In the Laboratory
Using Latex Elastomer to Illustrate Euler’s Chain Relationship
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Fred R. Hilgeman* and Ana Alcaraz Chemistry Department, Southwestern University, Georgetown, TX 78626; *
[email protected] Background The physical properties of an elastomer (latex) have been used in several physical chemistry laboratory experiments and demonstrations to illustrate various thermodynamic properties of the elastomers (1–4). These experiments focus on the relationship between tension (force) and temperature at constant length. This linear relationship shows that measuring tension with respect to temperature at constant length yields a straight line whose slope gives a measure of the entropic contribution to the restoring force. This experiment assumes that there is no change in volume; that is, PdV = 0 for the elastomer. It is also assumed that there is no significant deformation other than a onedimensional stretch, and possible “creep” is ignored. These contributions that result in nonlinearity of this elastomer are discussed elsewhere (5). If it is assumed that the length of a piece of latex can be expressed as a function of its tension and temperature, these variables of tension and temperature could be considered independent variables of the dependent variable length. Euler’s chain relationship (6 ) is a technique used to manipulate mathematical expressions, especially in the area of thermodynamics and hydrodynamics. The rule states that if z is a function of x and y, then dz = (∂z/∂y)x dy + (∂z/∂x)ydx. It can be shown then that (∂z/∂x)y (∂y/∂z)x(∂x/∂y)z = ᎑1. Considering the variables length (l), tension (t), and temperature (T ), the chain rule can be expressed as (∂t/∂T )l (∂l/∂t)T (∂T/∂l )t = ᎑1
device with sufficient accuracy, and the temperature of the water is observed, most easily by a digital thermometer. Beginning at room temperature one can heat the water to 50 °C and observe the change in tension with respect to temperature at constant length by observing the change in mass on the balance. Here temperature is considered to be the independent variable, tension the dependent one. The water can be quickly cooled to about 35 °C and held constant there while the length can be varied using the lab jack and measured with the cathetometer and tension observed with the balance. Now length is the independent variable and again tension is the dependent variable. At this point the water can be cooled again to room temperature and then heated to 50 °C to observe the change in length with respect to temperature; the position of the lab jack is varied to keep the mass on the balance the same, and the length of the latex is measured with the cathetometer. Again temperature is the independent variable, but now length is the dependent variable. If the experiment is done in this manner it is possible for all three measurements to be carried out in one laboratory period. The latex used was from rubber gloves, which were cut into strips that were then stretched over a suitable metal frame. This frame with the stretched latex was immersed in a large beaker of water that was brought to a boil, then cooled slowly to room temperature. Two thickness of latex approximately 6 cm long were used for the measurements.
(1)
Table 1. Data from a Typical Experiment
In this experiment, each of the quantities in eq 1 is measured, and the product of these three slopes, within experimental error, is found to equal ᎑1.
36.2
94.1
95.2
6.6
6.690
29.3
Experimental Procedure
39.6
95.1
100.9
6.7
6.675
32.2
42.5
95.7
106.6
6.8
6.660
36.0
The apparatus used consists of a wooden support about 1 meter high on which a top-loading balance of capacity 600 grams is placed. A hole through the support allows a hook hanging from the balance pan to be attached beneath the balance to a wire that drops down into a large glass jar filled with about 20 liters of water. The jar is placed on a magnetic stirrer; the assembly of jar and stirrer is then placed on a laboratory jack. The latex rubber to be studied is held by pinch clamps on each end. By means of an appropriate wire frame, the latex is anchored at one end to the bottom of the jar and the other end of the latex is fixed to the hook hanging below the balance. By adjustment of the laboratory jack, tension on the latex may be controlled. The water is stirred with the aid of the magnetic stirrer. An immersion heater with a constanttemperature controller and a temperature measuring device is also put into the water in the jar. By reading the mass on the balance the tension is determined, the length of the latex is determined using the cathetometer or some other convenient length measuring
45.4
96.4
113.1
6.9
6.640
38.8
78
T/°C
t/g
t/g
l/cm
l/cm
29.7
91.1
87.2
6.5
6.720
T/°C 25.0
33.1
92.9
89.9
6.5
6.715
27.0
48.0
97.2
121.5
7.0
6.635
41.3
50.6
97.1
126.6
7.1
6.630
45.3
129.5
7.2
6.620
49.5
135.2
7.3
142.2
7.4
145.8
7.5
152.0
7.6
156.7
7.7
162.9
7.8
168.3
7.9
174.0
8.0
182.2
8.2
187.7
8.3
191.9
8.4
y = 0.2852 x + 83.361 m = 0.2852 ± 0.0255 R 2 = .9542
y = 0.0178 x + 94.607 m = 0.0178 ± 0.0002 R 2 = .9977
Product of slopes m = ᎑1.11
y = ᎑219.23 x + 1497.2 m = ᎑219.23 ± 26.91 R 2 = .9383
Error in slopes m = 0.17
Journal of Chemical Education • Vol. 79 No. 1 January 2002 • JChemEd.chem.wisc.edu
In the Laboratory
Hazards There are no chemical hazards associated with this experiment. The only hazards to be considered are those associated with the presence of heaters, hot water, and ungrounded appliances around the water.
∆slope = ((slope(1) * slope(2) * ∆slope(3))2 + ((slope(1) * slope(3) * ∆slope(2))2 + (slope(2) * slope(3) * ∆slope(1))2)1/2
Results and Conclusions The data were analyzed using the Excel spreadsheet. A linear fit using the method of least squares gave a slope, and the standard error analysis was done to determine the error 99 98
Tension / g
97 96 95 94 93 92 91 90 25
30
35
40
45
50
55
Temperature / °C
The largest error typically comes from the constant-tension experiment. On more than one occasion the first experiment of the laboratory gives poor results. Redoing that experiment at the end of the laboratory gives a much better linear relationship. This situation occurred no matter which of the three experiments was done first. Latex gloves gave by far the best results; toy balloons and rubber bands were used with much less success. Typical data collected from an experiment are shown in Table 1. Each set of data generated one of the graphs shown in Figures 1–3. For the data in Table 1, the tension was about 100 grams, the length of latex used was about 6.5 centimeters and the strip was stretched 20%, and the temperature was 40 °C. The equations for the three lines were: tension = 0.2852 * temperature + 83.361; ∆slope = 0.026: R 2 = .9542 length = 0.0178 * tension + 94.607; ∆slope = 0.00021: R 2 = .9977
Figure 1. Plot of data from columns 1 and 2 of Table 1.
temperature = ᎑219.2 * length + 1497.2; ∆slope = 27: R 2 = .9383
For this experiment, (0.2852) * (0.0178) * (᎑219.2) = ᎑1.11 ± 0.17. This is typical of the data collected.
9
Length / cm
in slope for each of the three experiments. The error estimates were used to estimate the overall error in the product of the three slopes using the conventional propagation of errors (7). Using this technique one arrives at the following estimate of error in this product:
8
Table 2. Data from Seven Student Groups Group Experiment
7
6 80
100
120
140
160
180
200
Error
Conditions
Constant l Constant t Constant T Combined
0.3787 77.50 0.04203224 ᎑1.234
0.0247 5.71 0.00071 0.123
6 cm 120 g 36 °C —
.9968 .9513 .9294 —
2
Constant l Constant t Constant T Combined
14.40 0.002253 ᎑24.43 ᎑0.7932
5.20 7.59E᎑08 0.125 0.2863
6 cm 190 g 33 °C —
.5229 .9178 .9963 —
3
Constant l Constant t Constant T Combined
0.2109 356.08 ᎑0.01930 ᎑1.449
0.0236 42.08 2.03E᎑04 0.2366
6 cm 200 g 43 °C —
.9409 .9109 .9979 —
4
Constant l Constant t Constant T Combined
0.2852 219.23 ᎑0.01785 ᎑1.115
0.0255 26.91 0.00020 0.172
6 cm 100 g 40 °C —
.9542 .9383 .9976 —
5
Constant l Constant t Constant T Combined
0.2105 101.66 ᎑0.04865 ᎑1.041
0.0309 5.28 0.01137 0.292
6 cm 200 g 36 °C —
.9204 .992 .9968 —
6
Constant l Constant t Constant T Combined
0.1093 220.29 ᎑0.04334 ᎑1.044
0.0171 119.52 1.25E᎑07 0.590
6 cm 200 g 28 °C —
.953 .9878 .99989 —
7
Constant l Constant t Constant T Combined
0.2021 86.57 ᎑0.03812 ᎑0.6671
0.0193 4.08 5.42E᎑04 0.0717
6 cm 225 g 39 °C —
.9648 .989 .9968 —
Tension / g Figure 2. Plot of data from columns 3 and 4 of Table 1. 60
Temperature / °C
50 40 30 20 10 0 6.60
6.62
6.64
6.66
6.68
6.70
6.72
Length / cm Figure 3. Plot of data from columns 5 and 6 of Table 1.
6.74
R2
Slope
1
JChemEd.chem.wisc.edu • Vol. 79 No. 1 January 2002 • Journal of Chemical Education
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In the Laboratory
Shown in Table 2 are data from several groups of students in the Physical Chemistry laboratory course. The slopes and errors are calculated as described above. All samples were approximately 6 centimeters long when unstretched. For these data the average of the products of the slopes is ᎑1.05 ± 0.26. In addition to illustrating Euler’s chain relationship, the laboratory helps emphasize that error propagation analysis is important in drawing conclusions about the “correct” answer for an experimental value. Acknowledgments We are indebted to the Robert A Welch Foundation (grant no. AF 0005) for their support of this project and to the Southwestern University students who were involved in developing this experiment.
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Supplemental Material
Instructions for students and notes for the instructor are available in this issue of JCE Online. Literature Cited 1. 2. 3. 4.
Mark, J. E. J. Chem. Educ. 1981, 58, 898–903. Byrne J. P. J. Chem. Educ. 1994, 71, 531–533. Gilbert, G. L. J. Chem. Educ. 1977, 54, 754–755. Halpern, A. M. Experimental Physical Chemistry, 2nd ed.; Prentice Hall: Englewood Cliffs, NJ, 1997; 509–519. 5. Flory, P. J. Principles of Polymer Chemistry; Cornell University Press: Ithaca, NY, 1953. 6. Thomas, G. B.; Finney, R. L. Calculus and Analytic Geometry; 7th ed.; Addison Wesley: Reading, MA, 1988; p 8917. 7. Sime, R. J. Physical Chemistry; Saunders: Orlando, FL, 1990; Chapter 7.
Journal of Chemical Education • Vol. 79 No. 1 January 2002 • JChemEd.chem.wisc.edu
