Download Determining the Density of a Polystyrene Sphere

CHEMISTRY 117
University of Massachusetts Boston
Laboratory Exercise #1
Determining the Density of a Polystyrene Sphere
*There are two parts to this lab exercise. Please make sure to read and prepare for both parts
before coming to lab.*
PART I: LENGTH AND VOLUME
I. LEARNING GOALS
Almost everything that we do in Chemistry Lab this semester is going to involve measuring
something. This first laboratory exercise has been designed around the following learning goals.
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•
•
•
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Provide practice in performing typical measurements.
Instill a deeper understanding of the concepts of precision and accuracy of measurement.
Learn to calculate the average and standard deviation of replicate measurements.
Begin to learn about significant figures.
Appreciate how the precision of a measuring tool determines the precision of the
measurement and thus the precision of the results.
II. INTRODUCTION
Accuracy and Precision
Precision refers to the degree of variability in the values obtained from repeated
measurements. This variability can come from the inherent precision of the measuring device,
as well as the technique of the experimenter making the measurements. Accuracy is a measure
of how closely the mean (average) of the repeated measurements matches the true value of
the “thing” being measured. One of the tried and true teaching strategies for addressing the
concepts of precision and accuracy is the use of the dartboard analogy. If a dart player aims
some darts at the bull’s eye, the following outcomes are possible.
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∙
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A
not accurate
or precise
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∙ ∙ ∙ ∙
∙ ∙
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∙
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∙∙∙∙∙ ∙
∙ ∙∙ ∙∙
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B
accurate, but
not precise
C
precise, but
not accurate
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∙ ∙∙ ∙
∙∙∙ ∙∙∙∙∙∙∙
D
accurate
and precise
CHEMISTRY 117
University of Massachusetts Boston
Laboratory Exercise #1
For outcome A the darts are scattered over a wide area (low precision) and the average
position of the darts is far above and to the right of the bullseye (low accuracy). For outcome B
the darts are scattered over a wide area (low precision) and the average position of the darts is
centered close to the bullseye (high accuracy). For outcome C the darts are focused in a tight
area (high precision) and the average position of the darts is far above and to the left of the
bull’s eye (low accuracy). For outcome D the darts are focused in a tight area (high precision)
and the average position of the darts is centered close to the bull’s eye (high accuracy).
Relating this analogy to the process of making a measurement in a chemistry lab, the bull’s eye
represents the true value of the thing being measured and the average position of the darts
represents the mean of the repeated measurements. This experiment will focus on evaluating
the precision of three different methods for measuring the volume of a small sphere; that is,
the variability in the data obtained from repeat measurements. To evaluate the accuracy of
these methods one would have to know the true volume of the sphere (more on this later).
The Precision of the Measurement Tool
What is the diameter of the circle below? It’s larger than 0.5 units and less than 1 unit. It’s
closer to 1 unit than 0.5 units. Could it be said that the diameter is 0.7 units, 0.8 units or maybe
0.9 units? How about 0.85 units? How precise is this measurement tool?
Below is the same measurement with a “better” ruler. It is clear that the circle is larger than
0.8 units and less than 0.9 units. It looks about half way between the two. It could be said that
it is close to 0.85 units, but the last digit is really just an estimate.
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CHEMISTRY 117
University of Massachusetts Boston
Laboratory Exercise #1
Below is the same circle enlarged and measured with a more precise ruler. It looks as if the
circle is at least 0.86 units. It appears to be a little larger, maybe 0.861 units or 0.862 units. The
measurement tool indicates that the eight and the six are correct, but the last digit is really an
estimation. It is generally accepted (and should be commonly practiced in this laboratory) to
estimate a single graduation to the nearest tenth (Thus if the graduation is to the hundredths
place, the thousandths place should be estimated). This estimated digit determines the place
of least significance when the measured value is reported.
The precision of the diameter measurement can have a great impact on calculations that are
made later with them. In this quick experiment the diameter of a polystyrene sphere will be
measured. Using this measurement the volume of the sphere can be calculated using the
following equation:
Then the volume of the sphere will be measured by the method of water displacement using a
graduated cylinder. It is important that the proper technique be used for reading volume with
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CHEMISTRY 117
University of Massachusetts Boston
Laboratory Exercise #1
this instrument. After measuring the volume by this method, the results will be compared to
those of five other groups.
As the measurements are recorded and calculations are carried out, it is important to include
the units and the correct number of significant digits.
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CHEMISTRY 117
University of Massachusetts Boston
Laboratory Exercise #1
III. PROCEDURE
A. Measurement of the volume with a ruler
Measure the diameter of a polystyrene sphere in centimeters using the metric ruler
and write it in the data table below (1). Check with your TA whether you have
reported this value with the proper significant figures. You may need three rulers,
three hands, paper and a pencil to do this with the best possible precision.
B. Measurement of the volume using a graduated cylinder (one ball)
Fill a 25 mL graduated cylinder with about 12 mL of water. Measure the volume of
water in milliliters by setting the graduated cylinder on the lab bench at eye level and
carefully reading the bottom of the meniscus of the water level to the nearest 0.02
mL. Note it in the data table below (3). The increments of the graduated cylinder are
every 0.2 mL, so the uncertainty in the measurement is ± 0.02 mL. Add one
polystyrene sphere into the graduated cylinder (carefully! No Splashing!). Measure
the total volume in milliliters by setting the graduated cylinder on the lab bench at
eye level and carefully reading the bottom of the meniscus of the water level to the
nearest 0.02 mL. Note it in the data table below (4).
C. Measurement of the volume using a graduated cylinder (ten balls)
Add nine more polystyrene spheres [10 total] (carefully! No Splashing!) to the
graduated cylinder. Measure the total volume in milliliters by sitting the graduated
cylinder on the lab bench at eye level and carefully reading the bottom of the
meniscus of the water level to the nearest 0.02 mL. Note it in the data table below
(6).
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CHEMISTRY 117
University of Massachusetts Boston
DATASHEET (Pg 1 of 6)
Laboratory Exercise #1
NAME:
Measurement of the volume with a ruler
1. Diameter of a polystyrene sphere using the metric ruler (in
centimeters).
2. Calculate the volume of the polystyrene sphere in cm3 from
the measured diameter (V = 4/3πr3, where r = d/2)
Measurement of the volume using a graduated cylinder (one ball)
3. Volume of water added to the graduated cylinder.
4. Volume of water plus one polystyrene ball.
5. Calculate volume of one polystyrene sphere:
[Volume of sphere = total volume – volume of water]:
Measurement of the volume using a graduated cylinder (ten balls)
6. Volume of water plus ten polystyrene balls.
7. Calculate volume of ten polystyrene spheres:
[Volume of ten spheres = total volume - volume of water]:
8. Calculate average volume of spheres:
[Volume of one sphere = (1/10)volume of ten spheres]:
You now have three different measurements of the volume of a polystyrene sphere from #2, #5 and #8.
From different members of your lab class, obtain five other values for each of the three methods. If you
should find that your group completed the experiments quicker than the rest of your classmates,
DON’T JUST SIT THERE WAITING FOR THEM TO FINISH. REPEAT THE EXPERIMENT and then you can
use both sets of your data in Table 1.
Table 1
#2 volume of sphere
#5 volume of sphere
#8 volume of sphere
from ruler
from water displacement of
one sphere
from water displacement of
ten spheres
Your results:
Labmate #1
Labmate #2
Labmate #3
Labmate #4
Labmate #5
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CHEMISTRY 117
University of Massachusetts Boston
Laboratory Exercise #1
DATASHEET (Pg 2 of 6)
Calculate the mean for the three measurements.
Table 2
Calculating the Means
From Ruler
Mean
From Water displacement of one
sphere
cm3
From Water displacement of ten
spheres
mL
mL
The most commonly used measure of uncertainty is called the standard deviation. The standard
deviation is a measure of the differences between the actual numbers and the mean.
Calculating the standard deviation using this equation is easiest if you break it down into steps. The
following worksheets take you though each of the individual steps.
The first step is to calculate the difference between each of the values (from Table 1) and the mean
(from Table 2). Place these differences in Table 3:
Table 3
Calculating the Deviation from the Mean
#2
#5
from Ruler
from water displacement From water
of one sphere
displacement of ten
spheres
Your number – mean:
Labmate #1 – mean:
Labmate #2 – mean:
Labmate #3 – mean:
Labmate #4 – mean:
Labmate #5 – mean:
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#8
CHEMISTRY 117
University of Massachusetts Boston
Laboratory Exercise #1
DATASHEET (Pg 3 of 6)
The second step is to square each of the differences calculated in the first step (Table 3):
Table 4
Squaring the deviation from the mean
#2
#5
#8
from ruler
from water
displacement of one
sphere
from water
displacement of ten
spheres
(Your number – mean)2:
cm6
mL2
mL2
(Labmate #1 – mean)2:
(Labmate #2 – mean)2:
(Labmate #3 – mean)2:
(Labmate #4 – mean)2:
(Labmate #5 – mean)2:
The third step is to add together each of values in the columns of Table 4:
Table 5
Summing the squares of the deviations
from Ruler
from water
displacement of one
sphere
cm6
Total of Squares:
from water
displacement of ten
spheres
mL2
mL2
The final step is to divide the numbers in Table 5 by the number of data points minus 1 (or 5 in
our case) and take the square root to get the standard deviations:
Table 6
Calculating the standard deviations
from ruler
from water
displacement of one
sphere
from water
displacement of ten
spheres
Standard Deviation:
Often the results are reported as as the mean ± standard deviation. It is intended to represent
a range of possible “values” based on the precision of the measurement.
Table 7
Reporting the mean with the standard deviation
from ruler
Overall Result
from water
displacement of one
sphere
±
±
8
from water
displacement of ten
spheres
±
CHEMISTRY 117
University of Massachusetts Boston
Laboratory Exercise #1
PART II: MASS AND DENSITY
I. LEARNING GOALS
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Learn proper techniques for weighing substances using a triple beam balance and an
analytical balance.
Instill a deeper understanding of the concepts of density.
Continue to learn about significant figures.
Appreciate how the precision of a measuring tool determines the precision of the
measurement and thus the precision of the results.
II. INTRODUCTION
Density
Density is the mass of a substance divided by its volume. It is an inherent property of a
substance. For instance, measuring the density of a metal can help you identify it. In chemistry
applications, density is most often expressed in the units of g/mL (read “grams per milliliter) or
g/cm3 (1 mL = 1 cm3).
III. PROCEDURE
Learn the technique for determining mass with a triple beam balance and an analytical balance
from your instructor. We will be measuring the mass of the polystyrene balls using four
strategies. The first two strategies involve weighing one polystyrene ball on each of the
balances. Which balance is more precise? The other two strategies involve weighing ten
polystyrene balls of each of the two balances. Each measurement and calculation should be
recorded with the appropriate unit and the correct number of significant figures.
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CHEMISTRY 117
University of Massachusetts Boston
DATASHEET (Pg 4 of 6)
Laboratory Exercise #1
NAME:
Measuring Mass on the Triple Beam Balance.
It is critical in this experiment to report the measurements with the correct precision (to the
nearest 0.01 g for the triple beam balance).
1. Determine the mass of a clean small 50-mL beaker on a triple
beam balance:
2. Place one polystyrene sphere in the beaker. Determine the
mass of the beaker + the sphere on the triple beam balance:
2a. Calculate the mass of one sphere on the triple beam balance:
3. Add 9 more spheres to the beaker (10 total). Determine the
mass of the beaker + 10 spheres on the triple beam balance:
3a. Calculate the mass of 10 spheres on the triple beam balance:
3b. Calculate the average mass of a sphere by dividing the mass
of 10 spheres by 10 (remember that 10 is an exact number in this
case):
Measuring Mass on an Analytical Balance.
It is critical in this experiment, as well as future experiments, to report the measurements
with the correct precision (to the nearest 0.0001 g for the analytical balance).
4. Determine the mass of the beaker on an analytical balance:
5. Place one polystyrene sphere in the beaker. Determine the
mass of the beaker plus the polystyrene sphere on an analytical
balance:
5a. Calculate the mass of one sphere on an analytical balance:
6. Add 9 more spheres to the beaker (10 total). Determine the
mass of the beaker plus 10 spheres on the analytical balance:
6a. Calculate the mass of 10 spheres on an analytical balance:
6b. Calculate the average mass of a sphere by dividing the mass
of 10 spheres by 10 (remember that 10 is an exact number in this
case):
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CHEMISTRY 117
University of Massachusetts Boston
Laboratory Exercise #1
DATASHEET (Pg 5 of 6)
Add your data and the data from FIVE other lab groups to the table below and calculate an average and
standard deviation for the mass of a sphere obtained from the four methods.
MASS OF ONE SPHERE
Measurement
method
1 ball method/
triple beam
(2a)
1 ball method/
analytical
(5a)
10 balls method/
triple beam (3b)
10 balls method/
analytical
(6b)
Your Data
From members of your lab, obtain five values for each of the four methods:
Labmate #1:
Labmate #2:
Labmate #3:
Labmate #4:
Labmate #5:
Calculate the mean ± standard deviation of these calculations.
Mean ± sm
**Note: Excepting PART I, it is not necessary to calculate standard deviation long-hand. You are
encouraged to seek out the owners manual of your calculator to determine how to calculate
standard deviation quickly.**
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CHEMISTRY 117
University of Massachusetts Boston
Laboratory Exercise #1
DATASHEET (Pg 6 of 6)
Density Calculation
Using the method that you deem most precise, record the
mean and standard deviation of the volume of a
polystyrene sphere from PART I above.
±
Calculate the density using the average masses from the above table and data from PART I for
the volume (D = m/V).
METHOD
DENSITY
1 ball / triple beam (2a)
1 ball /analytical (5a)
10 balls /triple beam (3b)
10 balls /analytical (6b)
Measurement
method
1 ball method/
triple beam
(2a)
1 ball method/
analytical
(5a)
Density
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10 balls method/
triple beam (3b)
10 balls method/
analytical
(6b)
CHEMISTRY 117
University of Massachusetts Boston
Laboratory Exercise #1
IV. Lab Report
The initial submission of your lab report will be worth up to 50 points. Use next page as a guide
for a well-written lab report. The report will be graded during the lab writing workshop session.
You will receive a 0, 30(√-), 40 (√), 50 (√+) based on how well you follow the guidelines.
1. TITLE
2. ABSTRACT
3. INTRODUCTION
4. MATERIALS AND METHODS
5. DATA and ANALYSIS
6. DISCUSSION
7. CONCLUSIONS
There is no minimum page limit for a lab report, however if all sections are written
appropriately and thoroughly, you will find that it will typically 2-4 pages in length.
DISCUSSION POINTS
As mentioned above, precision refers to the variation in data obtained from repeated
measurements. Accuracy refers to the difference between the mean obtained from repeated
measurements and the true value. Accuracy can only be evaluated if one has verified the
accuracy of a measuring tool. This verification is usually accomplished by using a primary
standard that has been accurately measured previously by some other validated method (in
other words, the true value must be known).
To receive full credit for your discussion sessions, the following questions should be answered.
•
According to your class data (considering all five sets), which measurement method for
volume of the spheres is the most precise? Least precise?
•
According to your class data (considering all five sets), which measurement method for
mass of the spheres is the most precise? Least precise?
•
Are the most and least precise measurement methods the ones that you expected them
to be? Explain.
•
Explain why the precision is different when you measure ten balls to get an average
volume and mass.
•
If the true value for the density of a sphere is 1.05 g/mL , which method is most
accurate?
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CHEMISTRY 117
University of Massachusetts Boston
Laboratory Exercise #1
WHAT IS A LAB REPORT?
A written lab report is a common scientific exercise that helps to convey why an experiment
was performed, how an experiment was performed, what the results of the experiment were,
and why those results are of significance. For this experiment, you are being tasked with writing
a complete lab report. A complete lab report consists of:
•
•
TITLE: A summation of the experiment in ten words or less. It should describe the
main point of the laboratory work. For example: Effects of Atmospheric CO2
Concentrations on the Global Climate.
ABSTRACT: A summary of what is contained within a lab report. This allows the
author to highlight key information contained in the following sections and provides
the reader with key points so they can determine if the lab report contains
information they seek. The abstract should be the last thing that is written for the
lab report, as it contains elements of all the other sections. An abstract should be
roughly three to six sentences. This is more difficult than it sounds, so allow time for
several revisions.
•
INTRODUCTION: In a scientific research article this section of the paper is devoted
to making the case for why the work is important and significant and for discussing
the previous work reported in the literature that has led up to the work being
reported in this paper. In a lab report the nature of the introduction section is a
little bit different than in a research article. In a lab report you should focus the
introduction on the learning goals of the experiment. Discuss how the experiment is
designed to achieve these learning goals and how the experiment fits in with the
broader curriculum of the corresponding lecture course.
•
MATERIALS AND METHODS: This section should provide the details of how the
experiment was carried out. It should not be written as a recipe but more as a
journal entry; a fairly detailed account of what was done in lab. A description of
how the data was processed should also be part of the Methods section.
•
DATA AND ANALYSIS: When appropriate you data should be displayed in tables and
figures. The figures and tables should have captions that describe what they are
illustrating. You should also prepare sentences that introduce the tables and figures
and describe what they show. To an extent these sentences and the captions will be
and should be somewhat redundant.
•
DISCUSSION: A description of what the data means. Points of discussion could
include: The discussion section will discuss the significance of the findings from the
data analysis section. It is also in this section the questions that are being asked are
addressed in the context of a well-written paragraph.
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CHEMISTRY 117
University of Massachusetts Boston
Laboratory Exercise #1
(1) Was the goal of the experiment achieved?
(2) If the results obtained are poor, what could have happened during the
experiment to explain this?
•
CONCLUSION:
The conclusion should be a separate paragraph that briefly reiterates what
happened in the experiment, what the results are, and what those results mean.
In the upper right hand corner of your first page, be sure to include your name, your TA’s name,
and the date the experiment was performed.
It is imperative that scientists are able to communicate clearly in their writing. Thus, the lab
report will be graded both on the content of the report and the quality of the writing.
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