”
Metric Unit
mm
cm
m
Approximately the same length as:
Using these benchmarks, estimate the following distances in the
appropriate metric unit. These are referred to as experimental values.
After you have written your estimates, measure the distances(we will
refer to these as theoretical values) and calculate the percent error for
each
Percent Error = lexperimental value – theoretical valuel
theoretical value
x 100
Table 2: Estimates of Length
Length of room
Width of room
Your height
Height of door
Length of pencil
Thickness of 50 sheets of
paper
Estimate Measurement Percent
Error
(m)
(m)
(m)
(m)
(cm)
(cm)
(cm)
(cm)
(mm)
(mm)
(mm)
(mm)
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General Chemistry Unit 1: Measurement and Atomic Theory
Activity 2: The importance of measuring
We have the ability to use our senses to estimate measurements but it is
important to recognize that this method has limitations. Our senses can
be fooled and in science, we need to support our estimates with accurate
measurements.
For each of the illustrations below, answer the questions in the first
column of Table 3:
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General Chemistry Unit 1: Measurement and Atomic Theory
Table 3: Estimating and measuring distance
Assumption based
on observation
Which appears the
tallest?
A
B
C
Which line appears
longer?
DF
GE
Do the tracks
appear parallel?
YES
NO
Which line appears
longer?
KL
LM
Which line appears
longer?
NO
PQ
Measurement to the nearest tenth of a mm
Pole A
Pole B
DF
GE
Width at H
Width at I
KL
LM
NO
PQ
Pole C
Width at J
Conclusion based on
measurement
Which pole if the
tallest?
A
B
C
Which line is
longer?
DF
GE
Are the tracks
parallel?
YES
NO
Which line is
longer?
KL
LM
Which line is
longer?
NO
PQ
Once you have completed the first column, measure the length of each
line as accurately as possible and record in the table.
Were your original assumptions correct? Why or why not?
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General Chemistry Unit 1: Measurement and Atomic Theory
Activity 3: Determining the density of liquids
In this lab activity, you will determine the density of water, ethanol and
oil. It is most important that you practice careful lab skill and take very
accurate measurements.
Density is the amount of matter in a given volume. It is often measured in
units of g/ml.
Procedure:
1. Record room temperature
2. Mass an empty graduated cylinder and record the measurement.
3. Carefully fill the cylinder with water until it reads 10ml, using a
dropper to help when you are near the 10ml mark.
4. Mass the cylinder with the water and record this measurement.
5. Repeat steps 2 to 4 for ethanol and then again for oil.
6. Calculate the density of each liquid, showing all calculations.
Density Predictions
For this next step, it is most important that your group
makes a prediction before moving on to the experiment.
1. Predict the density of a mixture of 5ml of ethanol and 5ml of water.
2. Explain your thinking. Draw a diagram to illustrate how the density of
the mixture will compare to each of the original solutions.
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General Chemistry Unit 1: Measurement and Atomic Theory
3. When you have completed your hypothesis and explanation and
diagrams, proceed to mix the two substances.
4. Record all observations and measurements and calculations below.
5. How did your measured density compare to your predicted density? If
they are different, come up with a reasonable explanation for the
measured density?
Error and Uncertainty.
A. Percent Error – the density of water varies with temperature.
Below is a table listing the literature values for the density of
water at various temperatures. Look up the literature value for the
density of water at the room temperature you recorded.
Temperature (C)
17.0
18.0
19.0
20.0
21.0
22.0
23.0
Density (g/ml)
0.99880
0.99862
0.99843
0.99823
0.99802
0.99780
0.99757
Temperature (C)
24.0
25.0
26.0
27.0
28.0
29.0
30.0
Density (g/ml)
0.99733
0.99708
0.99681
0.99654
0.99626
0.99598
0.99568
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General Chemistry Unit 1: Measurement and Atomic Theory
Literature (theoretical) value for the density of water =
Your calculated experimental value for the density of water =
Find your percentage error by using the following formula:
% Error =
Theoretical value - Experimental value
Theoretical Value
x 100
If the percent error is large, then we conclude that the experimental
procedure was flawed and we need to work out which parts of the
procedure could be changed to produce to a result closer to the
literature value. Such errors in the experimental procedure are called
Systematic Errors.
B. Uncertainty – Because all physical measurements are estimates, there
is always some uncertainty about the final figure in a recorded
measurement. For example, imagine trying to measure the length of the
following line segment using a cheap metric ruler:
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General Chemistry Unit 1: Measurement and Atomic Theory
Is the length of the line between 4 and 5 cm? Yes, definitely. Is the
length between 4.0 and 4.5 cm? Yes, it looks that way. But is the length
4.3 cm? Is it 4.4 cm?
Given the precision of the ruler and our ability to estimate where
between a set of marked graduations a measurement falls, we are
somewhat uncertain about what number to record after the decimal. So,
what we can say is that the actual length is around 4.4 cm, but it might be
closer to 4.3 cm, or it might be closer to 4.5 cm. In other words, we think
the length is 4.4 cm but we might be off by 0.1 cm in either direction. We
would record this measurement in this way:
Length of line = 4.4 ± 0.1 cm
If you measured a mass and found it to be 2.0000 ± 0.0001g, it would be
wrong to record the mass as:
2 g (Wrong)
Activity 4:
Recording Uncertainty
There are several stations around the classroom and at each station you
will be asked to take a physical measurement, determine the uncertainty
in that measurement and record the measurement and its uncertainty
appropriately.
Station
Object to be measured
1
Volume of water in 25ml cylinder
2
Volume of water in 100ml cylinder
3
Mass of beaker using triple beam
balance
4
Mass of beaker using electronic balance
5
Length of object using 30cm ruler
6
Length of object using meter rule
Recorded measurement with
uncertainty and unit
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General Chemistry Unit 1: Measurement and Atomic Theory
Determining uncertainty in a calculated value
In the experiment to determine the density of water, you needed to
divide the mass of water by the volume of water. Each of those
measurements has a recorded uncertainty. Below are some sample
measurements for the mass and volume of water:
Mass of water = 5.67 ± 0.01 g
Volume of water = 10.0 ± 0.1 ml
The density would then be calculated as
Mass/volume = 5.67 /10.0 = 0.567 g/ml
BUT what do we record as the uncertainty in the density of water?
A simple rule to follow when dealing with uncertainty in calculated values
is as follows:
When measurements are added or subtracted in a calculation, then the
uncertainty values are simply added.
When measurements are multiplied or divided in a calculation, then we
cannot just add the uncertainty values. Instead we must determine the
percent uncertainty in each measurement and then these are added
together to give the uncertainty in the final calculated value.
So in summary,
If Calculation is:
Then:
+
or
-
Add uncertainty
÷
or
×
Add percent uncertainty
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General Chemistry Unit 1: Measurement and Atomic Theory
Activity 5: Uncertainty in a calculated value
Using your own results for the density of water from Activity 3, and the
information on the previous page, determine the uncertainty in the
calculated density of water.
Which is greater for your results, the percent error or the
percent uncertainty? This is a helpful question for any
experimental procedure. Systematic errors in the procedure
contribute to percent error. Random errors in measurement
readings contribute to percent uncertainty.
Think of this –
If there is an error in the experimental procedure, and you
simply repeat the experiment, that error remains and may
actually give a result that is further away from the theoretical
value.
So why do we do repeats in experimental procedures? We
know it won’t help to reduce systematic error. But repeating a
procedure may help to reduce the random error that comes
about from taking any measurement.
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General Chemistry Unit 1: Measurement and Atomic Theory
Two concepts that relate to this topic of measurements are accuracy and
precision.
The accuracy of the measurement refers to how close the measured
value is to the true or accepted value. For example, if you used a balance
to find the mass of a known standard 100.00 g mass, and you got a
reading of 78.55 g, your measurement would not be very accurate. One
important distinction between accuracy and precision is that accuracy can
be determined by only one measurement, while precision can only be
determined with multiple measurements.
Precision refers to how close together a group of measurements
actually are to each other. Precision has nothing to do with the true or
accepted value of a measurement, so it is quite possible to be very
precise and totally inaccurate.
Check your understanding by working through the examples below.
Justify each of your answers fully.
1. A metal rod about 10 cm long has been passed around to several groups
of students. Each group is asked to measure the length of the rod. Each
group has five students and each student independently measures the rod
and records his or her result.
Student
Group
Group A
Group B
Group C
Group D
Group E
Student 1 Student 2 Student 3 Student 4
10.1
10.135
12.14
10.05
10
10.4
10.227
12.17
10.82
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9.6
10.201
12.15
8.01
10
9.9
10.011
12.14
11.5
10
Student 5
10.8
10.155
12.18
10.77
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Which group has the most accurate measurement?
Which group has the most precise measurement?
Which group has the greatest error?
Which group has the greatest uncertainty?
2. We now receive a report from the machine shop where the rod was
manufactured. This very reputable firm certifies the rod to be 10.160 cm
long to the nearest thousandths of a cm. Answer the questions below
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General Chemistry Unit 1: Measurement and Atomic Theory
given this new information. Note that the questions are slightly different.
Which group has the least accurate measurement?
Which group has the least precise measurement?
Which group has the smallest error?
Which group has the smallest uncertainty?
3. The area of a rectangular metal plate was found by measuring its
length and its width. The length was found to be 5.37 ± 0.05 cm. The
width was found to be 3.42 ± 0.02 cm. What is the
area and the uncertainty in the area?
Answers:
1.
Which group has the most accurate measurement? Unknown – accuracy
can only be decided when a true or accepted value is available for
comparison
Which group has the most precise measurement? Group C – the
measurements are close to each other in Group C. The individual
measurements in Group B have greater precision but they don’t agree as
closely as those in Group C.
Which group has the greatest error? Unknown – this can only be
calculated given an accepted or true value
Which group has the greatest uncertainty? Group D – the average
variation is nearly 1cm which is three times greater than any other group.
2.
Which group has the least accurate measurement? Group C- results are
the most consistent even though the values are far from the accepted
value
Which group has the least precise measurement? Group D – the data
for Group D do not agree
Which group has the smallest error? Group A – there is a wide scatter
but the average is close to the accepted value
Which group has the smallest uncertainty? Group C – the uncertainty is
not changed by knowledge of the accepted value.
3. 18.4 ± 0.3 cm2
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General Chemistry Unit 1: Measurement and Atomic Theory
Activity 6
The process by which scientists acquire knowledge varies between
textbooks, teachers, scientists and the type of investigation but
commonly includes, a) defining the problem, b) gathering background
information, c) forming a hypothesis, d) make observations, e) testing the
hypothesis, and f) drawing conclusions. Research that is presented for
publication in journals is typically presented in a particular format but in
reality scientists don’t normally work to a standard format. Imagination,
creativity and prior knowledge are important aspects of the plan and
these elements can cause the process to take interesting twists and
turns.
TASK: Design an experiment to accurately determine how dense salt
water must be in order for a golf ball to float. Describe the steps in you
plan.
Where density is measured in g/ml, mass in g and volume in ml.
The materials available are: electronic balance, 100 mL & 500 mL
graduated cylinder, glass stirring rod, golf ball, sodium chloride, 250 mL
beaker, medium and large weighing dish
Activity 7
Read the PowerPoint presentation on how data should be reported in
Chemistry and complete tasks 1 to 4.
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