# Density Determinations and Various Methods to Measure

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1 Density Determinations and Various Methods to Measure Volume 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 brass-aluminum cylinder and the volume of empty space within a hollow cylinder will also be found. Objectives of the Data Analysis Determine volume by three different methods Use measured volumes and masses to calculate densities Use the relationship between mass, volume, and density to find desired unknown quantities Evaluate results using error analysis 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. ρ = m / V (1) 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 three-dimensional 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 ± 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 c Advanced Instructional Systems, Inc. and the University of California, Santa Cruz 1

2 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 1-3, 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 (mass-based) In parts 4-5, 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 mixed-metal 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 to calculate its volume. V = π ( ) d 2 l = π 2 4 d2 l where d = diameter and l = length. (2a) Figure 1 Density would be calculated in one step to minimize rounding errors: ρ = m V = 4m πd 2 l (2b) The uncertainty in the volume must be determined by error propagation. Mass, length, and diameter measurements contribute to the overall uncertainty. σ ρ = ±ρ [( σd ) + d ( σd ) + d ( σl ) ( σm )] + l m (2c) c Advanced Instructional Systems, Inc. and the University of California, Santa Cruz 2

3 Volume by water displacement For less defined shapes, volume can be determined by water displacement. Volumes of liquids such as water can be readily measured 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 V cyl = V final V initial = V water+cyl V water (3a) The uncertainty of the volume is based on the two volume readings. σ Vcyl = σ Vfinal + σ Vinitial (3b) The density is calculated using m/v. There is not a straightforward way to find density in one step (as with geometry). The uncertainty in the density would be given by: c Advanced Instructional Systems, Inc. and the University of California, Santa Cruz 3

4 [( σv ) ( σm )] σ ρ = ±ρ + V m (3c) Volume by pycnometry Pycnometry is a technique that uses the density relationship between volume and mass, and the vessel used is called a pycnometer. To perform pycnometry measurements, the mass of the cylinder and the mass of a flask filled with water to a mark (A, Fig. 3) are recorded. The cylinder is then inserted into the flask. Water is displaced when the cylinder is inserted. The volume of water displaced is removed by pipet, thereby restoring the water level to the mark (B). The combined mass of the flask, remaining water, and cylinder is then measured. 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. mass A + mass cylinder = mass B + mass displacedwater mass displacedwater = mass A + mass cylinder mass B (4a) 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. V displacedwater = V cylinder = mass displacedwater / densitywater (4b) 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. c Advanced Instructional Systems, Inc. and the University of California, Santa Cruz 4

5 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. m A = m A,trial1 + m A,trial2 + m A,trial #trials σ ma = ± ( m A m A,trial1 ) 2 + ( m A m A,trial2 ) #trials 1 (4c) (4d) 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: σ mwater = σ ma + σ mb + σ mcyl = σ ma + σ ma + σ mcyl (4e) 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 the density can be determined. [( σv ) ( σm )] σ ρ = ±ρ + V m (4f) 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. V cyl = V metal + V void V void = V cyl V metal (5a) c Advanced Instructional Systems, Inc. and the University of California, Santa Cruz 5

6 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 g/cm 3 for Al; between 8 and 9 g/cm3 for brass. V metal = m cyl ρ metal No error propagation is required (5b) 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 X m cyl and (1 - X )m cyl, respectively, where X is the Al fraction and (1-X ) is the brass fraction (the remainder). The cylinder volume is determined by pycnometry and is the sum of the volumes of the two metals: V cyl = V Al + V brass (6a) Replace each volume by its mass divided by its density using V=m/ρ: V cyl = m Al ρ Al + m brass ρ mass (6b) Replace the masses by the equivalent expressions in terms of X and m cyl : V cyl = Xm cyl ρ Al + (1 X)m cyl ρ brass (6c) Divide through by m cyl and replace V cyl /m cyl with 1/ρ cyl : 1 = X (1 X) + ρ cyl ρ Al ρ brass (6d) Collect terms on the right-hand side that contain X : ( 1 1 = X 1 ) + 1 ρ cyl ρ Al ρ brass ρ brass (6e) Solve for X, the mass fraction of aluminum in the mixed cylinder. c Advanced Instructional Systems, Inc. and the University of California, Santa Cruz 6

7 X = ( ) 1 ρ cyl 1 ρ ( brass ) 1 ρ Al 1 ρ brass (6f) 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 b Calculate each fraction in the equation, then the differences, and then the final ratio. 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 density near 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 1-3. 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 g (the uncertainities in your cylinders masses are ± g). You will use these masses throughout the experiment. c Advanced Instructional Systems, Inc. and the University of California, Santa Cruz 7

10 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 Mixed-metal 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 1-X, 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 1-3 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 1, 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 g/ml. 1 images/figure6.png c Advanced Instructional Systems, Inc. and the University of California, Santa Cruz 10

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