Goal and Overview
This lab provides an introduction to the concept and applications of density measurements. The densities of brass and aluminum will be calculated from mass and volume measurements. To illustrate the effects of precision on data, volumes will be determined by three different methods: geometrically (measuring lengths); water displacement; and pycnometry. The composition of a mixed brassaluminum cylinder and the volume of empty space within a hollow cylinder will also be found.
Objectives and Science Skills
•
Use three methods to determine the volumes of solid aluminum and solid brass cylinders and assess the relative merits and limitations of each method.•
Use massbased pycnometry measurements to find the volume of a void in a hollow cylinder and the mass fractions of aluminum and brass in a plugged (mixed) metal cylinder.•
Calculate resultsbased (values and uncertainties) experimental data, known mathematical relationships (e.g., between mass, volume, and density), and statistical methods (e.g., error analysis and propagation).•
Evaluate the experimental methods and their outcomes in terms of parameters such as reliability, difficulty, accuracy, and precision.
Suggested Review and External Reading
•
Data analysis introduction, textbook information on density
Introduction
The density, ρ, of an object is defined as the ratio of its mass to its volume. Density can be useful in identifying substances. It is also a convenient property because it provides a link (or conversion factor) between the mass and the volume of a substance.
( 1 )
ρ = ^{m}/_{V}
Mass and volume are extensive (or extrinsic) properties of matter  they depend on amount. Density, an intensive (or intrinsic) property, is a kind of "heaviness" factor. In macroscopic terms, density reflects how much mass is packed into a given threedimensional space. Typically, densities are reported g/ml or g/cm^{3} (which are equivalent because 1ml ≡ 1cm^{3}). Experimentally, mass and volume measurements are required to calculate density.Masses are measured on electronic balances. Pan balances, which are accurate to ±0.01 g, are used for quick measurements where greater precision is not required. Analytical balances (accurate to ±0.0001 g) are used for more precise measurements.Volume is an amount of space, in three dimensions, that a sample of matter occupies. The number and the phase of the molecules in the sample primarily determine the volume of a substance. Volume will be measured in many ways in this course, but the units are usually milliliters (mL) or cubic centimeters (cm^{3}). Methods for determining or delivering precise volumes include volumetric pipets and pycnometers; less precise methods include burets, graduated cylinders, and graduated pipets.In this experiment, you will measure masses and volumes to determine density. Four different metal cylinders are investigated.In parts 13, three different methods are used to find volume of two solid metal cylinders (Al and brass). Each method has its own degree of precision.
(i)
volume by geometry
(ii)
volume by water displacement
(iii)
volume by pycnometry (massbased)
In parts 45, one method for volume determination is used to find:
(i)
the volume of a void inside a hollow cylinder; and,
(ii)
the percent composition of a mixedmetal cylinder.
Volume by geometry
A cylinder is a standard geometric form. In this case, you can measure the dimensions of the cylinder and apply the formula tocalculate its volume.
( 2a )
V = π
d 
2 
2  
π 
4 
Figure 1
Density would be calculated in one step to minimize rounding errors:
( 2b )
ρ =
m 
V 
4m 
πd ^{2} l 
The uncertainty in the volume must be determined by error propagation. Mass, length, and diameter measurements contribute to the overall uncertainty.
( 2c )
σ _{ρ} = ± ρ
σ _{d} 
d 
σ _{d} 
d 
σ _{l} 
l 
σ _{m} 
m 
Volume by water displacement
For less defined shapes, volume can be determined by water displacement. Volumes of liquids such as water can be readilymeasured in a graduated cylinder.To use the water displacement method, an object (in this case, a small metal cylinder) is inserted into a graduated cylinder partially filled with water. The object's volume occupies space, displacing liquid and raising the water level. The difference between the two volumes, before and after the object was inserted, is the object's volume.
Figure 2
( 3a )
V_{cyl} = V_{final} − V_{initial} = V_{water + cyl} − V_{water}
The uncertainty of the volume is based on the two volume readings.
( 3b )
σ _{Vcyl} = σ _{Vfinal} + σ _{Vinitial}
The density is calculated using m/V. There is not a straightforward way to find density in onestep (as with geometry).The uncertainty in the density would be given by:
( 3c )
σ _{ρ} = ± ρ
σ _{V} 
V 
σ _{m} 
m 
Volume by pycnometry
Pycnometry is a technique that uses the densityrelationship between volume and mass, and the vesselused is called a pycnometer .To perform pycnometry measurements, the mass ofthe cylinder and the mass of a flask filled with water to amark (A, Fig. 3) are recorded. The cylinder is then insertedinto the flask. Water is displaced when the cylinder isinserted. The volume of water displaced is removed bypipet, thereby restoring the water level to the mark (B). The combined mass of the flask, remaining water, and cylinder is thenmeasured.
Figure 3
The sums of the masses before and after are equal. The mass_{A}, the mass_{B}, and the mass_{cylinder} were all measured on the balance. There is only one unknown in the equation  the mass of the displaced water.
( 4a )
mass_{A} + mass_{cylinder} = mass_{B} + mass_{displaced water} 
mass_{displaced water} = mass_{A} + mass_{cylinder} − mass_{B} 
The volume of water removed is equal to the volume of the cylinder. Mass_{water} can be converted to volume using the density of water.
( 4b )
V_{displacedwater} = V_{cylinder} = ^{massdisplaced water} / _{densitywater}
The density of the cylinder is calculated using m_{cyl}/V_{cyl}.The uncertainty calculation requires a few steps and assumptions. The volume of the cylinder was equal to the volume of the water. V_{water} was based on the three mass measurements  the mass of the cylinder, of A, and of B.The uncertainty in mass_{cylinder} comes from the balance reading.The uncertainty associated with mass_{A} and mass_{B} depends on your ability to precisely adjust the level of the water to the mark at the exactly same place every time (calibration). By repeatedly filling the flask to the mark and taking the mass readings, the average mass of A and the standard deviation (the fluctuation in the mass due to variations in the exact liquid level) can be found.
( 4c )
m_{A} =
m_{A,trial1} + m_{A, trial2} + m_{A, trial3} + 
# trials 
( 4d )
σ_{mA} = ±

Assume the uncertainty in the mass of both A and B is the same: m_{A} ± σ_{mA}; m_{B} ± σ_{mA}.The uncertainty in the mass of water displaced is determined by error propagation:
( 4e )
σ_{mwater} = σ_{mA} + σ_{mB} + σ_{mcyl} = σ_{mA} + σ_{mA} + σ_{mcyl}
The density of water at room temperature is known quite precisely and is assumed to contribute negligible error (see table at the end of the lab), so dividing σ_{m, water} by the density of water to give σ_{V, water} is adequate. Since σ_{V, water} = σ_{V, cyl}, the uncertainty in thedensity can be determined.
( 4f )
σ_{ρ} = ± ρ
σ_{V} 
V 
σ_{m} 
m 
You will use pycnometry in parts 4 and 5 to determine the volume and/or density of a hollow cylinder and of a mixed cylinder.
Volume of a void inside a hollow cylinder
A hollow cylinder has an empty space inside.
Figure 4
The volume of the cylinder is comprised of the volume of metal and the volume of the void inside.
( 5a )
V_{cyl} = V_{metal} + V_{void} → V_{void} = V_{cyl}  V_{metal}
V_{cyl} is determined by pyncometry. The volume occupied by the metal can be determined using the mass of the cylinder (which is due to only the metal, not the void) and the density of the metal, which was determined previously in the lab (either Al or brass, depending on the cylinder). Use the value for density that is closest to the literature values  2.70 g/cm^{3} for Al; between 8 and 9 g/cm3 for brass.
( 5b )
V_{metal} =
m_{cyl} 
ρ_{metal} 
Percent composition of a mixed cylinder
The total mass of the cylinder, m_{cyl}, is the sum of the mass of Al and brass (m_{Al} + m_{brass}). In terms of fractional composition, this would be Xm_{cyl} and (1  X)m_{cyl}, respectively, where X is the Al fraction and (1X) is the brass fraction (the remainder).The cylinder volume is determined by pycnometry and is the sum of the volumes of the two metals:
( 6a )
V_{cyl} = V_{Al} + V_{brass}
Replace each volume by its mass divided by its density using V=m/ρ:
( 6b )
V_{cyl} =
m_{Al} 
ρ_{Al} 
m_{brass} 
ρ_{mass} 
Replace the masses by the equivalent expressions in terms of X and m_{cyl}:
( 6c )
V_{cyl} =
X m_{cyl} 
ρ_{Al} 
(1−X)m_{cyl} 
ρ_{brass} 
Divide through by m_{cyl} and replace V_{cyl}/m_{cyl} with 1/ρ_{cyl}:
( 6d )
1 
ρ_{cyl} 
X 
ρ_{Al} 
(1 − X) 
ρ_{brass} 
Collect terms on the righthand side that contain X:
( 6e )
1 
ρ_{cyl} 
1 
ρ_{Al} 
1 
ρ_{brass} 
1 
ρ_{brass} 
Solve for X, the mass fraction of aluminum in the mixed cylinder.
( 6f )
X =
 

This is the equation to use. Density of the cylinder is found by pycnometry. The densities of Al and brass have already been determined.When finding X:
a
Calculate each fraction in the equation, then the differences, and then the final ratio.
b
Use the densities of brass and aluminum determined experimentally.
c
Find X, and use X to determine the mass fraction of brass in the mixed cylinder, 1  X.X has a range of possible values from zero to one (0  100%). If your mixed cylinder's density is between that of aluminum and of brass, you should calculate a percent of aluminum that makes sense. For example, if the mixed cylinder has a densitynear that of Al, X should be near one.
Equipment List
 cylinders: brass, aluminum, mixed brass/aluminum, and hollow
 Vernier caliper
 50 mL Erlenmeyer flask, 100 mL graduated cylinder, 400 mL beaker
 lab marker
 Pasteur pipet
 thermometer
Procedure
NOTE: IF YOU WORK WITH ANOTHER SET OF PARTNERS, MAKE SURE YOU RECORD ALL DATA. YOU WILL NOT BE ABLE TO COMPLETE THE DATA ANALYSIS IF YOUR DATA TABLES ARE INCOMPLETE. ALSO CHECK THAT DATA MAKES SENSE.Parts 13. Density of aluminum and brass cylinders using three different methods of volume measurement
Part 0: Measure metal cylinder masses.
1
Obtain four cylinders  brass, aluminum (solid cylinders marked S), hollow (marked H), mixed brass/aluminum (marked P for "plugged"). Return cylinders to the stockroom at the end of lab.
2
Record the cylinders' numbers.
3
Record the masses of the cylinders on the analytical balance to the 0.0001 g (the uncertainities in your cylinders' masses are ±0.0001 g). You will use these masses throughout the experiment.
Part 1: Volume by Geometry
1
Measure the diameter and length of each cylinder using the Vernier calipers. Your TA will help you if you need it. Record the values to the 0.01 cm (each measurement is ±0.01 cm).
2
Determine the density of the cylinder. Find the uncertainity using error propagation.
Part 2: Volume by Displacement
1
Put enough water to cover the metal cylinder into a 100mL graduated cylinder and record the volume. The graduated cylinder is not very precise; readings will be ±0.5 mL (the digit in the tenths place will either be a 5 or a 0).
2
Carefully slide the metal cylinder down the side of the graduated cylinder into the water. Tossing it in can break the bottom of the graduated cylinder.
3
With the metal cylinder completely submerged, record the new volume reading (to ±0.5 mL).
4
Determine the volume of the metal cylinder. Calculate the uncertainty in your volume using error propagation.
5
Determine the density of the cylinder. Calculate the uncertainty using error propagation.
Part 3: Volume by Pycnometry
1
Fill a 400 mL beaker with water and measure its temperature. Use this water throughout the experiment. Assume that the density of water makes a negligible contribution to the overall uncertainty in the values calculated.
2
Make your pycnometer.
a
Draw a ring midway up the neck of a 50 mL Erlenmeyer flask with a waterproof marker or wax crayon, as shown below.b
Invert the flask on the table; hold marker on top of something solid; and, rotate the flask while marking the neck at a constant height.
Figure 5
3
Calibrate your pycnometer. How well can you adjust the water's meniscus to the top of the line drawn? Precise filling to that mark increases reproducibility (and data quality). Practice with the pycnometer before making measurements. Your TA will demonstrate.The pycnometer filled with water to the mark is called 'A'.
a
Use a disposable pipet to add and remove drops of water to adjust the meniscus to the top of the line.b
Record the mass of the flask and water. No drops should appear on the neck of the flask above the water line.c
Pour out the water into your 400mL beaker; refill to the mark; reweigh.d
Repeat step c until you have three similar values for the mass of 'A'.e
Determine the average m_{A} and its standard deviation (σ_{mA}).The standard deviation, σ_{mA}, reflects your ability to reproducibly fill the pycnometer to the same place every time you use it; σ_{mA} is the uncertainity of the pycnometer and should be read with the average mass of 'A' as well as the mass of 'B' (parts 3, 4, and 5).f
If you share data with another set of students, make sure to record their calibration data as well. You must use the correct calibration result with the appropriate data. In your lab notebook, label which cylinders go with which pycnometer calibration.
4
Indirectly measure the mass of water displaced by your solid cylinders. The pycnometer containing the metal cylinder with water filled to the mark is called 'B'.
a
Carefully insert a metal cylinder, fill with water to the mark, and record the mass (the flask with water and cylinder).b
Repeat filling and weighing several times until the data appears reproducible.c
Calculate the mass of the water removed. Convert this mass to volume by dividing by the density of water (use a precise value, specific to the water's temperature). This volume equals the volume of the metal cylinder. Calculate the uncertainty in the mass of water removed using error propagation. Convert this mass to volume units by dividing by the density of water (use a precise value, specific to the water's temperature). This value equals the uncertainity in the volume of the metal cylinder.
e
Repeat with the other cylinders as instructed.
5
Determine the density of each cylinder. Include the uncertainties.
Please do not throw the metal cylinders away. Please return them to the reagent bench.Please put disposable pipets (and any broken glassware) in broken glass containers, not the trash can. You will lose points for inappropriate disposal.
Part 4: Determine Void Volume in a Hollow Cylinder by Pycnometry
1
For the hollow cylinder, record identity of metal of the hollow cylinder (either aluminum or brass). You recorded its mass at the beginning of the experiment.
2
Insert the cylinder into the pycnometer; remove the water above the line, and record the new mass.
3
Determine the volume of the cylinder and calculate the volume of the void. No error propagation.
Part 5: Mass Fraction of Al and Brass Determination for Mixedmetal Cylinder by Pycnometry
1
You recorded its mass of the mixed metal cylinder at the beginning of the experiment.
2
Insert the mixed Al/brass cylinder into the pycnometer, remove the water above the line, and record the mass.
3
Determine the density of the cylinder and the mass fractions of Al and of brass (X and 1X, respectively). No error propagation.
Reporting Results
Complete your lab summary or write a report (as instructed).
Results / Sample Calculations
 Masses and volumes for solid cylinders by each method
 %error relative to literature values (ρ_{Al} = 2.70 g/cm^{3}; ρ_{brass} = 8.44 g/cm^{3}, depending on the alloy's composition)
 Void volume
 Mass fraction
 Error analysis for parts 13
Discussion
 What you found out (refer to results tables) and how for all 5 parts
 What were the major experimental sources of error?
 Compare the three methods used to determine volume  which method was more accurate and why? Which was most precise?
 What could be done to improve the precision in any or all of the methods?
 How does the instrument error compare to standard deviation error?
Review Questions
Whole degrees are listed down the left hand side of the table, while tenths of a degree are listed across the top. So to find the density of water at 5.4 °C, find the whole degree by searching down the left hand column to '5'. Then slide across that row to '0.4'. The density of water at 5.4 °C is 0.999957 g/mL.
Copyright © 2011 Advanced Instructional Systems, Inc. and the University of California, Santa Cruz  Credits
FAQs
What are some lab methods for determining density? ›
Common laboratory gravimetric procedures for determining density are the buoyancy technique, the displacement principle and the pycnometer method.
What are the 3 methods of determining volume to calculate density? ›To illustrate the effects of precision on data, volumes will be determined by three different methods: geometrically (measuring lengths); water displacement; and pycnometry. The composition of a mixed brassaluminum cylinder and the volume of empty space within a hollow cylinder will also be found.
How do you measure density and volume? ›The formula for density is the mass of an object divided by its volume. In equation form, that's d = m/v , where d is the density, m is the mass and v is the volume of the object. The standard units are kg/m³.
What method is used to measure volume? ›The volume of these objects can be found by water displacement. A volume of water sufficient to cover the object is placed in a graduated cylinder and the volume read. The object is added to the cylinder and the volume read again.
What is density determination? ›Density is a measure of how compact the mass in a substance or object is. The density of an object or substance can be calculated from this equation: density in kilograms per meter cubed is equal to mass in kilograms, divided by volume in meters cubed (p = m / v).
What are two methods of determining density of an object? ›Students use two different methods to determine the densities of a variety of materials and objects. The first method involves direct measurement of the volumes of objects that have simple geometric shapes. The second is the water displacement method, used to determine the volumes of irregularly shaped objects.
What 3 tools can be used to measure liquid volume? ›Volumetric flasks, burettes, and pipettes made for measuring small amounts of liquid are the most accurate, with tolerances of less than ±0.02.
What are the 3 formulas for volume? ›...
Volume Formulas of Various Geometric Figures.
Shapes  Volume Formula  Variables 

Rectangular Solid or Cuboid  V = l × w × h  l = Length w = Width h = Height 
Density is the mass per unit volume of a substance. The formula for density is: density = mass/volume. The unit for mass is kilograms (kg)/grams (g) and volume in centimetres cubed (cm^{3}) or metres cubed (m^{3}).
What is volume and density? ›Volume refers to the measurement of the amount of threedimensional space occupied by an object. Unlike mass, volume changes according to the external conditions. Density refers to the mass contained in a substance for a given volume.
What determines the density of an object? ›
The mass of atoms, their size, and how they are arranged determine the density of a substance. Density equals the mass of the substance divided by its volume; D = m/v. Objects with the same volume but different mass have different densities.
How is volume measured in laboratory? ›The volume of a liquid can be directly measured with specialized glassware, typically in units of milliliters (mL) or liters (L). In this lab, a beaker, two graduated cylinders and a burette will be used to measure liquid volumes, and their precision will be compared.
Which is the most accurate method of volume measurement? ›Volumetric flasks, burets and pipets are the most accurate with tolerances of less than 0.2%. To achieve these accuracies the person using the device needs to use the proper technique and the measurements need to made at the temperature for which the glassware was calibrated (usually 20 degrees C).
What are the two types of volume measurements? ›The two most common measurements of volume are: Milliliters. Liters.
What are the 3 types of density? ›There is arithmetic density, physiological density, and agricultural density.
How do you determine density example? ›The formula for density is d = M/V, where d is density, M is mass, and V is volume. Density is commonly expressed in units of grams per cubic centimetre. For example, the density of water is 1 gram per cubic centimetre, and Earth's density is 5.51 grams per cubic centimetre.
Why determination of density is important? ›Density is an important concept because it allows us to determine what substances will float and what substances will sink when placed in a liquid. Generally, substances float so long as their density is less than the density of the liquid they are placed in.
Which method of determining density is more accurate? ›Answer and Explanation: Archimedes's principle method is more accurate in the determination of the density. According to this method, the buoyant force that is there on the object is equal to the weight of the fluid that it displaces.
What is the best way to measure density? ›One of the basic terms used in many natural sciences is density, a physical property defined as an object's mass divided by its volume. This means that to measure density, you typically need to measure an object's mass and volume separately, then calculate its density by dividing the mass by the volume.
What are essential tools for measuring by volume? › Beakers and Flasks. ...
 Graduated Cylinders. ...
 Burets. ...
 Pipets.
What are the 5 units of volume? ›
...
SI Unit of Volume.
cubic kilometer  km³  1,000,000,000 m³ 

cubic decimeter  dm³  0.001 m³ 
cubic centimeter  cm³  0.000001 m³ 
cubic millimeter  mm³  0.000000001 m³ 
Liquid volume is usually measured using either a graduated cylinder or a buret. As the name implies, a graduated cylinder is a cylindrical glass or plastic tube sealed at one end, with a calibrated scale etched (or marked) on the outside wall.
How do you explain volume? ›Volume refers to the amount of space the object takes up. In other words, volume is a measure of the size of an object, just like height and width are ways to describe size. If the object is hollow (in other words, empty), volume is the amount of water it can hold. Try this at home: Take a large cup and a small cup.
What is the measuring instrument for density? ›hydrometer, device for measuring some characteristics of a liquid, such as its density (weight per unit volume) or specific gravity (weight per unit volume compared with water).
What is density explain with example? ›Density means that if you take two cubes of the same size made out of different materials and weigh them, they usually won't weigh the same. It also means that a huge cube of Styrofoam can weigh the same as a tiny cube of lead. Examples of dense materials include iron, lead, or platinum.
What is called volume? ›In mathematics, 'Volume' is a mathematical quantity that shows the amount of threedimensional space occupied by an object or a closed surface. The unit of volume is in cubic units such as m^{3}, cm^{3}, in^{3} etc. Sometimes, volume is also termed capacity.
Why is density and volume important? ›Knowing the density of materials is very useful. Designers will estimate the weight of a product by multiplying the volume by the density of a material, this is very useful if a product had to be designed within a certain weight limit. You can also identify a material by measuring its volume and mass.
What factors influence density? ›Density: Why It Matters
Pressure, temperature and humidity all affect air density. And you can think of air density as the mass of air molecules in a given volume.
Density of an object depends on its mass and the space or volume occupied by it. d e n s i t y = M a s s V o l u m e. SI unit of volume is k g m  3 .
What are two ways scientists measure volume? ›Scientific Measurement of Volume
It is most often measured by the liter (L), 1.057 qt. in conventional units, or the milliliter (mL), 1/1000 of a liter, about 0.0338 of an ounce. It is often measured by cylinders, flasks, pipettes, or syringes in and out of the laboratory.
How do you measure volume example? ›
For example, if a cup can hold 100 ml of water up to the brim, its volume is said to be 100 ml. Volume can also be defined as the amount of space occupied by a 3dimensional object. The volume of a solid like a cube or a cuboid is measured by counting the number of unit cubes it contains.
How do you measure volume of water in a lab? ›For liquids, volume can be measured using volumetric flasks, graduated cylinders, volumetric pipettes, or burets. In this experiment we will use graduated cylinders to measure the volume of water. Density is an example of an intensive property, meaning that it doesn't depend on the amount of the material.
What is the accurate measured volume example? ›Example: The markings on a 100mL graduated cylinder are every 1mL, so the volume can be measured to ±0.1mL. In the measured value 74.3mL, the value 74 is known with certainty and the .
What lab equipment is used to determine the density of a liquid? ›hydrometer, device for measuring some characteristics of a liquid, such as its density (weight per unit volume) or specific gravity (weight per unit volume compared with water).
What is the standard method to test density? ›The more common being Method A, can be used with sheet, rod, tube and molded articles. For Method A, the specimen is weighed in air then weighed when immersed in distilled water at 23°C using a sinker and wire to hold the specimen completely submerged as required. Density and Specific Gravity are calculated.
What equipment is best for density? ›Today, we can determine density with a lab balance and a density kit. Most lab balances feature builtin software to measure density, so the addition of a density kit makes it much simpler to calculate than with the traditional method.