Simple Method for Derivative Differential Thermal Analysis Eli S. Freeman and David Edelman, Pyrotechnics Chemical Research Laboratory, Picatinny Arsenal, Dover, N. J.
differential thermal D analysis (DDTA) is of interest as a complement to differential thermal
I
ERIVATIVE
analysis (DTA) (22). The temperatures of peaks of bands as well as details often overlooked in DTA curves, such as inflections which appear as peaks in DDTA curves, make this technique useful for characterization of substances as well as for studying reaction kinetics (1, 8, 7). Derivative differential thermal analysis may be carried out electronically ( 5 ) . The method described involves using a duplicate sample as the reference material and maintaining a constant temperature differential in the applied temperatures between both samples. Conventional DTA apparatus (6) is used. The principle of this technique permits automatic recording of the DDTA curve and is the basis for the derivative thermogravimetric curves obtained with a De-Keyser differential by thermobalance (manufactured Sartorius, A. G. Werle, Gottingen, Germany), where two furnaces and duplicate samples are employed, the temperature of one system lagging the other. The apparatus is identical to that described previously ( 6 ) , except that Nichrome blocks are used. Borosilicate glass tubes 18 X 90 mm., into which thermocouple wells 5 mm. in outside diameter are fused near the open end, contained the samples. The bottoms of the thermocouple wells are centered and located 5 and 8 mm. from the bottom of the tubes. The increase in furnace temperature was regulated by a Model JPG Gardsman indicating pyrometric stepless program controller (West Instrument Co.). B. & S. No. 28 gage Chromel-Alumel thermocouples mere used. All the chemicals were obtained from the J. T. Baker Chemical Co. and were of reagent grade. Ten-gram samples are placed in each of two tubes and packed to the same height. The sample whose temperature is recorded is placed in the tube, with the thermocouple well 5 mm. from the bottom. A duplicate sample is located in the reference tube, with the thermocouple well 8 mm. from the bottom. These tubes are set into the Nichrome block, which is placed slightly off-center in the furnace, 1.0 to 1.5mm. to one side, so that the sample whose temperature is recorded is closer to the furnace coils. The temperature differential between the two samples was continuously recorded as a function of sample temperature a t a heating rate of 15" C. per minute. Conventional differential thermal analyses were conducted independently, using aluminum oxide as the reference material. 624
ANALYTICAL CHEMISTRY
SAMPLE TEMP.,.C.
Figure 2. DDTA and DTA curves of potassium perchlorate
SAMPLE TEMP.,%
Figure 1.
DDTA and DTA curves of potassium nitrate
Figure 3. DDTA and DTA curves of calcium o x a l a t e monohydrate
SAMPLE TEMP.,.C.
RESULTS A N D 1)ISCUSSION
An experiment i i a s carried out in which aluminum oride was placed in both sample and refcience tubes. Prom 100" to 700" the tenqierature difference between the samples m1s3.3" f 0.3" C. Over the temperatuic ranges of 150" to 375", and 375" to 6CO", deviations were considerably less, about f0.10" from the mean tempera1lire differential of each region. If the vertical distance of the thermocouple wclls from the bottom of the tubes containing the sample and reference material is che same, although the Nichrome block F off-centered in the furnace, the temp xature differential decreases with incrcuing temperature. When the thermocouple well of the sample tube is slightly lower ( 5 nim.) than that of the reference tube (8 mm.), the measured temperature differential is relatively constant. Figures 1, 2, and 3 illustrate the fusions and crystalline transitions of po-
tassium nitrate and potassium perchlorate and the decomposition of calcium oxalate monohydrate to calcium carbonate. Up to temperatures corresponding to the minima on the DTA curves, the derivative curves obtained by measuring and plotting the corresponding slopes are inagreement with the experimentally recorded DDTA curves. The minima of the DDTA curves correspond to the maximum rates of change in differential temperature (Table I). From the minima to completion of t h e DTA bands agreement is qualitative. The temperature differential between duplicate samples prior to physical o r chemical reaction ranged from 5" to 10" c.
Theoretical Basis. Consider the simultaneous differential thermal analyses of two substances, 1 and 2 , giving temperature differentials reprcsented by Equations 1 and 2 : ATs =
2'8
- T,
(1)
Table I. Comparison of DDTA and DTA Curves
Compound
Pheuomena Crystalline transition Melting Crystalline transition Dehydration Decomposition
KNOI KC104 CaC204.H20
A17s' = lT8' - lJr'
(2)
where T, is tlie temperature of sample 1; T8', temperature of sample 2; T,, temperature of inert reference material used with saniple 1; and Tr', temperature of inert reference material used with sample 2.
If the temperature applied to both systems differs, subtracting Equation 2 from 1 gives:
- AT,'
AT,
=
T, - Ts'
+ Tr' - T ,
(3)
Because the heating rates of both reference saniples are the same, dAT,.,' dt
-
dT,.,' dt
(4)
From Equation 4 it is apparent that derivati1.e curves may be obtained by using a single furnace and Mock, and a duplicate sample in place of the inert reference material. The temperature differential varies with the geometric arrangement of the system and depends on the thermodynamic properties of the substance under study. If it is too high, two DTA curves may be obtained with a conunon base and opposite deflections. Where
Temp., Max. slope tDTA) 129 335 300 235 493
O
C. . Minima (DDTS) 120 335 300 235 495
the temperature diff erentinl approaches zero, the true derivative plot is approached. This is accompanied by loss in sensitivity which may be partially overcome by an increase in sample size. The seriousness of loss in sensitivity depends on the endo- or exothermicity of the reactions involved and the sensitivity of the apparatus. A temperature lag ivhich is satisfactory for one system may be unsatisfactory for another. The fact that the experimentally observed temperature difference between the aluminum oxide samples was essentially constant with time indicated the feasibility of conducting derivative differential thermal analyses with tlie apparatus and experimental arrangement used. This is borne out by the DDTA and DTA curves shown in Figures 1, 2, and 3, and data summarized in Table I. For potassium nitrate and potassiuni perchlorate the minima on the derivative curves appear a t the temperatures of fusion and crystalline transition. This is in agreement with the findings of Gordon and Campbell (4). The maximum rates of change in differential temperature (DTA) occur a t the minima shown in the DDTA curves for the dehydration and decomposition of cal-
cium oxalai,e to calcium carbonate and carbon mor oxide (Table I). Combination derivative differential thermal andysis and differential thermal analysis ii rolves using a three-hole nietal block, containing duplicate samples and an inert material, such as aluminum oxide. Thl: sample, whose teniperature is to be rec$xdecl,is slightly closer to the coils of the furnace and contained in the tube with the lower thermocouple well. Differential thermal analysis may be conducted using one of the samples and aluminum oxide, while the derivative curve is ok tained simultaneously by recording the differential temperature bctween the two samples. Two furnaces may also be used, each containing a sample and inert reference material. Differential thermal analysis is carried '>ut in the usual manner and the derivative curve is obtained by recording the differential temperature between samples. This method is potentially more exact. ACKNOWLEDGMENT
The au1,hors thank Saul Gordon for reviewing this manuscript. LITERATURE CITED
(1) Borch:trdt, H. J., Daniels, F., J . A m Chem. SOC. 79, 41 (1957). (2) Fredrihon, A. F., Am. Xineral. 39, 1023 (1954). (3) Freeman, E. S., Carroll, B., J . Phys. Chenz. 62, 394 (1958). (4) Gordon, S., Campbell, C., BN.4L. CHEM.27, 1102 (1955). (5) Gordon, S., Campbell C., Division of Ana1:itical Chemistry, 131st Meeting, SCS, Miami, Fln., April 1057. ( G ) Hogan, V. D., Gordon, S., Campbell, C., Ibid , 29, 306 (1957). (7) Kissipger, H. E., Ibid., 29, 1702 (1957).
Theoretical Curves for Variably loaded Countercurrent Distributions from Tables of the Cumulative Binomial Gordon Alderton, Western Regional Research Laboratory, U. S Department of Agriculture, Albany, Calif.
countercurrent distribution Ithemethod developed by Craig (1) substances being separated and N
THE
tested for homogeneity are distributed by the machine according to the binoniial probability distribution, (p 9)" where p is the fraction in the upper phase, q = 1 - p is tlie fraction in the lower phase, and n is the number of transfers. Three general methods are available for constructing the theoretical (binomial) curves which must be tested for fit with the experimental data when judgments of homogeneity are desired: direct calculation of the individual terms of the binomial distribution (1, S), approximation of the binomial by an exponential espres-
+
sion either by substitution ( 7 ) or by use of table of ordinates of the normal distribution ( B ) , copying the appropriate values from tables of the binomial distribution (6). The first method is exact, but very laborious (1) for more than a few transfers. The second method is approximate and also involves calculation, though much less than the first. The approximation to the binomial is good except a t lorn (0.8) valucs of p or a t low ( < 2 5 ) numbers of transfers. The third method proposed by Tr'ay and Bennet (6),which makes use of a table (4)of the individual terms of the binomial distribution, is esact and convenient for countercurrent dis-
tribution in which a single tube is loaded. This table, however, extends only to 11 = 50. The introduction to these tahles mentions the existence of another ;able of the cumulative terms extendinj; to n = 150 prepared by the Ballistic Research Laboratory, Aberdeen Pro uing Ground, AId. Subsequently in 1955, the Harvard Comput2,tion Laboratory published a table (5"t of the cumulative binomial distribution in which values for n are given up to 1000. The entry intervals for n arc' unity from 1 to 50, two from 50 to 101, ten from 100 to 200, twenty from 200 to 500, and fifty from 500 to 1000. 'I'he 71 grid is 0.01. Stopping the countercurrent distribution instruV O L 31, NO. 4, APRIL 1959
625
