The Density of Metals and Equipment Precision
Roger Sandwick and Thomas Moffett Jr., Plattsburgh State University, 2007
Edited by E. J. Miller and T, Moffett Jr. (Version Fall 2010)
I. Learning Objectives
The scientist will be able to:
1. Use the top-loading balance, graduated cylinder, and volumetric pipette in a competent
fashion.
2. Articulate the differences between intensive and extensive properties.
3. Explain the concept of density.
4. Use density to convert between mass and volume of a material.
5. Explain the concepts of accuracy and precision in scientific measurements.
6. Use experimental data to calculate percent error and standard deviation.
II. Introduction
A) Density
Density is defined as the mass of a substance per unit volume.
d=
m
v
(1)
In the first part of this experiment, you will determine the density of an unknown cylinder of
metal by dividing its mass (determined on a laboratory balance) by its volume (determined using
the formula for volume from the length of a cylinder and its radius). This will be done in a team
fashion where your lab bench partners constitute your team.
In the second part of the experiment, you will again calculate the density of the same piece of
metal using a volume obtained from difference in water levels in a graduate cylinder filled with
water before and after immersing the cylinder. This procedure will be done again in a team
fashion.
Using two different methods, as described above, to make measurements on the same piece of
metal, will allow you to compare the accuracy of the two methods. Having different people
make the measurements on the same piece of metal will allow you to determine a real life
standard deviation.
We perform these procedures in this fashion so that you will learn how scientists
compare the quality of measurements. The standard deviation represents the dispersion in a
set of measurements of a property of an object. The dispersion is due to randomness of the
process. The percent error, or percent difference from a known accepted value for a property,
provides a measure of the accuracy of the methods.
You will identify the unknown piece of metal by comparing its measured density with values in
Table 1. Once you have identified the piece of metal by choosing the metal whose density is
closest to your measured value you will then calculate the percent error.
Fundamental Principles of Chemistry I
Density
Table 1. Densities of some metals (g/cm3)
Metal
Aluminum
Lead
Magnesium
Density
2.70
11.35
1.74
Metal
Zinc
Copper
Iron
Density
7.13
8.96
7.87
B) Intensive versus Extensive Properties
Certain properties can be used to identify a material since they are unique characteristics of the
material. These properties, called intensive properties, depend only on the identity of the
material and not on the amount of material that you have. Density is an example of such a
property. Extensive properties depend on the amount of material that you have. An example of
such a property is volume. Metal cylinders such as those that you will use in Part 1 of this
experiment, have their own characteristic densities. Since density is an intensive property, we
will use the measured densities to identify the material or its concentration.
C) Error and the Experimenter
Chemists deal with numbers and physical quantities almost everyday. As examples, chemists
may need to know how much heat is given off in a particular reaction; how distant two atoms are
from each other in some molecule; or want to measure the difference in volume of a gas before
and after it is heated. The uncertainties in the numbers with which the chemist deals are also
important. Scientists should always tell others how reliable a particular number is that they have
measured. There are various ways of signifying uncertainty or reliability.
At this point in your career as a scientist, it may be difficult to grasp the physical significance of
the uncertainty in a particular measurement. Hopefully, after using some of the various ways in
which scientists communicate uncertainty and error, you will understand the importance of this
in your scientific work.
The main terms related to error are: precision, accuracy, standard deviation, and percent error.
To understand accuracy and precision, refer to Figure 1. As can be seen in the left portion of
Figure 1, all of the measurements, i.e., where darts landed, were close together. If these had
been real measurement, then you would have what is known in science as good precision.
However, since you missed the bulls eye you would not have very good accuracy.
On the other hand, you could have the darts spread out as in the top-center portion of the figure.
Then you would have poor precision because the dart landings were not close together but the
average point would be about a bull’s eye. So you would have good accuracy but since the
darts were spread out, your precision would be poor. The bottom center is of course the ideal
situation; good accuracy and good precision. The drawing on the right side indicates a very bad
day.
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Fundamental Principles of Chemistry I
Density
Figure 1. Visual display of accuracy versus precision.
Good Accuracy – Poor Precision
Less
Ouch
Ouch
Poor Accuracy – Good Precision
Good Accuracy – Good Precision
Poor Accuracy – Poor Precision
Scientists communicate precision and accuracy using mathematical quantities. The easiest one
to understand is the measurement of percent error. The calculation for this is shown in equation
3. It is a measure of how far off you are from the accepted value.
Percent error calculation:
% Error =
(Measured Value - Accepted Value) X 100
(2)
Accepted Value
The way that scientists communicate the precision of their measurements is through the use of
standard deviation. The way that standard deviation is calculated will be shown with the aid of
Table 2 below.
Table 2. "Made-up" set of data
Run(i)
1
2
3
4
Measurement
(xi)
5.50 lbs
5.10
5.60
5.70
Absolute Deviation
(x-xi)
0.25 x 10-1
3.75 x 10-1
1.25 x 10-1
2.25 x 10-1
(x-xi)2
6.25 x 10-4
1.40 x 10-2
1.56 x 10-2
5.06 x 10-2
The Average value is equal to the sum of all of the measurements divided by the number of
measurements or:
x1 + x 2 + x 3 + x 4 + + x n
(3)
n
where n is the number of measurements where the x’s represent the values for the individual
measurements.
Average =
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Fundamental Principles of Chemistry I
Average for the made-up data =
Standard Deviation =
5.50 + 5.10 + 5.60 + 5.70
4
Density
= 5.48
( x1 − Ave.) 2 + ( x 2 − Ave.) 2 + ( x 3 − Ave.) 2 +
n −1
+ ( x n − Ave.) 2
(4)
Standard Deviation for the made-up data
(5.50 − 5.48) 2 + (5.10 − 5.48) 2 + (5.60 − 5.48) 2 + (5.70 − 5.48) 2
0.2076
=
=
= 0.0692 = ±0.26
4 −1
3
A small standard deviation indicates better precision in a set of measured numbers. When you
move to your more advanced classes, you will learn about the 95% confidence interval. This is
based on the standard deviation and gives a more meaningful message about the precision
relative to an average value.
II. Experimental Procedure (Review the Lesson, Introduction to the Laboratory and
Keeping a Laboratory Notebook, for this laboratory prior to the lab period)
1) Density of Solids
a)
Volume from Dimensions (All of the individuals at a single bench should do a
and b of the “Density of Solids” as a team – please share your data with each
other before you leave the laboratory)
(1) Obtain a piece of unknown metal from the side bench. Weigh it on the toploading balance (Review and study the Document on Angel: Demonstration
on the Use of Laboratory Balances prior to the lab) to as many significant
digits as possible and then measure its dimensions with a ruler. Pass the cylinder
to the next member of your team.
(2) The next person should INDEPENDENTLY repeat step 1 of this section.
(3) When the second person on the team has completed the measurements, it
should be passed to the third person and so on until each member of the team
(your bench mates) have completed the measurements.
(4) All members of the team or bench should share their data at this point – see
Table 1 in Section III of this document for a sample data table.
(5) Calculate the density of the metal for each set of measurements (the volume of a
cylinder is πr2h). Calculate the average density and the standard deviation for
the set of measurements (you may use your calculator or Excel to do this).
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Fundamental Principles of Chemistry I
Density
b) Volume by Displacement (All of the individuals at a single bench should do
this section as a team – please share your data with each other before you
leave the laboratory)
(1) Determine the volume of your unknown piece of metal by carefully sliding it into a
graduated cylinder (Don’t drop it in since it could break the graduate
cylinder) that is filled with a known volume of water (be sure there is enough
water to completely cover the metal piece). The difference in volumes before
and after adding the metal is the volume of the metal piece.
(2) Pass the cylinder to the next person on your team, they should then
independently do step one.
(3) When the second person on the team has completed the measurements, it
should be passed to the third person and so on until all members of the team
(your bench mates) have completed the measurements.
(4) All members of the team or benchmates should share their data at this point –
see Table 2 in Section III of this document for a sample data table.
(5) Calculate the density of the metal for each run using the average weight from
“Volume by Dimensions” and the volumes determined here. Calculate the
standard deviation for this approach.
2)
Equipment Limitations
In this section you will examine the accuracy and precision of various pieces of
laboratory equipment. Again each member of the bench will do this section
independently and will share their data with the other members of their bench at the
conclusion of the experiment. The same glassware items must be used for all
measurements.
(1) Fill a 250 mL beaker with deionized (DI) water.
(2) Use a balance to determine the mass of an empty 150 mL beaker.
(3) Measure 10 mL of DI water with a 10 mL graduated cylinder. (Use a disposable
pipette to transfer the DI water from the 250 mL beaker to the graduate.)
(4) Transfer the 10 mL of water into the empty 150 mL beaker. Record the mass of
the beaker with the water. Determine the mass of the water. Calculate the
density.
(5) Repeat steps 3 and 4 using the following items in place of the graduated cylinder:
• 50 mL beaker
• 10 mL volumetric pipette
• 100 mL graduated cylinder
Make sure that the beaker that you are using to measure masses in is completely
dry for each trial.
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III. Data Recording and Analysis
1) Sample data Tables: For your notebook, you should set-up appropriate data tables for
each experiment. You will need to think about this in order to do it effectively.
Presenting data is a key part of good scientific work. For this experiment, samples of
what the data tables should look like are provided below.
2) In your notebook make sure that you record all measurements with the appropriate
number of significant digits and units (Review Lesson: Uncertainty and Significant
Figures before the laboratory). Indicate the equipment used for a particular
measurement. Show all calculations and make observations.
Density of Solids
Part 1a
Unknown Code: ___________
Table 1. Density of a Solid Using a Ruler to Measure Dimensions
Person’s
Name
Person’s email
Mass
( )
Diameter
( )
Length
( )
Volume
( )
Average Mass ______
Average Density______
Standard Deviation______
Observations:
Part 1b
Table 2. Density of Unknown Metal by Displacement
Person’s
Name
Final Reading
( )
Initial Reading
( )
Volume
( )
*Use average weight from part a.
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Density*
( )
Density
( )
Fundamental Principles of Chemistry I
Density
Average Density______
Standard Deviation______
Observations:
Identity of the Metal __________
Percent Error in Density (from both part a and b) _________
_________
Equipment Limits
Part 2
Mass of empty beaker: _________
Table 3. Measurements of water made with a 10 mL Graduate
10 mL Graduate Person 1
Person 2
Person 3
Mass of beaker +
water
Mass of water
Person 4
Volume of water
Density of water
Average density _________
Accepted density: ________ - Your instructor will write this value on the board (it is temperature
dependant)
Standard deviation: _______
Percent error: _________
* You should have similar tables for the other glassware items.
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IV. Prelab Questions
1.) What value should be recorded from the measurement in the diagram below?
2 mL
1 mL
2.) Define accuracy, precision, standard deviation and percent error.
3.) Why should you make sure that the balance has all zeroes and the tare is all the way to
zero before you weigh anything. Be explicit in your answer.
4.) A student measures the density of a solution to be 1.36 g/mL. Determine the percent
error if the accepted density was 1.25 g/mL.
V. Post Lab Questions
1.) Which method of determining volume of the metal cylinder is more accurate? Explain
why.
2.) Calculate the mass of soda in a two-liter bottle (d = 1.07 g/mL).
3.) A sample of lead has a mass of 825 g, how much water would the lead displace?
4.) Determine the density of a 32.5 g metal sample that displaces 8.39 mL of water.
5.) In part 2, what glassware item is the most accurate for measuring volume? Which is the
least accurate?
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