Measurements.doc
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Measurements, Significant Figures and Graphing
The objective of this exercise include:
•
•
•
•
•
To become familiar with some common general chemistry laboratory equipment, its uses and
limitations.
To learn the difference between precision and accuracy.
To learn about types of errors.
To learn how to determine or estimate uncertainties in measurements.
To learn the rules of significant figures and apply them.
Pre–laboratory Reading:
Brown, LeMay
& Bursten
Chapter 1: sections 4, 5 and 6
Appendix A: Mathematical Operations
Laboratory Manual
Appendix A: Treatment and Representing Experimental Data
Appendix B: Laboratory Equipment and Use: Calibrated Glassware, The Buret,
The Volumetric Pipet, The Graduated Pipet and The Volumetric Flask
Background:
Measurements are a central part of the science of chemistry. Your textbook contains a large number of
scientific facts and several scientific theories. The facts were obtained and the theories are supported by
carefully made measurements. In order for experimental results to be meaningful, the experimental chemist
must:
1.
Record the results of his or her measurements carefully.
2.
Repeat the measurement to increase its reliability. Since individual results from a series of repeated
measurements will seldom be the same, a “best” value is obtained by taking the average, or mean,
x.
3.
Establish the probable limits of uncertainty that can be placed on the measurement.
When a measurement is made, one of the goals is to achieve as high a degree of accuracy and precision as
possible. Do not confuse precision with accuracy. Precision is defined as how close a series of measurements
of the same quantity are to each other or, in other words, how reproducible the results are. The precision of a
series of measurements of the same quantity can be quantitatively expressed in terms of the range in the
results:
Range = Highest Value – Lowest Value
The smaller the range the better the precision of the data.
Accuracy refers to how close the measurement is to the true or accepted value. Whereas precision can be
determined from the measurements themselves, without knowledge of the true or accepted value, in order to
determine accuracy you must know the true or accepted value. Accuracy is often quantitatively reported in
terms of percent error:
Percent Error =
where
Error in measurement
x 100%
Accepted value
Error in Measurement = Experimental Value – Accepted Value
The sign of the calculated error, either plus or minus, is retained when reporting a percent error,
because it indicates whether the result was either too high or too low, respectively, from the true
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or accepted value. The smaller the percent error, regardless of the sign, the better the accuracy.
Errors associated with making measurements can be divided into two types: systematic and random. A
systematic error causes a measurement to always be too low or too high. Systematic errors arise from a
faulty measuring device, a defect in the procedure or a consistent mistake in using the device. One example
is an improperly zeroed balance. A systematic error affects the accuracy of each measurement. Random
errors result in individual measurements that are just as likely to be too high as too low. The magnitude
of random error depends on the precision of the measuring device and the skill of the person making the
measurement. A small random error results in a high degree of precision. Random errors cannot be avoided,
since there is always some degree of uncertainty in every measurement. This means that the average of
several repeated measurements will be more reliable than any single measurement. As a final note, although
good precision is often an indication of good accuracy, this need not always be the case. It is possible to have
low accuracy due to a systematic error, but still have high precision.
Each instrument (e.g., ruler, beaker, thermometer, balance, etc.) you use in the laboratory has a precision
that determines the uncertainty of measurements, due to random error, taken with that instrument. The
precision of a measuring device is usually expressed in terms of a ± value indicating the limitation of the
device. The common instruments you will use in General Chemistry can be divided into two types: those that
have a graduated scale and can make measurements over a range of values (e.g., ruler, thermometer,
balance, graduated cylinder, graduated pipet, beaker) and those that measure a single, fixed volume of a
liquid (e.g., volumetric flask, volumetric pipet).
The distance between graduation marks on a ruler, thermometer, buret or other glassware may be subdivided
into ones, tenths, hundreds or other divisions depending on the precision of the device. A 50 mL graduated
cylinder, for example, has graduation marks at each mL. Since the experimenter can estimate between the
graduation marks, the volume can be measured and recorded to the tenth of a mL, as illustrated below. A
buret, on the other hand, has graduation marks at each one-tenth mL and the hundredth place can be
estimated. Therefore, an extra digit to the right is gained when the buret is used and we say that the buret is
more precise.
Rule of Thumb: For instruments with graduation marks, record the measurement to 1/10 or 0.1 of
the SMALLEST division.
50 mL
⇐An illustration showing the top part of a 50 mL graduated cylinder containing a liquid is
shown here. Let’s say that a person records the volume of the liquid in the cylinder as
48.5 ± 0.2 mL. How many significant figures does the measured volume of
48.5 ± 0.2 mL have? The answer is three, the “48” we know for certain and the “5” that
was estimated. In a number representing a scientific measurement, the last digit to the
right is taken to be inexact (it is estimated) and is counted as a significant figure. The
recorded uncertainty of ±0.2 mL indicates the precision of the device. The uncertainty of
±0.2 mL indicates that the volume actually lies somewhere in the range of 48.3 to
48.7 mL. You may read the volume as 48.3 mL, your friend may read it as 48.7 mL; both
40 mL are within the estimated ±error range. Recording a measurement with the correct
number of significant figures is critical in order to reflect the precision of the
measuring device correctly. How was the uncertainty of ±0.2 mL determined? It was determined based
upon the smallest scale division of the instrument and how sure the person making the measurement believed
they were about the value of the estimated digit. Usually the uncertainty is within the range of ±1/10
of the smallest scale division up to ±1/2 of the smallest scale division. In this example that would be
from ±0.1 mL up to ±0.5 mL and the person making the measurement decided that ±0.2 mL was a
reasonable uncertainty. If the uncertainty in a measurement is not stated, then it is assumed to be ±
one unit in the estimated digit. For example, a reported value of 48.5 mL would imply an uncertainty of
± 0.1 mL (48.6 ± 0.1 mL). If the uncertainty in the measurement is not ± one unit in the estimated digit,
then the recorder has a responsibility to report the actual uncertainty. Note that an uncertainty, by definition,
has only one significant figure. In this example, since we are uncertain about the reading in the tenths place
(±0.2 mL), it does not make sense to report an uncertainty in any digit further to the right.
In the previous graduated cylinder example, the person making the measurement estimated the uncertainty
and reported it as an absolute uncertainty. Absolute uncertainties carry the same units as the
measurement itself. For some instruments the uncertainty is conveniently given on the device as either an
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absolute uncertainty or a percent uncertainty. You should always check the measuring devices you use
to see if the uncertainty is given on the device and, if given, record this uncertainty with your measurement.
For example, a graduated pipet may have an absolute uncertainty of ± 0.03 mL written on it. For volumetric
flasks and volumetric pipets the absolute uncertainty is often indicated on the instrument. If it is
not given, a good rule of thumb for the volumetric flasks and pipets most commonly used in the
General Chemistry lab is an uncertainty of ±0.2 mL for flasks and ±0.02 mL for pipets. The volume
delivered from a 20 mL volumetric pipet with an uncertainty of ±0.02 mL is recorded as 20.00 ± 0.02 mL,
indicating that the volume is somewhere between 19.98 mL and 20.02 mL. If reported as a percent, the
uncertainty in this pipet would be ±0.1%. As another example of a percent uncertainty, consider a 200 mL
beaker with a uncertainty of ±5% written on the beaker If you measure liquid using this beaker, the
measured volume of liquid has an uncertainty of ±5% of 200 mL, or ±10 mL. Thus, if you fill the beaker to
the 150 mL calibration line, the measured volume is 150 mL with an absolute uncertainty of ±10 mL. Note
that the recorded volume in this case has two significant figures, not three. When using this device it would be
incorrect to report the volume measured to a decimal place that is smaller than the “tens place” since this
digit is the inexact digit. For instance, it would be incorrect to indicate a volume of 152 mL, even if you
observe that the liquid level is slightly above the 150 mL calibration line; the precision of the device does not
allow you to read the volume to the “ones place”.
It should now be clear that the concept of significant figures is directly linked to precision and NOT to
accuracy. Consider measuring the mass of a piece of gold several times with a digital balance. You will
obtain nearly the same measurement each time, within a small random error in the rightmost digit that is
characteristic of the balance. You will have determined the mass of the gold with high precision. However, if
the balance is not calibrated correctly, reading either consistently too high or too low (a systematic error), the
mass of the gold will be inaccurate. The accuracy of the balance can be checked using standard weights of
known mass, something you would definitely want to do if you were buying or selling the gold! Well-trained
analytical chemists also have methods for checking the accuracy of their volumetric glassware. Remember
also that glassware that is designed to accurately measure volumes is calibrated for a certain temperature. In
such cases, the temperature at which the glassware is calibrated is given on the glassware itself and the most
accurate measurements will be obtained at the indicated temperature.
Glassware that is designed to accurately measure volumes of liquids is calibrated to either contain an amount
of liquid measured in the glassware or to deliver an amount of liquid from the glassware You will find that
glassware is often marked with the letters TC (to contain) or TD (to deliver). To obtain accurate volume
measurements, we must understand the meaning of the notation “TC” and “TD”. For example, a 50 mL
graduated cylinder that is marked TC and is filled to 25.5 mL CONTAINS 25.5 mL of the liquid. When the
liquid is poured out of the cylinder, some of the liquid will adhere to the walls of the cylinder so that less than
25.5 mL is actually poured out, an inaccurate result. If it is essential that the experimenter know the amount
of liquid poured out of the measuring device accurately, then a TD device is preferred. A TD device DELIVERS
the measured volume.
Laboratory Exercise
Safety:
Remember to always wear your safety glasses while in the chemistry laboratory.
Equipment:
Each bench top will be given an identifying number (1 through 8) and will have the
following equipment set up.
100 mL volumetric flask filled to the mark with water
10 mL volumetric pipet
5 mL graduated pipet
250 mL beaker partially filled with water
100 mL graduated cylinder partially filled with water
10 mL graduated cylinder partially filled with water
50 mL buret partially filled with water
Procedure:
Using the equipment provided on your lab bench top, complete parts A through C on the data and report
sheets that follow. DO NOT change the volumes of water in the glassware. Your instructor will also record the
volumes in order to check your answers.
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Measurements.doc
Lab Section: MW or TTH
Data and Report:
Lab Bench Number:
Part A: This part can be completed together with the other people at your lab bench.
For each piece of glassware given complete the following table. Make sure you include units where applicable!
Some spaces may be left empty if the information is not given or not applicable for a particular device. In
some cases, the uncertainty may be given on the glassware, in other cases you will have to either estimate
the uncertainty or refer to the background reading for this exercise. Remember, you must include a unit on
your uncertainty.
Data Table I: Information About Some Glassware Used in Chemistry
Measuring
Device
Is the glassware
Is it To
For graduated
Temperature at
graduated or
Contain (TC)
glassware give the
which the glassware
designed to
or To Deliver
smallest scale
is calibrated.
measure a fixed
(TD)?
division.
volume?
What is the
uncertainty
(precision) of the
glassware
(± x)?
250 mL
beaker
50 mL buret
10 mL
volumetric
pipet
100 mL
volumetric
flask
100 mL
graduated
cylinder
10 mL
graduated
cylinder
5 mL
graduated
pipet
Questions:
1.
Which of the given measuring devices would you use to do the following:
a.
Measure approximately 150 mL of a liquid?
b.
Measure 50.0 mL of a liquid?
c.
Deliver 10.00 mL of a liquid?
d.
Deliver between 0.00 and 5.00 mL of a liquid?
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2.
Lab Section: MW or TTH
Why is the temperature at which laboratory glassware is calibrated important? More specifically, how
does the temperature affect the volume measurements?
Part B: Complete this part individually. You will compare your results with the results of the other
people at your lab bench when finished.
1.
2.
Report the volume of water in each of the following devices to the correct number of significant
figures. Include the uncertainty in your measurement.
a.
100 mL volumetric flask filled exactly to the calibration mark:
b.
250 mL beaker:
c.
100 mL graduated cylinder:
d.
10 mL graduated cylinder:
Record the reading for the water level in the 50 mL buret to the correct number of significant figures.
Include the uncertainty in your measurement. (Remember that in this case you are recording
the level of the water and that the scale increases numerically downward. You are not recording the
actual volume of water in the buret. This can seem strange at first. Reading about the buret in
Appendix B of the Lab Packet will help you understand.)
Part C: Compare your measurements in Part B with the other people at your lab bench.
1.
One of the measurements should be the same for everyone at your lab bench. Which one is it and
why?
2.
The other measurements made should vary slightly within your group. Why do they vary?
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Lab Section: MW or TTH
Part D: Graphing
In an experiment, Bud N. Chemist measured the mass of a clean, dry beaker. He then used a buret to
measure water into the beaker and reweighed the beaker plus water. He repeated this, measuring out
additional water and recording the total mass of water plus beaker each time. The results of his
measurements are given below:
Temperature of Water:
20.0 °C
Total Volume of Water
Added (mL)
Total Mass of Beaker
Plus Water (g)
5.8
84.6
10.0
88.2
20.7
99.1
25.8
103.5
1.
Using suitable graph paper, make a graph of total mass of beaker plus water (y-axis) versus total
volume (x-axis) of water in the beaker following the rules for good graphing given in Appendix A.
Plan the range of your axes so that you can extrapolate to find the y-intercept. Do not forget
to label your axes (with units) and title your graph. Draw the best straight line through your four
points being sure to extrapolate to the y-intercept. Staple your graph to the end of this lab when
finished.
a.
Using two points near the ends of your line that are not your data points determine the
slope of the line. (Remember, the slope of the line will have units of Y units/X units). Show
your calculation with units for the slope below. Use correct significant figures in your answer.
Slope of line =
b.
From the graph read and record the y-intercept of the straight line (do not forget the units).
Y-intercept
2.
Both the slope of the line and the y-intercept have physical meaning in this simple exercise.
a.
What physical property of water does the slope of the line represent? (Hint, let the units of
the slope help you.)
b.
What does the y-intercept represent?
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3.
Lab Section: MW or TTH
Compare your slope value to the accepted value given in the CRC Handbook of Chemistry located
in the laboratory. Calculate the percentage error in your experimental value. Show the calculation
below. This gives the accuracy in Bud’s results.
Accepted Value:
Calculate the percent error:
Follow-Up Questions:
Be complete in your answers. Show units in set-up and answers and report all answers to the
correct number of significant figures.
1.
Give the absolute uncertainty (implied precision) as a (± value) that is indicated in the following
recorded values:
a.
1.608 mL
b.
0.0910 g
c.
1.30x10–3 atm
2.
A 50 mL beaker has an uncertainty of ±10% written on it. What is the absolute uncertainty for this
beaker?
3.
Do the following calculations.
4.
a.
6.19 cm x 2.8 cm =
b.
78.07 g − 75.77 g =
c.
5.19x10–2 cm + 1.83 cm + 219.2 cm =
d.
[(2.841 x 104 mL) – (1.2 x 103 mL)] x 2.8959 g/mL =
e.
Convert 925°C to Kelvin =
Express the results of the following to the proper number of significant figures.
a.
log (6.19) =
b.
antilog (−7.01) =
5.
A block of metal with dimensions 5.2 cm x 2.1 cm x 4.6 cm has a mass of 109.82 g. Calculate the
density of the metal.
6.
A sample of a certain compound weighing 2.040 g is found to contain 0.721 g of carbon and 1.269 g
iodine. The only other element present is hydrogen. What is the percent by mass of hydrogen in the
compound?
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7.
Lab Section: MW or TTH
The accepted value for the melting point of pure aspirin is 135°C. You measure the melting point of
aspirin and obtain 132°C, 133°C, 139°C and 140°C in four separate trials. Your partner finds 138°C,
137°C, 138°C and 139°C. Remember to show your work for the following questions:
a.
Calculate the range in the measurements for both you and your partner.
Range in Your Measurements =
Range in Your Partner’s Measurements =
b.
Calculate the average of your measurements and of your partner’s measurements.
Your Average =
Your Partner’s Average =
c.
Calculate the percent error in your average and in your partner’s average.
Percent Error in Your Average =
Percent Error in Your Partner’s Average =
d.
Whose results are most precise? How did you decide?
e.
Whose results are most accurate? How did you decide?
f.
Does either data set appear to have a systematic error? Explain how you decided.
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Pre-laboratory Exercise (2 pages):
Lab Section: MW or TTH
Name:
Complete the following and turn it in at the start of lab lecture. Your answers are expected to be
complete and based upon the background information provided in this experiment.
1.
Define the term precision:
2.
Define the term accuracy:
3.
Give the following formulas: (You must memorize these.)
a) The formula for calculating the range of a series of measurements of the same quantity.
b) The formula for calculating the percent error of a measurement.
c)
Which of the above, range or percent error, is used to quantitatively describe:
i)
accuracy?
ii)
precision?
d) What does a negative percent error indicate about a measured value?
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4.
Lab Section: MW or TTH
The mass of a piece of gold is measured twice, once using a top loading balance and once using an
analytical balance, with the following results: Top loading 1.012 g
Analytical 1.0115 g
a) Which of the two measurements is MOST PRECISE?
(i) top loading
(ii) analytical
(iii) cannot determine without more information
b) Which of the two measurements is MOST ACCURATE?
(i) top loading
(ii) analytical
(iii) cannot determine without more information
5.
Define the term random error:
a) What does the magnitude of random error depend upon?
b) Consider an average value obtained from repeated measurements of the same quantity. Does random
error primarily affect the precision of the average, the accuracy, or both?
6.
Define the term systematic error:
a) What factors can cause a systematic error?
b) Consider an average value obtained from repeated measurements of the same quantity. Does a
systematic error primarily affect the precision of the average, the accuracy, or both?
7.
Give the number of significant figures in each of the following:
0.0432 g
8.
1.60 x 10–9 m
5680.2 cm
52.0 °C
Round each of the following to 3 significant figures: (Use scientific notation where needed to avoid
ambiguity.)
6.167 g
0.02245 m
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2.1349 cm
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3133 mL
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