a
THE USE OF THERMISTORS IN CRYOSCOPIC MEASUREMENTS' RICHARD K. McMULLAN and -JOHN D. CORBETT Iowa State College, Ames, Iowa -
THERMISTORS, semiconductors possessing rather large temperature coefficients of resistance, have been applied in several instances to the measurement of temperature and small temperature differences (1-4). However, the adaptability of thermistors to accurate measurement of freezing point depression and concentration without the use of elaborate or expensive equipment does not seem to have been emphasized. The purpose of this article is t o point out the practicability of cryoscopic measurement of concentration by means of a thermistor with an uncertainty less than that obtainable with the conventional ~ e c k d a n nthermometer techniques (6). A d.-c. Wheatstone bridge, such as would be available in any undergraduate physical chemistry laboratory, is used to follow the change in resistance of the element. I n addition, a t temperatures below -3g0, the freezing point of mercury, the thermistor appears unexcelled for measurements in low-melting solvents, such as in the study of weak association reactions. The usefulness of thermistors as temperature-sensing devices results from their relatively large negative temperature coefficient of resistance. This changes from one per cent per degree a t 300' to 4 per cent a t 25", and 6-8 per cent a t -50°, in contrast to about 0.35 per cent for platinum a t all temperatures. Commercial varieties composed of sintered transition metal oxides are available from 0.1 X 0.05 to 1.0 X 0.24 inches in size, and from 10 to 105 ohms in resistance (6). The temperature-resistance relationship of the individual element must be obtained by calibration. Although a t elevated temperatures thermistors slowly increase in resistance with time, this change a t room temperature and below is usually very minor, and its effect on the results can he eliminated.
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requirement for these fixed resistors was that their ratio be constant, two of the more inexpensive 2-watt wire-wound resistors wrapped together were a suitable substitute. For the variable resistance, two decade boxes (I-, lo-, loo-, 1000-ohm steps) were used in parallel so as to increase the sensitivity of the bridge without the use of decades with smaller resistance iucrements. Since the apparatus was to be used for the study of complexing reactions in nonaqueous solvent, the cryometric cell employed was of all-glass construction - .
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APPARATUS
For the measurements described herein the Wheatstone bridge shown in Figure 1 was used to measure the resistance of the element. The thermistor used was a Carbaloy type D-102 (about $3) (6), with resistances of 1010, 2300, and 22,000 ohms at 25', 5', and -40°, respectively. For measurements in the range of 0'-10' the bridge circuit included three 1.5-volt dry cells in parallel with a 500-ohm resistor in series as the current source, a Leeds and Northmu 2420-d galvanometer for the detector, and two 100b-ohm s&ndard resistors
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~i-
1. whsatStone id^ circuitand rmring.pointWI
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314
(Figure 1). Evacuation of the intervening space in the double wall of the cell gave much more satisfactory results, although the usual double cell with only an air gap (5) would for many purposes be adequate. The thermistor, connected t o the bridge with #22 copper wire, was positioned in the bottom of a thinwalled glass tube projecting into the solution from above. The bottom inch of this tube was filled with petroleum ether for better thermal contact with the system. Use of water-miscible liquids here gave erratic results, apparently due to conduction between the lead wires. The thermistor could, of course, be immersed directly in the solution if it were not conducting or reactive. Balance was achieved in a closed circuit so that the effect of the bridge current in heating the thermistor (about one milliwatt) would reach a steady state. The solution was stirred magnetically with a Teflon-covered stirring bar. With benzene, 1,4dioxane, and water a cooling bath approximately five degrees below the melting point of the solvent was used, while with pyridine the cell was cooled with a -SO0 dry ice-acetone mixture. With benzene solutions a constant resistance plateau was obtained for several minutes follaxing a supercooling of a few tenths of a degree. Since the freezing point would be expected to decrease as the solvent froze, the rate of removal of benzene from the solutions must have been extremely slow. With dioxane and pyridine the rather considerable supercooling observed was overcome by starting the freezing process with a few crystals of the solveut already present. Under these conditions, a supercooling of only a few hundredths of a degree was observed. With dioxane, the rate of change of temperature with time thereafter amounted t o less than one millidegree per minute. Under any circumstances, since the calibration and experimental observations are obtained in the same fashion it is necessary only that a consistent technique be used in making the measurements. The reagent grade or EII White Label benzene, pyridine, and 1,4dioxane solvents used were distilled from CaH2 or LiA1H4 into the cell in uacw, in most experiments and the cell and solvent weighed on a solution balance. The dioxane, as obtaiued from Wilfred Borduin, had been treated with HCI, NaOH, refluxed over sodium metal, and fractionated in a 30plate Oldershaw column. "Nonaq" stopcock grease (Fisher Scientific Co.) was used for ground-joint lubrication in the presence of these solvents. Over a temperature interval of several degrees, the resistance of a thermistor can be expressed quite accurately as: In I/T. = B ( l / T
- l/To)
(1)
where B is a constant of the thermistor (6). At the freezing point of benzene the sensitivity of the galvanometer enabled balancing to the nearest 0.03 ohm in r, so that a temperature difference of about 3.5 X lo-' degree could be detected with this circuit. Use
of different relative resistance settings on the parallel decades, or only a single decade, does not affect the sensitivity of the measurement provided the variable is equipped with sufficiently small resistance iucrements so that the full sensitivity of t,he galvanometer can be utilized. CALIBRATION IN BENZENE
As will be shown later, a differentialcalibration of the thermistor that may be adequate for some purposes can be obtaiued by comparison with a Beckmann thermometer. Direct calibration of the element could also be obtaiued with a standard resistance thermometer, if available. However, for molecular-weight studies, the simplest and most convenient method of calibration was taken to be the direct measurement of the resistance of the thermistor a t the freezing point of solutions of known concentration, using systems in which the solvent presumably behaved ideally. Phenanthrene or naphthalene (EK White Label, vacuum sublimed) was used as solute in benzene, dioxane, and pyridine, and c. p. urea or dextrose in water. Concentrations were calculated directly from the weights of the components. For a dilute solution in which the solvent obeys Raoult's law, the classical expression for the variation of the freezing-point depression AT with concentration is: where N , is the mole fraction solvent, and the other symbols have the usual significance. When allowance is made for the change in the heat of fusion nith temperature, the expression:
can be derived (7), where AC,, the difference in heat capacity of the liquid and solid solvent a t TO,is assumed constant over the temperature range involved. For this work terms higher than ATPin the polrer series are insignificant and can be dropped. Substitution for AT in terms of the thermistor resistance (equation (I)), with TTo = To2in the AC, term, gives:
A test of the suitability of this equation was made employing measured values of r a t the freezing points of 13 known solutions of phenanthrene in benzene. A measured value of B, obtained as described below, and a ACo of 1.82 cal. mole-' (8)were used in evaluation of the coefficient of the (log r/ro)2 term (7.67 X I n practice, an estimated B (6) would be sufficient. Within the concentration range where the solvent behaves ideally, the accuracy of the measurement varies directly with AT, or N2. Accordingly, a value of 0.361Z2for the constant U 0 / R B was obtained from a weighted average of the ratio of the left-hand
VOLUME 33, NO. 7, JULY, 1956
expression so obtained is shown in the table. The freezing-point depression for the largest concentration shown would be 0.542", as calculated by equation (1). The last column expresses the difference betrveen the known and calculated values of N2 in terms of the equivalent error in measurement of AT, the most, convenient reference scale for the comparison of the results with those of the thermometric method. The average and standard deviations of 0.96 X and 1.24 X degrees in AT, respectively, or a confidence range of 0.74 X degrees for a 95-per cent confidence level, are indicative of both the accuracv of the measurements and the suitabilitv of equation (4) as a description of the relationship between resistance and concentration. The deviations obtained here are from one-half to one-fourth of those reported in the literature for other similar measurements with thermistors (1, 3). The source of the apparently systematic trend in the deviations shown in the table is not known. The deviations observed cannot be explained over the entire concentration range by nonideal behavior of the solvent, unless an unhkely change in the sign of the deviation is allowed. Although the activity of benzene in this system apparently has not been reported in the literature, the probably similar henzene-naphthalene eystem has been studied a t higher concentrations by Campbell (9). Even a t the eutectic ( N , = 0.86) he found a negative deviation of only 0.93 per cent. In the second set of data shown in the table the largest differences in the deviations found (hetween the third and sixth concentrations) would correspond to a positiue deviation of the solvent in the more concentrated solution of less than 0.01 per cent relative to that in the reference solution. It is therefore not considered too likely that the deviations from the theoretical equation observed are a real indication of the nonideality of the benzene. No recognizable nonideal behavior was found in any of the systems studied. More likely sources of the discrepancies observed in the calibration data may have been small weighing errors, uncompensated lead resistances, or the need for additional small terms in the description of the resistance of the thermistor by equation (1) (10). Since these deviations show a regular trend, they could he reduced by curve fitting, although the desirability of having a theoretical expression to express the relationship may he considered more important. A value of 2345 calories for the molar heat of fusion of benzene was calculated from the constant AHo/RB obtained in the calibration, and a measured value of B (see below). This compared to 2351 cal. mole-I reported from calorimetric measurements (8). A heat of fusion of 2344 cal. mole-' was ohtained from the same data without the use of the AC, term, and the data fit the resulting equation with the same accuracy. The thermistor so calibrated has been used to determine molecular weights, to measure solubilities,
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- ~ , ~" ,- - - - . .. -. . - . ... . -.. ~~
bration Data for Phenanthrene in Benzene log ?/TO N1 N*calc. AN% A(AT) To. ohn~s r ohms X loS X 1010' X 106 X 10" 2299.09 2305.39 1.1883 1.024 0.988 + 3 . 6 f2.39 2311.52 2.3415 1.950 1.946 +0.4 +032 2317.24 2.823 3.553 2.836 -f 10 ..31 -0.85 2321.86 3.4148 4.2801 3.554 +0.08 2327.18 5.2740 4.386 4.377 fO.9 +0.62 +0.23 2334.02 6.5486 5.435 5.431 f 0 . 4 2348.63 9.2588 7.689 7.670 f 1 . 9 +1.28 zz88.92 2296.45 1.4265 1.104 1.186 +0.8 +0.61 2303.81 2.8160 2.324 2.339 - 1 . 5 -0.97 2314.16 4.7628 3.920 3.953 - 3 . 3 -2.22 2321.80 6.1942 5.114 5.138 - 2 . 4 -1.57 2332.06 8.1090 6.710 6.721 - 1 . 1 -0.71 2341.75 9.9100 8.219 8.207 f1.2 +0.78 1.5 0.96 Av. dev. 1.24 Std. dev.
and to study the stoichiometry of reactions of gallium(I1) halides. For example, from measurements on ten concentrations of naphthalene in henzene up to N2 = 0.0078, or 0.101 molal, an average molecular weight of 128.08 0.66 average deviation (0.87 standard deviation) was calculated, as compared to 128.16 for the accepted value. The trend in the values obtained was the same as observed in the calihration. From the change in resistance observed when henzene was saturated with SbI,, the solubility a t 5.5O was calculated to he 6.00 0.06 gram per 1000 grams of solvent.
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The use of equation (4) for the calibration and subsequent evaluation of molecular weights may for many purposes be found to he somewhat lengthy, since evaluation of log Nl and antilog Nl for Nl close to unity requires use of the first two or three terms of the equivalent power series expansion. A more facile expression can be derived from equation (4) with only minor approximations. Taking -In Nl = mM J1000, where m is the molality and MI, the molecular weight of the solvent, and noting that r/ro is equal to 1 Ar/ro, expansion of ln (1 Ar/ro) in a power series gives:
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A calibration can then be obtained by fitting the data to this quadratic in Ar/ro. Actually the AC, term for benzene is so small that it can be dropped, giving the simple expression:
Fitting the same benzene calibration data to this oneconstant equation gave average and standard deviations of 1.11 X 10-3 and 1.43 X degrees in AT, and a A H o of 2349 cal. mole-', values not significantly different from those obtained with equation (4). Eqnation (6) is the resistance-concentration relation that
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has been found most convenient for subsequent applications of the thermistor in this laboratory. If the coefficients of the first and second powers of A r / n are allowed t o vary independently, the error in the approximation of molality for log NL, as well as any systematic errors, can be partially eliminated. A general quadratic in Ar/ro was fitted to the same data with average and standard deviations in AT of 0.83 X and 0.95 X degrees, respectively, as compared t o 0.96 X and 1.24 X lo-' degrees obtained by equation (4).
JOURNAL OF CHEMICAL EDUCATION
bridge was modified by substitution of 20,000-ohm resistances in the fixed arms and a single decade for the variable resistance. The resistance of the thermistor was reproducible to one to two ohms, the larger value corresponding t o 1.6 X degrees. With AC, = 0, the three concentrations used fit equation (4) with an average deviation of 2.8 X degrees in AT for N2 up to 0.0063. A heat of fusion for pyridine of 1880 + 8 cal. mole-' was calculated from the slope and an estimated B. A literature value for this constant was not found. DElZRMINATION OF B AND COMPARATIVE BECKMANN MEASUREMENTS
I n the calibration of a thermistor and subsequent use in concentration measurements there normally would be no need t o use a Beckmann thermometer. However, in order to obtain the values of B employed in the above heat of fusion calculations, as well as t o estimate the comparative accuracy of Beckmann thermometer readings in cryoscopic measurements, several series of experiments were made in which both a Beckmann thermometer and a thermistor were used. The thermometer readings were corrected for emergent stem, and for the setting factor. The latter allowed for the difference in the mass of mercury in the bulb a t temperatures other than that a t which it was calibrated, usually 20' (13). I n thxee separate runs, 17 solutions of phenanthrene in benzene gave a weighted average B of 3.267 X lo3 degrees (equation (1)) a t Actually for student use a large graph of Ar/r, 5.5'. Similarly, 11 solutions of urea in water gave a versus m or Nz is probably the most convenient pro- B of 3.233 X loa degrees. From these values, it can cedure as the result is very nearly linear (Figure 2). be seen that, although the "constant" B for the therThis behavior indicates the small contribution of the mistor does change slightly with temperature, this AT/^,)^ term in equation (6). Slightly better results change would result in an error of only 0.2 X lo-' ean be obtained if the deviations of the data from the degrees in AT for a 0.5-degree interval. The values straight line m = a (Ar/ro) are either plotted or de- of B used in the AHo calculations with dioxane and scribed analytically, and these applied t o the con- pyridine were estimated from the rate of change obcentrations calculated from the linear relationship. served with benzene and water, together with similar If a quadratic is used to fit the deviations the results data provided by the manufacturer (6), and are probare the same as those given in the previous paragraph. ably accurate to 0.5 per cent. The differences between the observed and calculated OTHER SOLVENTS values of AT (equation (I)) are shown in Figure 3 for Cryoscopic measurements in dioxane and in water both solvents. The average and standard deviations utilizing the thermistor gave similar results. In in AT for benzene are 2.6 X and 3.4 X dioxane, five concentrations of naphthalene up t o Nz = degrees, and for water 2.7 X and 3.5 X 0.009 fit equation (4) with an average deviation in degrees, respectively. It will he noted that the deviAT of 0.82 X degrees, using a AC, of one cal. ations found here are about three times as large as those mole-' (11). An estimated B gave 3021 cal. mole-' found for a similar temperature interval in the resistfor the heat of fusion, as compared t o 3070 aud 3017 ance calibration with equation (4) (see the table). cal. mole-' reported in the literature (11,1$). The heat Moreover, all systems studied with the thermistor of fusion of water similarly obtained differed from the showed about the same trend in the deviations, whereas accepted value by three calories. with the Beckmann these show quite different beSince measurements in low-melting solvents with haviors in the solvents water and benzene. AccordBeckmann thermometers are limited by the freezing ingly, the deviations shown in Figure 3 are interpreted point of mercury (-3g0), the thermistor appears par- to be essentially a calibration curve for the Beckmann ticularly useful for investigations a t lower tempera- in the (different) scale ranges employed, with a scatter tures. To confirm this, the element was calibrated in indicative of the reproducibility of such a measurepyridine a t -42', using naphthalene as the solute. ment, and with the smaller trend observed in the use Since ro a t this temperature was 22,250 ohms, the of equation (4) superimposed. The effect of the devi-
VOLUME 33, NO. 7, JULY, 1956
ations from the resistance-concentration expression found for benzene (see the table) is to reduce the largest differences shown and t o increase slightly those in the intermediate range; the over-all average is not affected significantly. Since all the benzene solutions used were of known concentration, comparison of the observed depressions with those calculated from concentration (equation (2)) confirmed the effect of the thermistor deviations. THERMISTOR STABILITY
One difficulty often mentioned in the application of thermistors to temperature measurement is that they tend t o undergo slow changes in resistance over a period of time (6), and a t a rate that varies with the individual element. However, for the type of work described herein, this factor has been substantially eliminated by the use of expressions relating the concentration and the ratio of the resistance r a t the freezing point of a given solution to the resistance ro for the pure solvent a t the start of the measurement. This then requires only that ro does not change during the run, a condition which from experience has been found to be quite practical. The two series of measurements shown in the table were purposely chosen to test this concept. The runs were carried out seven weeks apart, during which time, as a result of various treatments, ro had changed by several tenths of a per cent. Moreover, the first set was obtained with undried benzene a t one atmosphere pressure (not the usual procedure), while the second was under the vapor pressure of the solution using dry vacuum-distilled benzene. Although in the former the 0.003" depression of the freezing point reported for one atmosphere of air (14) affected ro only slightly, an unknown but apparently small concentration of water condensed from the air also gave an added increase in ro. Much smaller differences were found between separate sets of data obtained under similar conditions only a few days apart. When the resistance of a thermistor is first measured a t a new and somewhat different temperature, a slow drift is sometimes noted. However, if the element is allowed to age a t that temperature for a short time, the values become quite constant. Thermal shock, such as cooling to -190°, will change the resistance observed a t a reference temperature, although again, after a few hours, the element will settle down to a constant value. Cooling t o -80°C. was observed to have little effect on the resistance at 0". Johnson and Kraus (5) observed that, over a period of 15 days, the variations in the apparent temperature of the ice point as measured with a thermistor showed a standard deviation of only 1.1 X 10-3 degrees.
rectly relating concentration and thermistor resistance, has been shown t o give better results than those ohtained by the usual Beckmann technique. I n addition, once the element has been calibrated, the ease of the measurements is somewhat greater. The method also appears applicable to the study of molecular weights a t low temperatures, and, if the thermistor constant B is obtained by direct comparison to a temperatureinterval standard, it is applicable to the determination of the heat of fusion of the solvent. LITERATURE CITED (1) ZEFFERT, B. M., AND S. HORMATS, Anal. Chem., 21, 1420 (1949). Anal. C h m . , 24,348 (1952). (2) ZLMAN~, P. D., (3) JOHNSON, J. S., AND K. A. KRAWS, J. Am. C h m . Soe., 74, 4436 (1952). (4) MULLER, R. H., AND H. J. STOLTEN, A d . Chem., 25,1103 llOK7\ ,*""",.
( 5 ) DANIELS, F., J. H. MATREWS, AND J. W. WILLIAMS, "Ex-
perimental Physical Chemistry," 4th ed., MeGraw-Hill Book Co., Inc., NewYork, 1949, p. 79. ( 6 ) Manual TH-13, Carbaloy Dept., General Electric Co., Detroit, Michigan. (7) ROSSINI,F. D., "Chemicsl Thermodynamics," John Wiley & Sons, Inc., New York, 1950, p. 302. The first term given in equation (98) should he (1 - AT/T,)-'. The In (1 - AT/Td term was expanded in a power series, and similar terms combined to give equation (4). (8) "Selected Values of Properties of Hydrocarbons and Related compound^," API Res. Proj. 44, Carnegie Institute of Technology, Pittsburgh, 1952, Part 2, Table 5Z. (9) CAMPBELL, A. N., Can. J. Research, 19B, 143 (1951). Bell (10) BECKER, J. A,, C. B. GREEN,AND G. L. PEARSON, System Tech. J., 26,170 (1947). J. Am. Chem. Soe., 56, 1513 (11) JACOBS,C., AND G. PARKS,
.
(14Rd> , -..-,
(12) ROTA,W., AND I. MEYER,Z. Electrochem., 41,229 (1935). (13) buss^, J., "Temperature, Its Measurement and Control in Science and Industry," Reinhold Publishing Corp. New York, 1941, p. 245. AND W. C. SCAUMB, J. (14) RrcnAR~s,T. W., E. K. CARVER, Am. Chem. Soc., 41, 2019 (1919). WATER
t4
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CONCLUSION
The application of thermistors to cryoscopy, -.. or what is actually cryometry, using a aimpie expression di-
rn-n
Thumomater 6r.m Thw. C a l ~ h t e dh m AT = .ol?
r h
