In the Laboratory
Laser-Based Liquid Prism Sucrosemeter—A Precision Optical Method To Find Sugar Concentration V. Anantha Narayanan* Department of Physics, Drew-Griffith Building, Savannah State University, Savannah, GA 31404 Radha Narayanan (student) Windsor Forest High School, Savannah, GA When a light ray is incident on a prism, it undergoes refraction at the air–prism interface on face 1, travels inside the prism, refracts again at the prism–air interface on face 2, and exits into air (Fig. 1). The angle of deviation (D) (the angle by which the prism turns the incident ray as it emerges out of face II) is a function of the angle of incidence i1 at face I, A (the prism angle), and the refractive indices of the prism (N) and of air (= 1.00028). In general the values of D can be calculated and plotted as a function of i1 and i 2 (1) for a given prism of known A and N (Fig. 2). Data for such a curve can be generated by a computer program (2) and such curves are given in standard optics texts (e.g., ref 1). The calculations involve use of Snell’s law of refraction in faces I and II, and the relations between the angles of incidence, refraction, and deviation, and the apex angle (2). The computer program in BASIC (2) used to generate the data for Figure 2 is given as an appendix to this paper. For a prism of known A and N, the deviation angle is minimum (Dm) when i 1 = i2. At this position the incident and emerging rays are symmetrical with respect to the refracted ray. The refracted ray then will be parallel to the base of the prism (Fig. 1). For this position the formula for the refractive index, N, of the prism is
1.00028 × sin A + Dm / 2 N=
sin A / 2
(1)
60°. A He-Ne laser (1 mW) giving out 6328 Å red light was used as the incident light. The experimental setup is illustrated in Figure 3. The laser and prism were set up in separate stands on a sturdy table. The wall facing the prism and laser served as the screen. The perpendicular distance (L) from the apex of the prism to the wall measured 318.4 cm, when the base of the prism was set normal to the wall. The value of 318.4 cm is nothing
Figure 1. Refraction in a prism minimum deviation case. The refracted ray is parallel to the prism base, the incident and emerging rays are symmetrical, A = apex angle of the prism, Dm = angle of minimum deviation, i1 and i2 = angle of incidence in faces 1 and 2 (in the minimum deviation case i1 = i2).
Thus in this position, knowledge of A and Dm alone will give a value of N. If a precisely made equilateral prism is used, A = 60° and N = 2.00056 sin (30° + Dm / 2)
(2)
With the advent of small lasers, it is very convenient to make quantitative measurements on a prism without a precision spectrometer table and vernier scales, and often with comparable or better accuracy (3). We report in this paper a novel way to assemble a liquid prism sucrosemeter, calibrate it, and use it to measure sucrose concentration in a simple yet precise student project in optics. Experimental Details
Setup The hollow prism we used was an equilateral prism (5-cm side) made by EMD, Fisher Scientific (1991–92 Physics and Technology Catalog #S42451 A) with optically plane and polished walls and base. The size of the prism is not very important, except that a smaller prism needs less volume of the solution. The apex angle A = *Corresponding author. email:
[email protected].
Figure 2. Angle of deviation vs. angle of incidence (theoretical curve). The data were calculated using A = 60°, N = 1.45, and angles of incidence from 25° to 89° in 4° intervals, and using the refraction formula for a prism when i1 ≠ i2 (general case). Such figures are given in references 1 and 2 .
Vol. 74 No. 2 February 1997 • Journal of Chemical Education
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In the Laboratory
special, except that it was the maximum we could arrange in our setup. The larger is the value of L, the more accurately can Dm be calculated. Thus a hollow prism when filled with liquids of different refractive indices will give unique values for the angles of minimum deviation, which can be measured with an accuracy of as little as 0.01° by trigonometric formulas. It is seen that tan Dm = X/L (Fig. 3). The undeviated laser beam and the base of the prism in its normal position were adjusted to be perpendicular to the wall surface. The position of the laser was kept stable so that the laser spot after traveling through the empty prism always formed at C on the wall. This is a setup check that must be maintained for the duration of the project. The prism stand can be rotated delicately to turn the prism clockwise or counterclockwise in small gentle steps.
Figure 3. Liquid prism sucrosemeter. L = perpendicular distance from the apex of the prism to the screen, Dm = angle of minimum deviation, CB = X = linear deviation (when the prism is adjusted for minimum deviation), C = spot where the laser beam hits the screen when the prism is empty, and A = apex angle of the prism.
The wall screen was covered with several square meters of computer paper so the laser spots could be recorded with pencil.
Preparation of Samples Pure cane sugar was used at the 13 concentrations listed as items 2–14 in Table 1. Of these, the 17%, 33% and 47% solutions were used as controls to test the calibration graph. A large volume of each concentration (about 200 mL) was prepared. The weighings were done correct to 0.01 g and volumes were measured correct to 0.5 mL. The concentrations expressed as wt % of sucrose are accurate to within 0.3%. The refraction experiments were done in a climatecontrolled room maintained at 20 ± 1 °C. The temperatures of the solution, work table, and room were monitored with precision thermometers. It is necessary to do this because the refractive index varies with temperature. Procedure First the prism was filled with water. The emerging beam hit the point B on the wall screen (Fig. 3). By gently turning the prism clockwise or counterclockwise, the movement of spot B on the wall was followed till CB was the minimum. Turning the prism further will make CB larger. The minimum position was determined in 3 trials, and in each case spot B was marked on the screen and labeled. Next, the prism was emptied, washed, and then dried. Successively it was filled with all 13 sucrose solutions and the minimum deviation positions were marked and labeled in three separate trials for each one. The computer screen paper was then transferred to a long table and the values of X were measured to the nearest 1 mm for water and the 13 solutions. Results for the 3 trials were averaged and reported correct to 1 mm. Results
Table 1. List of Solutions Used with the % by Weight of Sucrose, X-Distances in cm, Calculated Dm Values in Degrees, and Calculated N Values Experimental Values X-Distances Dm Values Nb (cm)a (°)a
No.
Weight of Sucrose (%)
1
0
137.2
23.3
1.3295
2
5
141.1
23.9
1.3373
3
10
144.8
24.4
1.3438
4
15
149.4
25.1
1.3528
5
17
151.2
25.4
1.3567
6
20
153.5
25.7
1.3605
7
25
158.4
26.4
1.3695
8
30
163.8
27.2
1.3796
9
33
165.8
27.5
1.3834
10
35
168.7
27.9
1.3884
11
40
174.9
28.8
1.3997
12
45
179.3
29.4
1.4072
13
47
182.3
29.8
1.4121
14
50
185.4
30.2
1.4171
Table 1 lists the data for the 14 liquids: values of X, Dm (calculated from tan D m = X/L) correct to 0.1°, and N (calculated using eq 2). Figure 4 is a plot of wt % of sucrose vs. X (the linear deviation). For plotting this graph data for all the liquids in Table 1 except the control solutions (17, 33, and 47 wt% of sugar) were used. A straight line described the 11 data points very well, indicating a linear relation.
a
Average of three trials. X -values are correct to 1 mm. Dm values are calculated and rounded to the nearest 0.1°. Dm values can be calculated precisely from Dm = arctan ( X/L), where X and L can be measured correct to 1 mm. L is an experimental setup constant equal to 318.4 cm. b Experimental N values calculated from N = 2.00056 sin (30°+ Dm/2) and expressed to four decimal places as in literature values.
222
Figure 4. Plot of wt % sucrose vs. linear deviation.
Journal of Chemical Education • Vol. 74 No. 2 February 1997
In the Laboratory
Sources of Error Possible sources of error are (i) temperature fluctuation during measurements on the samples, (ii) error in making up the solutions, and (iii) error in measurements of X and L. For most liquids, a temperature rise of 1 °C will decrease the refractive index by 0.0004–0.0005 (4). The concentration uncertainty of 0.3% gives an uncertainty in N (5) of not more than 0.0005 for the values in Table 1. The estimated errors in X and L are at most 4 mm. This implies an error in Dm values of not more than 0.01°. Discussion The three control liquids gave the following experimental data: 17% sugar solution, X = 151.2 cm and Dm = 25.4°; 33% sugar solution, X = 165.8 cm and Dm = 27.5°; 47% sugar solution, X = 182.3 cm and Dm = 29.8°. The X values of these control solutions were used in Figure 4 to derive their respective concentrations as 16.3%, 31.3%, and 48.3%. Thus the concentrations on the average are predicted with an uncertainty of within 1.2% by weight of sucrose concentration. This experiment can also be used as a precision method to determine refractive indices, as the data in Table 1 illustrate. The setup can be extended to various soluble salts for precise concentration and refractive index determinations. Literature Cited 1. Jenkins, F. A.; White, H. E. Fundamentals of Optics, 3rd ed.; McGraw-Hill: New York, 1957; pp 20–25. 2. Anantha Naryanan, V. “Refraction in a Prism—A Computer Program to Collect Data for the ID Curve;” Savannah State College Res. Bull. 1973. (This program is given in the Appendix.) 3. Naba, N. Physics Teacher 1973, 11, 241. 4. Vogel, A. I. A Text-Book of Practical Organic Chemistry, 2nd ed.; Longmans, Green and Co.: London, 1951; p 898. 5. Handbook of Chemistry and Physics, 65th ed.; CRC: Cleveland, 1984–85; pp E-357–E358, E-360–E361.
Appendix. Computer Program to Generate the Data for an I-D Curve 1 PRINT “ANG OF DEV FOR VAR ANG OF INC IN A PRISM”
35 PRINT “____________________________________________” 100 PRINT “AFTER THE ? GIVE A VALUE OF U = FOR THE REF IND OF THE MATERIAL OF THE PRISM” 102 PRINT 200 INPUT U 740 LET A = 60 745 REM A IS THE PRISM ANGLE IN DEGREES 750 PRINT “REF. IND. OF THE PRISM =”;U 760 PRINT “PRISM ANGLE=”;A 800 PRINT “I=ANG OF INC. OF FACE 1;R1-ANG OF REF ON FACE 1;” 802 PRINT 804 PRINT “I3=ANG OF INC ON FACE 2;Q=ANG OF REF ON FACE 2;” 806 PRINT 808 PRINT “D=ANG OF DEV TOTAL” 809 PRINT “_____________________________________________” 811 FOR I = 25 TO 89 STEP 4 850 LET I1 = 0.01745329251 * I 880 REM I1 IS I CONVERTED TO RADIANS 900 LET P = SIN (I1) 950 LET Q = P/U 990 REM Q IS SIN (R) WHERE R IS REFRACTION ANGLE OF FACE 1 IN RADIANS 1020 LET R = ATN (Q/SQR (1-Qˆ2)) 1050 LET R1 = 57.29577951 * R 1090 REM R1 IS R CONVERTED TO DEGREES 1240 LET A1 = A*.01745329251 1300 REM A1 IS PRISM ANGLE CONVERTED IN RADIANS 1400 LET R2 = A1 - R 1600 LET Q = 57.29577951 * R2 1800 S2 = U *SIN(R2) 1801 IF S2 >1, THEN NEXT I 1900 LET I2 = ATN (S2/SQR (1 - S2ˆ2)) 2800 LET I3 = I2 * 57.29577951 2801 LET D1 = I1 + I2 - A1 2802 LET D = D1* 57.29577951 2810 PRINT “=======================================” 3000 PRINT 3200 PRINT “I=”;I 3202 PRINT 3204 PRINT “R1 =”;R1 3206 PRINT 3208 PRINT “I3=”;I3 3210 PRINT 3212 PRINT “Q=”;Q 3214 PRINT 3216 PRINT “D=”;D 3300 PRINT 3600 NEXT I 7000 END
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