Derivative Thermoa na Iytica I Tech niques Instrumentation and Applications to Thermogravimetry and Differential Thermal Analysis CLEMENT CAMPBELLl SAUL GORDON, and CHESTER L. SMITH Pyrotechnics Chemical Research laborafory, Picatinny Arsenal, Dover, N. 1.
b Thermoanalytical techniques provide continuously recorded curves that characterize a system in terms of the variations in its thermodynamic properties and reaction kinetics as a function of temperature. To define these thermal spectra more exactly and interpret the phenomena and reactions involved, particularly in overlapping temperature regions, a technique has been developed for simultaneously obtaining a continuous record of the derivative for the thermogram. A simply constructed differentiator consists of a battery energized precision transmitting potentiometer coupled to the balancing slide-wire of the recorder used to obtain the primary thermogram. The signal generated b y the transmitter circuit is differentiated b y a resistance-capacity circuit and recorded on another timebase or x-y potentiometric recorder.
R
advances in laboratory instrumentation have contributed to a great increase in the applications of thermoanalytical techniques such as thermogravimetry, differential thermal analysis, and thermovolumetry as anaECENT
lytical methods and physicochemical research tools. Data are obtained as continuously recorded curves which may be considered as thermal spectra. These thermograms characterize a system, single or multicomponent, in terms of the temperature dependencies of its thermodynamic properties and physicochemical reaction kinetics. Thermogravimetry involves continuously recording the change in weight of a system under investigation as the temperature is increased a t a predetermined rate. These data can be used to study any physical or chemical process which is accompanied by a gain or loss in weight. Differential thermal analysis consists of continuously recording the difference in temperature between the sample under investigation and a thermally inert reference compound as the two materials are heated to elevated temperatures a t predetermined heating rates. By this technique it is possible to detect physicochemical phenomena that involve the absorption or evolution of heat. Thermovolumetry is a more recent addition to thermo-
analytical techniques. It is a means of following a gas-absorbing or gasproducing reaction by continuously recording the change in the volume of gas consumed or evolved as the material is heated to elevated temperatures a t a constant rate. Differential thermal analysis (DTA) curves are made up of endo- and exothermal bands and peaks having characteristic points of inflection, maxima, and minima. Thermogravimetric and thermovolumetric curves generally are stepped curves with points of inflection and plateaus which are indicative of the formation of various intermediate and final reaction products. To define these thermal spectra more exactly and interpret the phenomena and reactions that occur, a technique has been developed for simultaneously obtaining a continuous record of the derivative for any thermoanalytical curve plotted on almost all types of electronic graphic recorders. This derivative curve is of value in the detection of crystalline transition, fusion, and boiling temperatures, intermediate reaction products,
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PO'-"-'-"---TRANSMITTING CONNECTED TO $ SHAFT OF X I AX1 I
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TO X r 40mfd AXIS OF RECORDER ~
u1°F
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100mfd.
X-Y RECORDER
6 VOLT
ITR SLIMWIRE ANSM~NG ON PRlMARY FUNCTION RECORDER.
Figure 1.
Circuit diagram for R-C differentiation
Top. R-C differentiating circuit for use with potentiometric recorders Bottom. R-C differentiating circuit for use with Moseley X - y recorder
1 188
ANALYTICAL CHEMISTRY
(b) Figure 2.
Circuit diagrams
a.
For u5e with potentiometric recorders
b.
For use with Moseley x - y recorder
Ca(N0, )* *4H20 0.2496 g.
KNO, /
, I -
,-'
0
Figure 3. Primary and derivative curves for calcium nitrate tetrahydrate
and other characteristic thermal phenomena, where they appear consecutively in overlapping temperature regions. When differential thermal analysis and thermogravimetry are used as complementary techniques for studying a given system, the derivative thermogravimetric curves may be more directly compared with the DTA curves because of the similarity of the bands and peaks corresponding to the changes in weight accompanying the thermal reactions. The value of derivative methods as applied to thermogravimetry has been shown by de Keyser ( I @ , Lambert (I@, Erdey, Paulik, and Paulik (4, and Waters (222) in describing differential thermobalances especially developed for these purposes. However, these latter techniques involved somewhat elaborate equipment ( I I ) , restricted solely to thermogravimetric applications. Frederickson (6) and Freeman and Edelman ('7) have applied the derivative technique to differential thermal analysis curves. The simply constructed differentiator employed for the derivative techniques described in this paper is a resistancecapacity (R-C) circuit ( I $ ) , similar to those reported by Kelley and Fisher (14) for derivative polarography and by Zenchelsky and Segatto (23) who investigated the techniques of derivative thermometric titrations. APPARATUS
Differentiator. The primary thermograms consisting of differential temperature, weight, or gas volume plotted as a function of time were recorded on either of two potentiometric strip-chart recorders, a twopen Leeds & Northrup Speedomax with adjustable zero and adjustable range, and a Westronic recorder also provided with a n adjustable zero and range (3). A precision single-
50
100
150
400
200 250 300 350 SAMPLE TEMPERATURE ('CJ
450
Figure 4. Derivative and normal differential thermal analyses curves for potassium nitrate
turn 500-ohm transmitting potentiometer, Helipot Model 5701, is coupled to the slide-wire shaft of the Leeds & Northrup recorder. The Westronic instrument was equipped with a 300-ohm transmitting slide-wire mounted parallel t o the balancing slide-wire with both contactors fastened to the pen carriage. A 6-volt battery connected across the transmitting slide-wire served as the voltage source for generating signals directly proportional to the displacement of the pen and therefore proportional to the input signal. Because these primary input signals were of the order of 5 mv., the effective electromechanical amplification attained was lo3. The circuit diagrams for these differentiators are illustrated in Figure 1. The R-C differentiator was calibrated by sequentially generating a series of linearly changing direct current signals, feeding them into the recorder used t o plot the primary thermoanalytical curve, and plotting the output of the differentiating circuit. A batteryenergized synchronous motor-driven precision potentiometer with a variable limiting resistor was used to obtain the linearly changing direct current signals. The rates of change were obtained from the slopes recorded on the primary timebase recorder. These primary signals are readily calibrated in terms of units of weight, temperature, or any other parameter being measured. Thennoanalytical Equipment. The apparatus for thermogravimetry, differential thermal analysis, and thermovolumetry have been described (8-10). Circuit Analysis. The simplest circuit for differentiating the output of a transmitting slide-wire is shown in Figure 2, a. An external resistance, R, is chosen having a value very much greater than that of the potentiometer slide-wire, Rs,so that the capacitor charging current, i, is essentially proportional to the rate of change of the voltage with respect to time. Because the slider of potentiometer R, is coupled to the contactor of the balancing slide-
wire in the primary curve recorder, the rate of change of the transmitted voltage, dE* E,-Le.,--is a linear function of the dt rate of change of the primary signal (12).
If R is shorted and R, is extremely small, the current passing through the capacitor, C, is
With R in the circuit the output voltage,
ED,obtained from this differentiating circuit is dE RC-a
Eo = iR
di
Equation 2 closely approximates the true derivative if dE,/dt remains constant for a period of time greater than the RC time constant (the product, ohms X farads) expressed in eeconds. If it is assumed that E, changes instantaneously from one voltage to a higher voltage, then
Eo = E, (e-'IRC) where E,, is the output voltage a t time dE, t . However, if - is constant and dt changes instantaneously to a new value, the expression for Eo as a function of time is
so that when t
=
-,
The simple circuit in Figure 2, a, produces satisfactory results with many types of potentiometric recorders. However, all recorders have some erratic VOL. 31, NO. 7, JULY 1959
1189
-
9 Figure 5.
Differential temperature vs. time for the
motions (hunting, overshoot) in the servo-system and also irregularities in the slide-wires which produce tracings containing a certain amount of "noiselike" irregularities. Therefore, it is necessary to filter the output by means of condenser Cz, so that these spurious values of dE./dt are minimized. Because certain recorders-e.g., Moseley Autografs-have fixed input resistances ranging from 1000 to 100,000 ohms, depending on the recorder sensitivity range, it is often not practical to vary RCl a t will. Therefore, the RC1 value must be established within limits, determined by the particular output voltage and recorder sensitivity. In the case of a 50-mv. full scale recorder range, with a 1-second full scale pen deflection time, the differentiator RC, time constant is chosen to produce the desired output voltage range, while the value of the filter RC2 time constant can be 21 second (Figure 2, b). The upper limit of the differentiator RC time constant is determined by the nature of the primary curve. Therefore, if E, is expected to be a slowly changing curve, then the value of RCI can be made relatively large so as to produce a dE,/dt relation free from spurious recorder pen deflections. The expression for the output, EO, a t any time, t, for the circuit in Figure 2, b, for the conditions as described for Equation 3 is
Again, when t = 0, Eo = E, times a constant For the circuit in Figure 2, b, if dE,/dt is constant and instantaneously changes to some new value, the expression for Eo is
1 190
ANALYTICAL CHEMISTRY
I
I
I
I
I
I
I
Figure 6. Normal and derivative DTA curves for 50% nitric acid and 50% potassium perchlorate mixture
Again, when t =
Eo,
Eo = dE,/dt times
a constant. APPLICATIONS
The value of the derivative technique
in thermogravimetry can be realized by comparing the primary curve with the simultaneously obtained derivative curve illustrated for calcium nitrate tetrahydrate in Figure 3. The electric signal from the thermobalance ( 9 ) , proportional to the change in weight, was simultaneously recorded on an x-y recorder and on the Westronic strip-chart recorder, the inputs of which were connected in parallel. A second 2-y recorder was used to record the R-C differentiator output from the strip-chart recorder with its transmitting slide-wire. The furnace temperature served as the common base for both abscissae. The two regions of interest, on these curves, are those involving dehydration of the tetrahydrated salt from 75" to 300" C. and decomposition of the anhydrous salt between 575" and 750" C. Although the stepwise dehydration of the salt is clearly shown in both the primary and derivative plots, the complexity and stepwise nature of the decomposition pattern is more clearly defined on the derivative curve than it is on the corresponding primary curve. Important points of inflection on the primary thermogravimetric curve can be identified and pin-pointed by means of the maxima on the corresponding derivative curve. The derivative curve, although plotted on the temperature base, is a derivative of the change in weight with respect to time. The areas of the bands on the derivative curve are proportional to the corresponding changes in weight. The derivative technique may be used to elucidate the kinetics of reactions by theoretical treatments of the thermo-
gravimetric data in accordance with equations such as those reported by Baur, Bridges, and Fassell ( I ) , Kofstad ( I Y ) , Freeman and Carroll (6), and van Krevelen, van Heerden, and Huntjens (18); or of differential thermal analysis data by the methods of Borchardt and Daniels ( 2 ) or Kissinger (16). The derivative as well as the normal differential thermal analyses curves for potassium nitrate (Figure 4) illustrate the application of these techniques to analytically characterizing endothermal physical phenomena such as crystalline transition and fusion. The points of inflection on the lower temperature side of these endothermal bands, comprising the primary DTA curves, appear on the derivative curve as well defined peaks in the downward direction whose maximum deflections occur a t temperatures corresponding to the respective crystalline transition of 128" C. and melting point of 340" C. for potassium nitrate. Because the derivative DTA curve is the calculus derivative of the primary DTA curve, with respect to time, each band on the latter curve will produce two bands on the corresponding derivative curve, one below and the other above the zero base line. In the case of a curve with an endothermal band consisting of a point of inflection, minimum, and a second point of inflection, the derivative curve will comprise two consecutive bands with their peaks corresponding to the inflection points and which join the base line a t the minimum of the primary curve. The area of a given thermal band is proportional to the amount of reactant(s) responsible for the physical or chemical reaction involved, and it has been applied to semiquantitative analyses of mixtures such as those containing potassium perchlorate (IS) and mineralogical substances (20, 21). The
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2 100
eo
,
1
,
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3
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20
60
40
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20 I
,
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40
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60
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80
1
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KNO 3
Figure 7. Derivative peaks vs. sample composition for both crystalline transitions
peak of the derivative curve occurs at the temperature of the point of inflection. The magnitude of this derivative 13 proportional to the amount of reactant(s) present for zero-order physical reactions such as crystalline transition and fusion, where the reactant is diluted with a thermally inert material such as the reference compound. The endothermal band for the crystalline transition of potassium nitrate in Jvhich differential temperature is plotted us. time, shown in Figure 5 , depicts the deviation from the base line a t a, with points of inflection b and d, minimum c, and the return to the base line at e . Assume that the thermal effect of the reaction is completed a t the minimum point, c, and that the portion of the endothermal band contributed by the change in enthalpy is represented by the area enclosed within the right triangle, fgh. This triangle is formed by the hypotenuse, fg, drawn tangent to the point of inflection b, meeting the base line ae a t a n angle 8, and forming a n apex at point g, by intersection with a line, hg, drawn perpendicular to the base line a t h and passing through the minimum point, c. Because the area of this previously defined part of the DTA band is proportional to the quantity of reactant involved, the following expression may be used to define this area, A :
9 = '/gBH
(7)
H tan 0 = B
where B is the base, fh; H the altitude of the triangle hg, and 8 is the angle subtended by the altitude. 1/,B2 tan 0
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\
220
S L "0.2I '0.4I "0.6' I0.8" '1.01 1.2 ' " 0 0 RATE Figure 8.
Calibration curve for
mental parameters are the same, then A = K tan
e
(10)
mhere K is a proportionality constant. Angle 0 is chosen at the point of inflection, because at this point the slope has a maximum value and consequently the maximum area is obtained. Experimental verification of this interesting and analytically useful application of derivative differential thermal analysis \vas made by obtaining normal as well as derivative DTA curves for the pure salts, potassium nitrate and potassium perchlorate, and their binary mixtures of 25 to 75, 50 to 50, and 7 5 to 25y0 composition as illustrated in Figure 6 by the curve for the 50 to 50 composition. The first two of the consecutive endothermal bands correspond to the crystalline transitions of potassium nitrate and of potassium perchlorate a t 128" and 300" C., respectively (8). The 300" C. transition band overlaps the third endotherm, which is the melting of potassium nitrate at 340" C. I n all cases the points of inflection, as defined by the derivative peaks, occur a t the temperature cited. The linearity of these derivatives as a function of sample composition for both transitions is shown in Figure 7 . A calibration curve for the differentiating circuit used in this quantitative evaluation of derivative DTX is presented in Figure 8, in terms of the recorded R-C differentiator output in millivolts as a function of the input signal, expressed in degrees centigrade per minute. ACKNOWLEDGMENT
The authors are indebted to Seymour
(9)
Z. Lewin for his suggestions concerning
If we now assume that the base B is essentially a constant for the system under study-where the diluent, granulation of materials, and the geometry of the DTA apparatus, and all other experi-
the use of resistance-capacitor circuits for the derivative techniques. Recognition is also given to John Ruscus for his aid in constructing prototype differentiator circuits and to Eli S. Freeman for
d
=
1.4
1.6
OF CHANGE OF PRIMARY SIGNAL (MV./MIN.) differentiating
circuit
a helpful contribution to the mathematical interpretation of the derivative curve. LITERATURE CITED
(1) Baur, J. P., Bridges, I ). W., Fassell, W. M., Jr., J. Electrochem. SOC. 102, 490-6 (1955). ('2) Borchardt, H. J., Daniels, F., J. Am. Chem. SOC.79, 41-7 (1957). (3) Campbell, C., Gordon, S., ANAL. CHEM.29,298-301 (1957). (4) Erdey, L., Paulik, F., Psulik, J., Nature 174, 885-6 (1934). (5) Frederickson, A. F., Am. Mineralogist 39, 1023-5 (1954). (6) Freeman, E. S., Carroll, B., J . Phys. Chem. 62, 394-7 (1958). (7) Freeman, E. S.,Edelman, P., AN.4L. CHEX 31, 624-5 (1959). (8) Gordon, S., Campbell, C., Ibid., 27, 1102-9 (1955). (9) Ibid., 28, 124-6 (1956). (10) Ibid.,29, 1706-8 (1957). (11) Ihid.,in press. (12) Greenwood, I. A,, Jr., Holdam, J. V., Jr., Macrae, D,, Jr., "Electronic Instruments," Chap. 4, M.I.T. Radiation Laboratory Series S o . 21, McGrawHill, New York. 1948. (13) Hogan, V. D., Gordon, S., Campbell, C., ANAL.CHEJI.29, 30(i-10 (1957). (14) Kelley, hI. T., Fisher, D. J., Ihid., 30, 929-32 (1958). (15) Keyser, W.L. de, Kature 172, 364-5 (1953j . . 29, (16) kissinger, H. E., h . 4 ~ CIIEY. 1702-6 (1957). (17) Kofstad, P.,Nature 179, 1362-3 ( 19 5 7). (18j Krevelen, D. SJ'. van, Heerden, c. van, Huntjens, F. J., Fuel 30, 253-9 (1951). (19) Lambert,, A,, Bdl. SOC. fran6. ceram. 28, 23-28 (1955). (20) biarel, H. W. van der, A m . Mineralogist41,222-4 (1956). (21) Vold, 11.J., ANAL.CHEX 21, 683-8 (1949). (22) Waters, P. L., J . Sci. I ~ s ~35, T .41-6 (1958). (23) Zenchelsky. S. T., Segatto, 1'. R., bS.4.L. CHEJI. 29, 1856-8 (19.57).
RECEIVED for revieir- Sovemher i, 1958. Accepted February 27, 1959. Division of Analytical Chemistry, 181st Meeting, .4CS, Miami, Fla., -4pril 1957.
VOL. 31, N O . 7, JULY 1959
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