33
Measuring Mass, Length, Volume and Density
Exactly how “good” is a measured number?
PRE-LAB ASSIGNMENTS:
Before doing this lab you should have completed the following prelab tutorials or taken the
corresponding fast-track quizzes demonstrating your proficiency.


Measurements and Units
Significant Figures
STUDENT LEARNING OUTCOMES:








How to properly make and record measurements taken with both analog and digital
measurement devices with the maximum precision.
The rules for the use and importance of significant figures.
How the choice of measurement device affects the quality of measurements.
Review calculations of volumes of regularly shaped objects from their dimensions.
Familiarity with the concept of density. Should know how it is used to identify samples
qualitatively and should be able to use it the interconvert between mass and volume.
How to properly use a volumetric pipette.
Define accuracy and precision.
Determine the accuracy and precision of different types of laboratory glassware so that in
future labs you can select the appropriate glassware for a particular application.
EXPERIMENTAL GOALS:
The goal of the first part of this experiment is to determine the densities of a number of
cylinders, and gain an understanding of how different measurement techniques can affect the
reliability of experimental results. In the second part of the experiment, you will measure the
volume of water delivered by several different types of volumetric glassware, and determine the
relative precision of their measurements.
INTRODUCTION:
Measurements are important in all areas of our lives. A great deal of effort goes into
making sure that the measurements we base our decisions on are accurate, but much of that effort
is hidden from our day-to-day experience.
What are some measurements made by you or others that affect you as a consumer?
What are some measurements that would be made by someone in a profession that you are
34
interested in pursuing? In both of these cases, what might the consequences be if those
measurements are off by 5%, 10 % or 25%?
In this lab, you will develop skills for measuring and recording measurements with the
maximum attainable precision, distinguish between accuracy and precision, and develop some
insight into what is required to make measurements that are truly accurate and precise. You will
also learn some tools for gauging the precision (and to a lesser extent, the accuracy) of a
measurement.
You will be required to choose appropriate glassware for future lab exercises based
on what you learn from this lab. Make sure you understand how the choice of glassware
and the proper use of significant figures affect the quality of experimental results.
Determining Volume from the Dimensions of an Object
The volume (V) of a cuboid (or rectangular prism), a box-like 6-sided shape with all
angles being right angles, is calculated from its length (l), width (w), and height (h) by the
equation
V=lwh
For any regular shape (cuboid, cylinder, or prism) in which the area (A) of a face can be easily
calculated, this simplifies to
V=lA
For a cylinder, the face is a circle. Since the area of a circle is πr2 or π(½D)2 (where r is the
radius and D is the diameter), the volume of a cylinder is πr2l [or π(½D)2l].
A
l
Determining Volume by Displacement of a Liquid
Another way to determine the volume of a solid is to measure the volume of a liquid that
is displaced by the solid. This is particularly useful for irregularly shaped objects. This type of
measurement is usually done in a graduated cylinder. (See Figure 1.) The solid object must fit
in the graduated cylinder, and be denser than the liquid.
35
50
50
mL
mL
40
40
30
30
20
20
10
10
(a)
(b)
Figure 1. Measuring the volume of a cylinder by displacement of water. The volume of water
in the cylinder in 1a is 35.5 mL1, and the volume of water in the cylinder in 1b is 47.8 mL,
therefore, the volume of the metal cylinder is 12.3 mL.
Enough liquid to completely cover the solid is added to the graduated cylinder. The volume of
the liquid is measured to the level of precision allowed by the glassware, and then the solid
sample is added to the cylinder, causing the liquid level to increase. A second volume
measurement is made, which gives the total volume of the liquid and the solid. The volume of
the solid is determined by difference:
Solid Volume = Total Volume - Liquid Volume
In this lab, we will apply this technique to a regularly shaped object in order to compare
precision of this method of volume measurement to the other methods used, and to identify an
irregularly shaped object.
1
Why is this number reported to one decimal place? (Consult Introduction and Review of Some Important
Concepts, part I-A, if you are unsure.
36
Density
Density (d) is defined as the ratio of the mass (m) of a sample to its volume (V):
d
m
V
Even though the mass and volume of a sample vary with the sample size, the density of a
particular substance is the same for any size sample; it is an intensive property (see Introduction
and Review of Some Important Concepts, section VI-B). This is true of all “ratioed” properties,
making them useful for sample identification.
Density is useful for a number of reasons. The density of a sample can be used by
scientists or technicians to find the identity of the sample. Density is also used to interconvert
between mass and volume.
Density is an example of a very powerful class of properties that found by dividing one
extensive property by another extensive property. Properties of this sort will be referred to
“ratioed properties” in this course. It is important to realize that the relatively simple
mathematics used in working with these properties is exactly the same, regardless of the
property. If you learn this, and learn how to recognize a “ratioed property,” you will be
able to perform a vast array of important calculations.
If we consider the formula above, we can see that if the density is known, the mass of an
object can be determined from its volume by doing some simple algebra:
m  dV
and the volume can be determined from the mass:
V
m
d
Example: One brand of commercial pancake syrup has a density of 1.23 g/mL.
What would be the mass of the syrup in a 12 ounce (355 mL) bottle?
Solution 1: Using Algebra.
We know the volume and density of the syrup and want to find its mass, so we
can use m = dV. Since the density is in units of g/mL, the volume must also be in
units of mL:
m = dV = (1.23 g/mL)(355 mL) = 436 g
or
37
m
1.23 g 355 mL

 436 g
mL
1
Note that writing the density as a ratio and putting the volume over 1 makes it
easier to see how mL’s cancel, and determine that the units generated from the
calculation is g. Since we are trying to determine mass and g is a unit of mass,
this result provides us with good evidence that the calculation was performed
correctly.
Solution 2: Using Dimensional Analysis.2
In the above example, the density was written as a fraction:
1.23 g
mL
The reciprocal can also be written as a fraction:
mL
1.23 g
Notice that the numerical value of the property, 1.23, is always in front of the unit
(g), the unit that is numerator (top of the fraction) of the property (g/mL).
The key is in knowing which of the two fractions we need to multiply by
in a particular problem: this is done by choosing the form which results in the
original units canceling out and the resulting units being what we want (and
where, in the top or bottom of the fraction, we want them).
The general approach is to start with the original quantity and to write a
fraction symbol next to it.

355 mL 




or
355 mL 

1 



We then write units in the fraction, placing them so that the original unit (mL) is
cancelled out, and the desired unit is where we want it. We want to end up with
mass in g, so g goes in the numerator:
355 mL  g 


1  mL 
Finally we add the numerical value of the property and calculate.
355 mL  1.23 g 

  436 g
1  mL 
Determining the Accuracy of Volume Measurements
2
See Introduction and Review of Some Important Concepts, part IV for a review of Dimensional Analysis.
38
When properly calibrated, mass balances are more accurate and precise than most other
measurement devices routinely used in chemistry laboratories. For that reason, we can use
balances to check the accuracy and precision of volumes delivered by different types of
volumetric glassware, such as beakers, flasks, or pipettes. This is most commonly done by
measuring the mass of water contained or delivered by a piece of glassware, and using the
density of water to calculate the volume of that sample of water. Since the density of water
varies with temperature, it is necessary to determine the temperature of the sample of water, and
look up the density of water at that temperature.
In this portion of the laboratory you will measure out volumes of deionized (DI) water
(why would we want to use DI water instead of ordinary tap water?) with various types of
glassware. You will then determine the accuracy and precision of these glassware types by
weighing the measured volume of water in the container and using the density of water to
calculate the actual volume.
Example: A beaker weighs 47.21 g and a graduated cylinder is used to deliver 22.5 mL water to
the beaker at 27oC. Afterwards, the mass of the beaker and water sample together have a mass of
70.15 g. What volume was actually delivered by the graduated cylinder?
Solution: The mass of water delivered was 70.15 g - 47.21 g = 22.94 g. From Appendix 2C of
the lab manual, the density of water at 27oC is 0.99654 g/mL.
The volume of water delivered is
22.94 g 
mL 

  23.02 mL
1  0.99654 g 
39
PROCEDURE:
A. Cylinder Measurements.
In this part of the procedure, you will work in groups of four. Each group will be given a tube
containing four cylindrical samples of the same material. These samples can be identified by
their color. All four cylinders have a different size. We will designate the cylinders by the terms
smallest, small (for the second smallest), large (for the second largest) and largest:
This procedure must be done by each person in your group. For each cylinder, make the
following measurements and record them in Table 1 (Measurements Made Individually) in
Section A of data sheet. All measurements should be recorded with the correct number of
significant figures. In order to accurately compare the precision of these measurements is
important that the replicate measurements are collected without bias. DO NOT share your data
with your team members until all measurements have been made.
1. Measure the mass of each cylinder, following the directions given in the Mass Measurement
Tutorial. Make sure to tare the balance (i.e., set it to zero) before making each measurement.
2. Using the plastic ruler in your equipment drawer, measure the length and radius of each
cylinder, to the appropriate number of significant figures. Calculate the volume of the
cylinders from these measurements, to the appropriate number of significant figures.
3. Put between 10 and 30 mL of water into a 50 mL graduated cylinder, and record the volume
of water in the cylinder. Holding the cylinder at a 45oangle, gently slide the clean dry
cylinder into the cylinder containing the water. Measure the volume of the combined water
and cylinder. Calculate the volume of the cylindrical sample from this data, to the
appropriate number of significant figures. Repeat this procedure for each cylinder.
4. Combine the data from everyone in your group in Table 2 on your data page (Group Data
Table). Make a copy for everyone in your group and one for your instructor (turn it in before
leaving lab). DO NOT CALCULATE densities, averages and standard deviations AT
THIS TIME. These calculations will be done outside of class. Assign each member in your
group to be responsible for doing these calculations for one of the cylinders. Make sure
calculations are done for all cylinders. Select a person in your group to coordinate the
compilation of these calculated results. These calculations and further analysis of these
results are discussed in Part D. Analysis of Results. You should exchange these calculated
results using email or Blackboard and email a copy to your instructor within 24 hours
finishing the lab work.
40
B. Identifying a Sample Based on Density
1. Each group will be given an irregularly-shaped unknown. Each person in your group should
determine the mass of the unknown, and find its volume by displacement, as you did with the
cylinders. Record this data in the table in Section B of your data page.
2. Use the data collected by your group to calculate the density of the unknown, and determine
its identity using the table of densities provided in Table 1.
C. Verifying Glassware Accuracy and Precision Using Mass Measurements.
1. Each group should put about 400 mL of deionized water into a clean 600 mL beaker, and
immerse a thermometer in the beaker for a few minutes to determine the temperature of the
water. Use the table in Appendix III-C in the back of your lab manual to find the density of
water at this temperature.
2. Each member in the group should follow the procedure below for each of the pieces of
glassware listed in Table 4 of the data sheet.
a. Weigh an empty 50 mL Erlenmeyer flask on the balance. The flask can have some
residual water from the previous run on the inside, but the outside must be dry.
b. Use the glassware item indicated in the first row of Table 4 of the data sheet, “Measuring
Device”, to measure as close to the indicated volume (“Volume to Deliver”, row 2, Table
4) of water as the glassware will allow. Use the markings on the glassware item to record
the volume of the sample to the correct number of significant figures in row 3, Table 4.
For example, if you were asked to deliver 50 mL with a 250 mL graduated cylinder
having 10 mL divisions, you should be able to measure the volume to 1 mL (1/10
smallest division) and the “Measured Volume” would be “50.” or “50” indicating two
significant figures. The value in “Measured Volume” should be the same as the “Volume
to Deliver”, but should be written with the correct number of significant figures.
c. Once the volume of the water has been measured, transfer the water to the Erlenmeyer
flask. Reweigh the Erlenmeyer flask, and determine the mass of water delivered.
d. Once the mass of the water has been measured, empty the flask and make sure the outside
is dry.
3. For each piece of glassware, use the density of water at the measured temperature to calculate
the actual volume of water delivered using the mass of the sample of water.
4. Tabulate all of the volume results into the group data table (Table 5) and determine the
average volume delivered and its standard deviation.
41
D. Analysis of Cylinder Measurement Results (To be completed outside of lab)
The dimensions, volumes and masses determined thus far are extensive properties and are
dependent on how the samples were prepared. When their values were compared, we could only
determine the precision with which we could measure one of the samples. Since all of the
samples your group measured have the same composition, they should all have the same
intensive properties. Since intensive properties are independent of sample dimensions and
history, they can be used without a standard to determine accuracy. Your group will examine
the densities of all of these samples.
Make sure you have gotten all of the results recorded in Table 2 from your team members.
1. Transfer the average densities in Table 2 into the appropriate locations in Table 3. Since all of
the cylinders are made of the same material, they should have the same density. If there are
any glaring discrepancies, you may want to double check the calculations of the outlier
values and consult with your instructor.
2. Calculate overall averages and standard deviations for the densities determined by the two
methods.
3. Use the Data in Tables 2 and 3 to answer the questions in Section D of your data sheet.
42
Table 1. Materials Sorted by Density in g/mL3
Category
Material
Wood
Wood
Chemical
Cherry
Mahogony
Gasoline
Density,
g/mL
0.433
0.705
0.721
Wood*
Maple
0.77
Metal
Liquid
Liquid
Ethyl Alcohol
Methyl Alcohol
0.802
0.809
Ceramic
Metal
Other
Ice
0.897
Mineral
Plastic*
Polypropylene
0.9
Mineral
Liquid
Plastic
Liquid
Liquid
Vegetable Oil
HDPE
Water, 100 °C
Water, 4 °C
0.93
0.955
0.9581
0.99997
Ceramic
Metal
Metal
Metal
Plastic
Polyurethane
1
Metal
Material
Ceramic
Metal
Ceramic
Density,
g/mL
2.6
2.643
2.755
Mineral
Coal, Coke
1.201
Metal
Plastic*
Polyruethane
1.23
Metal
Mineral
Plastic*
Mineral
Coal, Lignite
Phenolic
Coal, Bituminous
Polyvinyl
Chloride (PVC)
Kevlar 29
Coal, Anthracite
Magnesium, Pure
Beryllium
Graphite
Quartz Glass
Teflon (PFTE,
TFE)
1.25
1.32
1.346
Metal
Metal
Metal
Glass
Aluminum
Iron Slag
Aluminum, 2024T3
Alumina
Titanium
Iron Ore,
Magnetite
Iron Ore,
Hematite
Zirconia
Zirconium
Zinc, Cast
Iron, grey cast
Stainless Steel
27Cr
Steel, tool
Carbon Steel
Pure Iron
Nickel
Monel Metal,
rolled
High Speed Tool
Steel
Bronze, phosphor
Cobalt
Copper, Pure
1.37
Metal
Nickel, Pure
8.9
1.44
1.554
1.7
1.8477
2.163
2.2
Metal
Metal
Metal
Metal
Metal
Metal
Silver, Pure
Lead
Mercury
Uranium
Gold, Pure
Platinum
10.5
11.37
13.57
18.74
19.32
21.3
2.2
Metal
Iridium
22.16
Liquid
Plastic*
Plastic*
Mineral
Plastic*
Plastic
Mineral
Metal
Metal
Ceramic
Ceramic
Plastic*
3
Category
Seawater
1.025
Polyamide (nylon) 1.13-1.15
Acrylic
1.16-1.19
Coal, Coke
1.201
Metal
Metal
Metal
Metal
Densities from http://www.solarnavigator.net/material_densities.htm
2.77
3.9
4.5
5.046
5.206
5.75
6.3798
7.049
7.079
7.47
7.715
7.84
7.86
8.602
8.688
8.75
8.8
8.8
8.9
43
LAB REPORT
Measuring Mass, Length, Volume and Density
Name ________________________________
Date _________
Partner ________________________________
Section _________
Report Grade ______
Partner ________________________________
Partner ________________________________
A. Cylinder Measurements
SAMPLE ID: What is the color of your samples? ______________________
Table 1. Measurements Made Individually
Cylinder
Mass
Length
Diameter
Volume from
Measurements
Starting Volume
of Graduated
Cylinder
Final Volume
of Graduated
Cylinder
Volume by
Displacement
Smallest
Small
Large
Largest
44
Table 2. Group Data Table
Ruler Measurements
Volume
Density
Graduated Cylinder
Volume
Density
Mass
Ruler Measurements
Volume
Density
Graduated Cylinder
Volume
Density
Mass
Ruler Measurements
Volume
Density
Graduated Cylinder
Volume
Density
Smallest Cylinder:
Student
Mass
Average:
Stand. Dev.:
Small Cylinder:
Student
Average:
Stand. Dev.:
Large Cylinder:
Student
Average:
Stand. Dev.:
45
Largest Cylinder:
Student
Mass
Ruler Measurements
Volume
Density
Graduated Cylinder
Volume
Density
Average:
Stand. Dev.:
Table 3. Summary of Density Data
Cylinder
Average Density
from Ruler
Measurements
Average Density
from Graduated
Cylinder
Smallest
Small
Large
Largest
Average:
Stand. Dev.:
B. Identifying a Sample Based on Density
Student
Mass of Object
Volume of Object
Average:
Stand. Dev.:
Identity of Object: ____________________
Density of Object
46
C. Verifying Glassware Accuracy using Mass Measurements and Density.
Temperature of water: __________
Density of water:
__________
Table 4. Individual Measurement Data Table
Measuring Device
100 mL
Beaker
50 mL
graduated
cylinder
10 mL
graduated
cylinder
5 mL
volumetric
pipet
Volume to Deliver
20 mL
10 mL
10 mL
5 mL
Measured Volume
(Significant figures
from measuring device)
Mass of Erlenmeyer
flask + Water
Mass of Erlenmeyer
flask
Mass of Water
Volume of Water
Calculated from Mass
and Density
Difference between
Measured Volume and
Calculated Volume
47
Table 5. Group Data Table
Glassware
Used
Glassware
Used
100 mL
Beaker
50 mL
graduated
cylinder
10 mL
graduated
cylinder
5 mL
volumetric
pipet
Actual Volume (determined using mass and density)
Measured
Volume
(determined
from
glassware)
Student 1
Student 2
Student 3
Student 4
Student 5
Average
Standard
Deviation
48
QUESTIONS COMPARING DIFFERENT GLASSWARE TYPES.
1. For each type of glassware used, is the number of significant figures recorded based on the
glassware markings in step 2 consistent with the precision indicated by the standard
deviations of the volumes?
2. Discuss the relative accuracies and precisions of the different types of glassware.
3. Compare the time and effort needed for each type of glassware with the accuracy and
precision obtained.
49
D. Analysis of Cylinder Measurement Results (To be completed outside of lab.)
Consider the cylinder masses in Table 2.
1. Like significant figures, standard deviation is a measurement of measurement reproducibility.
Poorer precision (more scatter in measurements) results in a greater standard deviation.
Ideally, the standard deviation should indicate a variation in the measurement in the same
decimal place as the last significant digit of your measurement. What does the standard
deviation reveal regarding the precision of the mass measurements that you performed?
2. What does the standard deviation reveal regarding the accuracy of the mass measurements
that you performed?
3. Is the number of significant figures recorded for these mass measurements consistent with
the conclusions that you made from the standard deviations regarding precision? (Confirm
with your instructor and/or other students that your response to this question is correct
before continuing.)
50
4. Compare the cylinder diameter measurements made with the ruler. How much do they vary?
What does that tell you about the precision of these measurements?
5. Compare the averages and standard deviations for each of the volume measurements of the
cylinders. What does this tell you about the comparable precisions for the two methods of
determining volume? What does this tell you about the comparable accuracies for the two
methods of determining volume?
6. Please explain the following statement: “None of the data obtained in this lab gives any
conclusive way to determine which of the methods of measuring volume is more accurate.”
51
7. If lengths or volumes measured by different methods are different, what would have to be
done to positively determine which method was the most accurate?
8. Use your density to determine the composition of your samples using the density table
provided. Some materials can be determined readily from the density and knowing what
class of material it is plastic, metal, ceramic, etc. In some cases, it may be necessary to
examine other characteristics. If needed a collection of cube shaped samples is available in
the front of the room to help you distinguish between different materials that have similar
densities. (Dyes have been added to many of the samples we are using to identify the sample
type. All cylinders of the same color are made of the same material; however, cylinders of
the same material might have different colors. Since a very small amount of dye is required
to color the materials, the presence of the dye has no effect on the density of the sample.)
52
9. Compare the average density obtained with each method to the density given in the table for
the substance you identified your sample to be. Which methods for determining the density
is more accurate (closer to the accepted density)?
10. Based on your data does better precision always correlate with better accuracy?