Density Determinations

Identification of Metals and Plastics

by Density Determinations

Abstract: We will determine the density of samples of metal and plastic by different methods, and determine whether the data we collect are good enough to positively identify the metal or plastic. Plastics may be sorted for recycling by measuring their densities. An important goal is to report random error in measured and calculated quantities, and we will use significant figures to indicate the precision of measurements.

Introduction

I. Density

Density is the mass per unit volume of matter:

D(g/cm³)= m(g) / v(cm³)

Solids have densities ranging from extremely low values for Styrofoam to a maximum of 22.48 g/cm³ for osmium. The densities of some metals are given below. A much more complete listing of densities of many materials can be found on the web at http://www.matls.com.

Metal	Density (g/cc)	Metal	Density (g/cc)
Al	2.7	Zn	7.14
Fe	7.86	Cr	7.20
Cu	8.92	Pb	11.35
Mg	1.74	Au	19.3
Sn	7.28	Brass (Cu/Zn)	8.4-8.7
Pt	21.45	Bronze (Cu, Sn...)	8-9 (dep. on type)
		Steel (Fe, C, ...)	7.7-8.1

Density is easily determined by measuring the mass and volume of a sample. But how confident can we be in the measurements? What if our value does not match that of any of the metals in a table of densities--should we conclude that our measurement is in error, or that we have found a rare metal that is not listed in the table? Recording the correct number of significant figures, as explained below, is the key.

II. Errors in Measurement

A measurement is worthless (or at least not scientific) without a statement of its error, and every measurement has some error associated with it. An error is not a mistake, or something done incorrectly. An error is an uncertainty in measurements that results from the fact that instruments never give one precise value, but always give a range of values in the vicinity of the exact, correct value.

Think of it this way: An object might weigh exactly 1 g, which would be written as 1.00000000....g, with an infinite number of zeros. Many modern electronic balances have divisions down to about 0.001 g (0.1 mg), however, so all the remaining decimal places past the 3rd decimal place are uncertain. On this balance, the mass 1.000 really represents a range of measurements that are too small to register “1.001” and too large to measure 0.999. “1.000” on this balance represents masses from about 0.9995 to 1.0005 g, a range of +/- 0.0005, or +/- one-half the smallest measurable mass (smallest scale division). Measurements above 0.0005 g would show up on the balance as 1.001 g, and measurements below 0.9995 would show up as 0.999 g.

Because the balance represents any random weight between 0.9995 g and 1.0005 g as 1.000 g, we say that the random error is 0.0005 g, one-half the smallest scale division.

Since we are weighing 1.000 g, the percent random error is

Notice that a measurement like 1.000 g, which has 4 significant figures (as does 1.2345 g), has a very low percent error. Sometimes the percent random error is called “precision.”

What if we were to use a crude balance, with markings only every gram? This balance would record “1” for a mass larger than 0.5 g, and smaller than about 1.5 g. For the 1 significant figure measurement, “1 g”, then, the random error would again be 1/2 the smallest scale division, or 1/2 x 1 g = 0.5 g, and the percent random error would be

For this balance, which gives only 1 significant figure, the random error or precision is 50% (a terribly crude measurement)!

We see that there is a correlation between the number of significant figures in a measurement and its precision, with 1 significant figure giving 5% to 50% error (see Prelab Exercise 1):

# sig. figs: 5 4 3 2 1

__|___________|___________|___________|___________|___________|__

precision: 0.0005% 0.005% 0.05% 0.5% 5% 50%

Using the correct number of significant figures in reporting a measurement is a simple but effective way of communicating the approximate percent random error or "precision". Precision is a measure of the range of possible values that a measurement reflects. It will be necessary to make a clear distinction between precision and accuracy, which is defined in another experiment.

Number of Significant Figures from Non-Digital Devices (graduated cylinders, rulers, etc.)

The volume of the metal sample will be determined by either water displacement in a graduated cylinder, or by direct measurement with a ruler. How should the error in these measurements be determined, since they are analog (continuous) measurements in contrast to the digital balance discussed above?

Many "rules of thumb" can be used to estimate error in measurements from various devices. To simplify matters, we might use the same rule for all devices, not just digital ones; two rules have been suggested:

Rule I: Record only measurements made with smallest scale divisions.

An alternative is

Rule II: Estimate one more digit than can actually be read on the instrument. It is common practice, for example, to “estimate” a value that lies between two scale markings on a burette.

Usually Rule II tends to slightly underestimate error, while Rule I tends to overestimate it [It is best to experimentally determine the error in an instrument reading, and see which rule gives the best estimate under the conditions of our experiment, and another experiment in this book does so].

Example: Ruler with 0.1 cm (1 mm) divisions:

Correct measurements under Rule I Incorrect measurements under Rule I

.1 cm" (= "21 mm"), “2 cm”

3.5 cm" (= "135 mm"), “1.0 mm”,

"0.1 cm" (= "1 mm") or "2.10 cm"

Explanation:

"2 cm" indicates that the instrument is less precise than it actually is. Since the ruler can measure to the nearest mm or 0.1 cm, the length of an object that falls exactly at the 2 cm mark should be recorded as "2.0 cm".

"2.15 cm" or "1.1 mm" incorrectly implies that hundredths of cm (or tenths of a millimeter) can be measured (Note: as mentioned above, some instructors will say that it is better to slightly overestimate the precision by writing "2.15 cm" than it is to underestimate the precision by writing "2.1 cm"; nonetheless, we will generally write "2.1 cm").

Another Means of Expressing Random Error: The Standard Deviation

We saw above that one-half the smallest scale division of an instrument can be used as an estimate of the random error expected from that instrument. The “standard deviation” (symbolized s or “s“) is another means of estimating the “range” or uncertainty in a measurement, especially when several determinations have been made. It is defined so that about 68% of the measurements of a single quantity would lie within 1 standard deviation of the mean (m +/- 1 s), and about 95% of the measurements lie within 2 standard deviations of the mean.

In the Excel Spreadsheet, the standard deviation of values in a range of cells can be calculated by entering the formula “=STDEV(Cell 1...Cell 2)” in an empty cell, where (Cell 1...Cell 2) is the range of cells containing the measurements.

Calculation of the Standard Deviation: Since the "standard deviation" is such an important indicator of precision, an example of how it is calculated follows. Suppose we collect three measurements, 1 g, 2 g, and 3 g for the mass of an object (with a crude balance)! Then:

measurement,x average,x deviation, x‑x (x ‑ x)²

1 2 ‑1 1

2 2 0 0

3 2 1 1

_______________________________________________________________________

sum of deviations squared, S(x‑x)² 2

number of measurements, n 3

With very large sets of data, “n” is more appropriate than “n‑1” in the denominator, but with the limited datasets in the laboratory, we will use (n‑1).

III. Errors in Calculated Quantities:

The error in a calculated quantity can also be indicated by writing the correct number of significant figures for the quantity. When reading that result, we’d assume that the last digit is questionable, and that the error is therefore about ½ the smallest decimal place. That is, the error in “1.0g” is probably about 0.05 g.

Two rules are applied to arrive at the correct number of significant figures:

(1) When data are multiplied or divided to get the calculated quantity, it may have no more significant figures than the measurement with the fewest.

(2) When data are added or subtracted to give a calculated quantity, it may have no column smaller than the smallest column common to both measurements. Thus the sum of 1.12 cm and 21 cm is 22 cm, where the units column is the smallest column common to both measurements.

III. Density of a Plastic by Floatation

Plastics must be separated according to type for recycling, because mixing types can degrade their physical properties. It is possible to identify plastics by measuring their density, although infrared spectroscopy is used for a more positive identification, and a portable "Tribopen" which identifies plastics by the amount of static charge that develops on their surfaces is used for rapid field ID[1]. Common recycling symbols, the corresponding type of plastic, and its density are given below:

Alternatively, the Materials Information Resource on the Web (http://www.matls.com/search.htm) will do a search of virtually all known plastics based on density.

The density of plastics[2] will be determined by placing a sample of the plastic in water. If the sample floats, the water is more dense than the plastic. The density of water is 1 g/mL (density is temperature dependent, and the exact value for water at 25.0^oC is 0.998 g/mL). A less dense liquid, like methyl alcohol (D = 0.89 g/mL) can be added to the water until the plastic neither floats nor sinks. At this point, the density of the liquid is the same as the plastic. Because these densities are similar, a high precision method of determining the density of a liquid will be necessary, and that method involves linear regression.

Using Linear Regression to Improve Precision

We can determine the density of a liquid by measuring the mass of a single sample, but the volume measurement often limits the precision. Measuring the mass and volume of several samples, and plotting the volumes vs. the mass to get a straight line, can do much to reduce random error and thus improve precision.

The general equation for a straight line is

y = mx + b

where "m" is the slope, "b" is the y‑intercept, "y" represents a dependent variable (usually plotted on the vertical axis), and "x" represents the independent variable (usually plotted on the horizontal axis). The density equation, (1), has this form, because it can be rearranged as shown below so that if the volume is plotted on the horizontal axis and the mass on the vertical axis, a straight line will result with a slope equal to the density (m = D). The y‑intercept is 0, indicating that if something has zero volume it must have zero mass:

y = m x + b

m(g) = D(g/cm³) v(cm³) + 0

If a straight line is drawn that comes as close as possible to as many of the points as possible, its slope will be a very precise estimate of the density. Because the random error in individual points will cause about equal numbers to lie above the line and half to lie below the line, random errors tend to cancel out. The mathematical process of determining the best straight line is called linear regression. It is a built-in procedure in Excel, and you may want to review the Practice3 spreadsheet for practice in Linear Regression.

Prelab Assignment

1. Show that the percent random error in a 3 significant figure measurement ranges from about 0.05 to 0.5%, by calculating the implied percent error in measurements like 9.55 g and 1.00 g.

2. Give some examples of correct and incorrect measurements using Rule II for recording significant figures, and explain why they are correct or incorrect. See the similar examples for Rule I in the introduction.

3. In this experiment and in those to follow, we will automatically create graphs using the Excel Spreadsheet. Understanding how to create graphs is so important, however, that we will require you to create a graph by hand as a "ticket" allowing you to use Excel from now on.

a. Please graph the following data using volume as the independent variable plotted on the abscissa (x axis), and mass as the ordinate (y axis), draw the best straight line, label the and axes with its parameter and units in parentheses, and give the graph an appropriate title. Use one of the graphs in this book as a model. Make sure the divisions on the axes are spaced equally (eg. every 1.0 g).

b. Select two points and show a calculation of the “rise” (Dy), “run” (Dx) and slope of the line.

Volume (mL) Mass (g) of water

0 0

1.0 1.8

1.5 2.0

2.0 2.8

3.8 4.8

6.2 7.5

10.0 12

4. How should one record the measurement (with the proper number of significant figures) of a paper clip which is exactly two centimeters long if it is measured with (a) a ruler with 1 mm divisions (b) a micrometer with 0.001 cm markings as the smallest divisions?

5. What is the precision, or percent error, in the measurement of a paper clip exactly two centimeters long if measured with (a) a ruler with 1 mm markings and no smaller divisions and (b) a micrometer with 0.001 cm markings?

6. What is the density of a metal block weighing 109.00 g and having dimensions 12.0 cm x 0.8 cm x 4.2 cm? How many significant figures should the density value have?

7. If the true density of the metal block in Q6 above is 3.2 g/cm³, calculate the absolute error and the percent absolute error. Does either measuring device used in Q6 (the balance or the ruler) have a precision poor enough to explain the percent absolute error?

Equipment & Supplies

Metal density samples (cylindrical, spherical, or rectilinear samples of various solids).

100 mL graduated cylinder

25 mL graduated cylinder

interfaced balance

1 or 2 mL pipettes and bulbs

hacksaw, metal cutters for cutting plastic samples

Saturated solution of K₂CO₃·1.5H₂O (this is an inexpensive, innocuous, high density solution).

Methanol

Samples of plastic: soda bottles (a very tough, high density PET); milk containers (often HDPE); BIC^Ò pen barrels (brittle polystyrene); plastic beads; mixed anion/cation deionizing columns; “floatation” antifreeze checkers; etc.

Procedure

I. Density of Metal Bar:

A. Mass

1. Tare the balance; obtain a regularly-shaped metal bar, and record the mass three times in the table, tareing the balance between measurements.

Place the cursor on the avg. mass, and look at the formula in the edit line.

The standard deviation is calculated for you with another Excel Formula,

and you need not know the mathematics behind the calculation, just what the standard deviation means.

Random Errors:

1. If measurements are all the same, the estimated random error is sometimes

taken to be 1/2 the smallest scale division on the instrument.

2. If measurements differ, the estimated random error is taken to be the standard

deviation.

B. Method I Volume

1. Add enough water to a 100 mL graduated cylinder to cover the bar, record the volume on the spreadsheet (using the keyboard),

Remember to record the proper number of significant figures.

Computers are terrible at significant figures. Frequently we simply add a note

about the correct number of significant figures to a spreadsheet result,

because it is so difficult to have the computer correctly report the data to the

correct number of significant figures. The Format/Cells/Number menus can be

used to set the number of decimal places that show in a spreadsheet value, or the Toolbuttons below can be used:

2. Carefully add the metal bar to the (tipped) graduated cylinder with the water

and record the volume again.

3. Repeat the procedure with a different initial volume of water.

Calculate the volume of the bar for each trial, entering an Excel formula analogous to "=A2-A1".

4. Use the average volume and mass determined previously to calculate the density.

To enter an Excel formula in the box, double click the box (select it), enter "=",

then click on the cell containing the mass (this enters it into your equation.

Now enter the "/" sign, and point to the cell containing the volume.

C. Method II Volume

1. Use a ruler with millimeter divisions to measure the length and diameter of the metal bar, and enter the measurements (in cm) in the spreadsheet table.

The calculation for the volume of the metal bar depends on its shape.

for a cylinder: V = p r² h , where p = 3.14, "r" is the radius, and "h" is the height.

An Excel formula can be created using Insert/Funtion/Math and Trig/Pi, which

returns the value of p to 15 figures in the selected cell. Complete the formula

using cell references for the height and d/2, then copy the formula for repeat calculation.

2. Use a table of densities to identify the metal.

II. Identifying Plastics by Density

A. Preparing a solution equal in density to the plastic

1. Cut a piece of plastic small enough to easily fit in a large (15x150mm)

test tube or 25 mL graduated cylinder, and add it to the container.

2. Add 5 mL of water to cover the plastic. Dislodge air bubbles that cling to it.

3a. If the plastic floats, add methanol until it is just suspended.

3b. If the plastic sinks, add saturated K₂CO₃ until it is just suspended. Mix the solution after each addition! If you don't have enough solution to take 4 samples with the pipette you've chosen for part B below, add water and K₂CO₃ and/or methanol as needed to increase the volume of your solution. Determine the density of the solution as described below.

B. Density of Solution

1. Tare the Balance with the beaker on it before adding solution to the beaker.

2. Add an aliquot of solution with a 1 or 2 mL pipette, record the volume in the table, and

record the weight of the solution in the table.

Note: do not empty the flask or tare between additions, and record the cumulative volume.

3. Without emptying the flask, add another aliquot of solution, and weigh the beaker again. Repeat a third time.

III. Creating the Graph:

Highlight a range including the mass and volume measurements, and click on the Graph Wizard icon. Drag an outline for the graph at the bottom of the spreadsheet, choose Scatter chart, add a title, and label the axes. Format the Data Series as Markers but no Line.

IV. Linear Regression:

Activate the Data Series and Choose Insert/Trendline/Linear, and under the Options Tab, select Display Equation and Display R².

After you've done the regression analysis, show how the values in the

last column of the table above, "Lin Reg," are calculated. These are the "best" Y values (compare to your experimental results for each X value in mL) that are used to create the best straight line graph.

Projects:

I. Determine the density of paper clips to three significant figures.

II. Can you distinguish reliably between pre-1982 and post-1983 pennies by their densities (the earlier ones were pure copper; the later ones are copper-clad zinc alloy)?

III. Determine the thickness of a 10 x 10 cm piece of aluminum foil by using only a ruler, a balance, and its density.

IV. Deionizing columns usually contain two types of resin: an anion exchanger to remove CO₃^2- and other negative ions, and a cation exchanger to remove Ca²⁺ and Mg²⁺. The two exchangers may be separated by their density; determine the densities if material is available. A 35% sodium hydroxide solution is sometimes used to both separate the resin and partially regenerate anionic sites.

V. Some antifreeze checkers contain several balls made of different plastics to determine the density of the coolant mixture, and from it, the percent antifreeze[3]. Determine the densities of the plastic balls.

Plastic	Name	Density (g/mL)
HDPE	High Density Polyethylene	0.95-0.97
LDPE	Low Density Polyethylene	0.92-0.94
Teflon	polytetrafluoro- ethylene	2.2
PMA	Polymethylmeth-acrylate (Plexiglas)	1.24
PS	Polystyrene	1.05-1.07
TPX	Poly-4-methyl-1-pentene	0.83
PETE	Polyethylene Terephthalate	1.39
V or PVC	Polyvinyl chloride	varies
PP	Polypropylene	0.90-0.91