t h e addition of hydrogen ion, whereas one or a combination of several possible reactions might be responsible for t h e anode potential shift. Further work is indicated. CONCLUSION
Simplicity of end point detection and equipment needed makes this titration technique a valuable extension of the constant nonaqueous potentiometric titration of organic acids (8). The accuracy and precision of the method are comparable to, or better than, those of any of the zero-current potentiometric or indicator titrations of bases in nonaqueous solvents previously reported (2). This technique is adapt-
able to automatic titration with the type of instrument described earlier ( 7 ) . Thus, this technique is more convenient than zero-current potentiometric methods. Another valuable aspect of the technique is the ability to determine accurately total basicity of polyfunctional amines. ACKNOWLEDGMENT
The author is grateful to Irving Shain for helpful comments and assistance in the preparation of this manuscript. LITERATURE CITED
(1) Allen, M. J., “Organic
Electrodc Processes,” Reinhold, New York, 1958.
(2).Fritz, J. S., Hammond, G. S., “€Juan-
titative Organic Analyses,” Wiley, New York, 1957. ( 3 ) Harlow, G. A., Noble, C. M., Wyld, G. E. A., ANAL.CHEM.28,784 (1956). ( 4 ) Pifer, C. W., Wollish, E. G., Zbid.,
24.300 (1952). ( 5 ) Pifer, ‘c. ‘C. LV., LV., W Wollish, O~ E. G., J . Am. Pharm. Assoc. 40,609 (1951). ( 6 ) Reilley, C. N., Cooke, W. D., Furman, N . H., ANAL.CHEM.23, 1223 (1951). ( 7 ) Shain, I., Huber, C. O., Zbid., 30, 1286 (19581. (1958). ((88 ) Shain, I., Svoboda, G. R., Zbid., 31, 1857 185 (1959). ( 9 ) Siggia, si-- , S.,, Hanna, J. G., Kervinski, I. R., R., Ibid., 22, 22,1295 1295 (1950). (10) Wagner. Wagner, C. D.. B Brown, R. H., Peters, E. D., J . Am. Chem. SOC.69, 2611 (1947).
RECEIVED for review February 24, 1961. Accppted Augrwt 17, 1961.
Differential Thermometric Titrations and Determination of Heats of Reaction BRUCE C. TYSON, Jr.,l W. H. McCURDY, Jr.,2 and C. E. BRICKER3 Department o f Chemistry, Princeton University, Princefon, N. 1.
b An apparatus utilizing thermistors for conducting thermometric titrations, which measures the difference in temperature between the reaction vessel and a blank solution, is described. To eliminate a high impedance recorder and simultaneously to realize the sensitivity of high resistance thermistors, four 100,000-ohm thermistors are connected in parallel for each of the detectors in the differential circuit. The response of the electrical circuit is analyzed mathematically in order to achieve a linear response of the circuit to differences in temperature between the two sensing elements, Because of its linear response, electrical power can be used as a standard for calibrating the apparatus, so that heats of reactions can be measured rapidly and simply.
D
the past eight years there has been considerable interest in thermometric titrations (9, 15-16, 18, 20, 21, 28-50)-that is, titrations in which the course of the reaction is followed by observation of the heat absorbed or liberated. The literature on this type of titration has been well URIKQ
1 Present address, Strategy and Tactics Analysis Group, U. S. Army, Bethesda, Md. 2 Present address, Department of Chemistry, University of Delaware, Newark, Del. 3 Present address, The College of Wooster, Wooster, Ohio.
1640
ANALYTICAL CHEMISTRY
covered in two excellent review articles (12, 27). Many thermometric methods for following titrations have been limited because the temperature-sensing device has been too insensitive or because heats other than just the heat of reaction have been detected. It seemed reasonable t h a t thermometric methods could be applied more widely if a more sensitive circuit could be designed that would measure only the heat of reaction. With such an apparatus it should be comparatively easy to determine calorimetric data for calculating heats of reaction. A differential thermometric apparatus seemed to be the best solution to this problem. Muller and Stolten (28) and Higuchi et al. (11) have used differential thermistor circuits for thermometric determinations of molecular weights, and Pakulak and Leonard (24) have used a shunted differential thermistor circuit in differential thermal analysis. I n the differential apparatus designed for this work, temperature-sensing devices were placed in both the sample and blank solutions. Sensitivity greater than that previously reported for thermometric titrations has been obtained and extraneous heat effects, such as stirring and heats of dilution, have been greatly minimized. This apparatus has been used to follow 14 d 8 e r e n t reactions and the heat of the reaction has been calculated for 11 of these reactions.
EXPERIMENTAL
Chemicals. Common chemicals were of reagent or analytical grade. T h e hydroxyamines were supplied by the Commercial Solvents Co. and were purified by distillation except for tris(hydroxymethy1) - aminomethane which needed no purification and for 2 - methyl - 2 - amino - 1,3- propanediol which was recrystallized from acetone (26). Apparatus. Motor-driven burets similar to ones designed by Lingane (19) and by Jordan and Alleman (15) were used. -4 60-cycleJ 110-volt, 1r.p.m. synchronous motor from the Holtzer Cabot Co. was geared so that a 16-thread-per-inch screw was turned at 4 r.p.m. This screw, in turn, drove a threaded brass block which pushed the plungers of two horizontally mounted 5-ml. syringes. Only one size of syringes and one set of gears were used, but others could have been substituted to vary the delivery rate of the titrant. The delivery rate, which was 0.01110 ml. per second for each buret, was determined b y measurement of the weight of titrant of known density delivered during a determined number of revolutions of the gear driven by the synchronous motor. During each calibration, the delivery rates of the two burets were found to be the same within 1 part per thousand, although the syringes were not a specially matched pair. I n fact, a third syringe was substituted and no appreciable change in delivery rate was found. These delivery rates were checked periodically over a period of 7 months and the mean deviation of all of the values was 1 part per thousand. Each syringe was connected to a
?r=
Sensing C i r c u i t
Bucking V o l t a g e
L
To
Figure 1.
R.cord.r
50-ohm IO-turn potentiometer 1 000-ohm potentiometer 4000-ohm potentiometer 16,000-ohm resistor 18,000-ohm resistor Rd. 2000-ohm potentiometer R5 to Rg. 5000-ohm resistors V. Variable voltage from dry cells, 0. to 22.5 volts V ' . 6-volt dry cell 71 = T Z = sets of four 5 l A l thermistors in parallel. Resistance a t 2 5 ' C. 25,900 ohms
To utilizc a lower input impedance recorder and still have the advantage of the high temperature coefficient of high resistance thermistors, it \+as necessary t o use several thermistors connected in parallel. For a differential circuit, however, two sets of thermistors with the same resistancr-tcmperature response would be needed. Accordingly, the resistances of 11 thermistors were measured over the range 0" to 50" C. to the nearest 100 ohms. By combining
where 'T is the resistance of the thermistor, B is a constant, and T is the absolute temperature. The best value for B was 4018.9 for the sets in the range 21" to 34' C., and values of 7 calculated from the expression =
-1.44168
+ 1745.4/T
+ R D E+ RDF
ri '71
+ RAB + RAC
(3)
and
various thermistors, two sets of four thermistors connected in parallel were found which had resistances t h a t matched to within 1% over the 0" t o 50" C. range. Within the narrower range of 21" to 34" C., the resistance of the two sets agreed to 100 ohms. Thermistors are known to follow closely a resistance-temperature relationship of the form
T
72
+
PI. Pa. R1. Rz. Ra,
log
72
A may be made to vary linearly with temperature most simply if both terms on the right side of Equation 3 vary linearly with temperature. Because i t is conceivable that the temperature in the blank container (involving 'TJ will remain constant during a titration, the only way for A to vary linearly n i t h temoerature in this case is to have r1 +'RAB vary linearly with tempera+I RAC ture. However, in the most gmeral case and expressed mathematic:~lly,it is necessary for P to vary linearly n i t h temperature, so that
Temperature-sensing and bucking voltage circuit
three-way stopcock which, in turn, was joined to a storage vessel for titrant and to a capillary tip. One of the capillaries dipped in the solution where the reaction took place and the other in a blank solution. The solution of the sample and the blank solution or solvent n ere placed, respectively, in 100-ml polyethylene cups. Plastic pres-on lids for these cups were mounted on the underside of a block of Styrofoam 12 inches long, 8-1/pinches wide, and 4 inches thick, nhich was suspended in a rigid metal franie\!ork. For titrations, the cups \\ere pressed into the lids and were insulated by a second block of Styrofoam 12 inches long, 81/2 inches wide, and 8 inches thick, which had two hollowedout spaces to accommodate the cups. The two blocks of Styrofoam werc then fitted togethw snugly and the loner block, which had been raised around the cups, n d s supported by nooden hlocks. When the cups had been plated in position, each enclosed a temperature-sewing element, .i capillary buret tip, and the glass rod of a motordriven stirrer, all of which had been installed in holes drilled through the upper Styrofoam blork and through the cup lids. The sample cup also enclosed n heating coil. For the temperature-sensing elements, Type 51.41 thermistors from the Victory Engineering Co. were used. These have the following specifications' cold resistance a t 25" C., 100,000 ohms i 15%; temperature coefficient at 25" C., -4.6y0 per O C . ; thermal time constant in still water, about 1 second.
-v
-~ 71 + = a constant dT T I -k RAC
Since 71 = 72 over a given temperature range and since RAc = R D F , Equntions 4 and 5 can be written in the general form
D 7+-" F= a constant
(6)
Plots of this general expression us. T give parabola-like curves, the stationary region-Le., where the slope of the parabola is zero-of which shift positively along the temperature axis as R" is decreased. If the second derivative,
+ + R'
d2 __ T R'
-.
dT2 T
is set equal to zero and solved by use of Equation 1, the result is
(2)
B B
R"
agreed with measured values within 100 ohms. T o suspend the two somewhat fragile sets of thermistors reproducibly in the two cups (sample and blank, respectively), pairs of thermistors were mounted with paraffin in 6-mm. glass tubing so t h a t 2 cm. of each thermistor rod extended beyond the tubing. Thus, each sensing element consisted of two pairs of thermistors which were suspended on opposite sides of each cup to provide the best conditions for measuring the temperature of the solution in a cell. As shown in Figure 1, each set of four thermistors was used as one arm of a Wheatstone bridge circuit similar to those suggested by Linde, Rogers, and Hume (18) and by Jordan and Alleman (IS). This circuit was selected from several tried and a separate paper will deal with several thermistor circuits in more detail. If the current drawn by the recorder is negligible, the equation for the output voltage of this circuit, A , is
(5)
7-
- 2T
+ 22'
(7)
Because 7 and B are known, it is now possible to calculate the value of R" for a certain temperature which will provide the closest approximation to linearity in the voltage response of the circuit to temperature differences between blank and sample. Thus, the choice of resistance in the arms of the circuit is dependent on the temperature range t o be used. If linearity of output voltage us. temperature difference is required, the temperatures encountered must be in a small range about the temperature where the value of R' is stationary-Le., where the
a
T F T F
d T+R'
value of dT R,, is a maximum. With the particular circuit used, where R" = 20,000 ohms, a change of =t4" C. from the optimum temperature of 297.2' K. should cause a change of less than 1% from this stationary value, whereas if a decrease of less than 0.1% from the stationary value is desired, a deviation ~
+
VOL. 33, NO. 12, NOVEMBER 1 9 6 1
1641
of only f1.5" C. from the central temperature is allowable. Examination of Equation 7 shows that the temperature, where the sensitivity is a maximum and stationary, is independent of R'-that is, of the settings of the corner potentiometers in the circuit-but since I
I
..*A,
I
1
R3
Figure 2.
the stationary value and therefore the output voltage of the circuit may be changed by altering the settings of the corner potentiometers. This means that the sensitivity of the circuit to changes in temperature will be affected by changes in the corner resistances. The experimental effect of this is that the two halves of the bridge circuit can be made to have different sensitivities, so that even if r1 = 7 2 at a given temperature and RAC = RDF, then if RAB does not equal RDE, the bridge will become unbalanced and produce a n output voltage when r1 and 72 are changed to the same new value. This effect is desirable if the heat capacities of the blank and sample containers and,'or their contents are different. Because heats of dilution are to be cancelled by the differential circuit, it is desirable that the same amounts of heat generated in both blank and sample containers give the same apparent temperature change- even if the heat capacities of the two cells are different. If the corner potentiometers are carefully adjusted during trial titrations, this effect can be obtained easily. Usually in this work the two heat capacities were so nearly equal that the correction was not necessary. Therefore, the corner potentiometers were adjusted so that Rail and RDE wcre zero. This, as shown in Equation 8, resulted in maximum sensitivity. The sensitivity of the circuit is directly proportional to the input voltage, V . The usual input voltage was 12 1- and values higher than 15 V were unsatisfactory, apparently because of self-heating of the thermistors and resulting "noise." The self-heating of the thermistors is dependent on the wattage through the thermistors and on the heat dissipation constant of the thermistors. At 25' C. in the circuit used, with 12 volts applied, RAB = 0, and RDE = 0, the power through each individual thermistor is 0.44 mw. The dissipation constant is stated by the manufacturers to be 5 mw. per C. in unstirred water. I n this case, the self-heating of each thermistor would be about 0.09" C. I n Jordan and Alleman's circuit (IS), the amount of self-heating was calculated to be about 0.06" C. Actually, the dissipation "constant" depends on the rate of flow and on the nature of the fluid 1642 *
ANALYTICAL CHEMISTRY
Heating circuit
€. 6-volt storage battery R1. 2-ohm heating coil, &inch No. 30 constantan wire RB. 2-ohm, 25-watt resistor 1.981 -ohm resistor (constantan wire) RI. R,. 7-ohm resistor (constantan wire) Ra to R l z . 1 -ohm resistors (constantan wire) SI. Single-pole single-throw switch SI. Single-pole double-throw switch SI. Double-pole double-throw switch
around the thermistor. So the true amounts of self-heating are not known. Since the dissipation constants vary with stirring rate, the two stirrers used were equipped with Variacs, so that the rates could be matched. Or, better stated, the combined effects of heat of stirring and self-heating of the thermistors were matched so that a horizontal base line in titration curves could be obtained. The heating coil enclosed in the sample cup was a part of the circuit shown in Figure 2. The coil, Rt, was an 8-inch piece of KO.30 constantan wire having a resistance of about 2 ohms. The ends of the coil were soldered to low resistance S o . 12 copper wire. This wire was coated with paraffin and sealed in glass tubing which led through the cup lid and Styrofoam insulation. The tubing served to make the wire slip free in the apparatus. Since it was expected (and later found) that the resistance of the coil would change slowly after contact n-ith many solutions, a length of constantan wire having a resistance of 1.981 ohms was placed in series with the coil and used as a standard resistor. 3leasurement of the voltages across the standard resistor and across the coil was suffirient for determining the rate of heat production b y the coil. The current measured in the standard resistor is the same current flowing in the heater coil. Multiplication of E across the heater coil by I gives the energy dissipated by the coil. Variation of the amount of resistance in series with the coil allowed a choice of heating rates from 0.05 to 0.14 cal. per second. The constantan resistors, R3through R1,, were suspended in a glycerol bath to maintain them at a constant temperature and constant resistance. Rt was a dummy resistor and was substituted for the heating coil in the circuit when the components of the circuit were equilibrated before an actual heating curve was made. A Varian Model GllA, 10-mv., 5inch chart width, recording potentiom-
eter was used to follow changes in the unbalance potential of the temperaturesensing circuit. The chart was driven a t a rate of 6.00 inches per minute. -4t 28" C. with 12 T7 input voltage and R.AB and RDE set equal to zero, a change of 1" C. in the sample container produced an unbalance voltage of 133 mv. Thus, the entire width of the chart paper represented a change of 0.075" C. The sensitivity reported for Jordan and Alleman's ( I S ) circuit was 15.7 mv per " C. If the input voltage of our circuit Fere lowered to 9.5 volts, so that the amount of self-heating would be equal to that allowed b y Jordan and Alleman. the sensitivity of the circuit would still be 106 mv. per O C. Because the solutions used for the blank and the sample were not thermostated, the temperatures of these two solutions a t the beginning of a titration were not necessarily the same. Frequently the difference in temperature of these solutions initially was sufficient to throw the recorder reading off scale. The bucking circuit, s h o m in Figure 1, was used to adjust the output of the over-all circuit so that the base line for the titration was on the recorder scale. By changing the setting of the potentiometer resistors, P1 for fine adjustment and PI for coarse settings, this bucking circuit could supply a voltage of 0 to *290 mv.
Procedures. AH Determinations.
A 50- t o 75-ml. portion of a sample solution and a n equal volume of the solvent were placed in t h e sample and blank cups. The solvent contained the same components as the sample solution, b u t none of the chemical species to be titrated. After the cups had been pressed into their lids and the Styrofoam insulation had been placed around the cups, the stirrers were started. The bucking voltage was adjusted until a suitable base line was obtained on the recorder. Then the chart drive and burets were started in order. About 10 seconds after the titration was complete, as indicated by a change in slope of the chart recording, the burets were stopped. With the bucking circuit, the base line was readjusted to a suitable level on the recorder. The heating coil was then turned on and after a temperature change similar to t h a t observed in the titration had occurred, the heating coil was turned off. The base line was now readjusted and another recording of the heating rate was made. This procedure was repeated three or more times. During the recording of each heating curve, the voltages across the heating coil and the standard resistor were measured with a potentiometer.
The titration curves and the heating curves had the same general shape. Both of these curves were extrapolated in the same manner, as shown in Figure 3. The product of the two measured voltages, when divided by the resist-
Table 1.
Substance Titrated
Figure 3.
Thermometric titration curve
Titration of silver nitrate with 0.9698N hydrochloric acid 75 ml. of solution in sample cup, 75 ml. of water in blank cup 10 volts applied to bridge circuit AB. Base line. Addition of titrant started a t 6
ance of the standard resistor, gave the heating rate of the coil in joules per second. The slopes of all heating curves and of the titration curve were measured. The heat of the reaction could then be calculated from the equation, AH = $1
(4.185) R;NF
$9
=
calories per meq.
where
E1 = voltage across heating coil E* = voltage across standard resistor R, = resistance of standard resistor, ohms N = normality of titrant F = flow rate of buret, ml. per second SI = slope of titration curve SI = slope of heating curve For each titration a value of AH for each heating curve was calculated and the average was taken. If several titrations \+-ereperformed, the resulting values for all titrations were then averaged. DETERMINATION OF QUANTITY OF REACTANT PRESENT. The procedure was similar to that above, except that no heating curves were taken. If the samples taken were of such size or concentration that the temperature change produced was greater than that represented by the full range of the recorder, i t was necessary to adjust the bucking voltage so that the recorder was back on scale before the end point was reached. The response of the recorder could have been kept on scale during the entire titration by the use of a smaller input voltage to the bridge circuit. However, with the higher voltage applied to the bridge, the angle between the base line and the titration line was more acute and, therefore, a more accurate extrapolation to the point where the titration began and ended
Results of Thermometric Titrations
Amount Added, Meq. 0.996 (4)' 0.4989 (4) 0.1989 (3) i.o05
