2164
D. M. SPEROS AND R. L. WOODHOUSE
fractions of ai1 Angstrom: benzene, 0.22; hexane, 0.34; cyclohexane, 0.76; carbon tetrachloride, 0.70; chloroform, 0.61; methanol, 0.23; ethanol, 0.40; carbon disulfide, 0.11; nitrogen, 0.18; chlorine, 0.62; bromine, 0.07; argon, 0.24. These are the differences for the lowest T at which a, was calculated. Since a, and up are dependent upon temperature, whereas ag is not, a, or up could be arbitrarily made equal to a, for several liquids simply by choosing the appropriate temperature for the computation of a, or ag. For many liquids (e.g., benzene, Table I), a, is larger than ag at one temperature, but as the temperature is raised the difference vanishes and then up begins to exceed a,. I n a separate report the hypothesis mill be presented (with supporting data and calculations) that the dependence of a,u, and p on T arises from the change in the required diameter of the spherical in the liquid model used for eq. 2 and 3. I n any event, as the agreement in these tables indicates, eq. 5 can be applicable without postulating a hypothesis for the dependence of the parameter a on T . I n Table V, the results for the liquid metals can be seen to be poor relative to the results discussed previously. Inasmuch as the structure of liquid metals would not be expected to correspond well to the molecular (or ionic) structure of fluids implicit in the parameter a of eq. 1-5, it might be anticipated that the liquid metal results 71-ould not be good. Kevertheless, ionic-salt and atomic models for liquid metals were found6 to be consistent with eq. 3 for surface tensions. It might be hypothesized, therefore, that it is eq. 2 in particular which is not follolved well by liquid metals and that compressibility is affected more strongly than surface tension by the electronic aspects of liquid metal structure. The agreement found for water (Table V) becomes good only at the highest temperature examined. The compressibility of mater is considered to be anomalous
Vol. 67
because its temperature dependence exhibits a minimum near 323’K. The improved agreement with eq. 5 above this temperature could possibly be attributed to some continuing change (e,g., depolymerization) rvhich allows the intermolecular interaction potential better to approximate spherical symmetry. The typical molten alkali halides for which results are summarized in Table VI illustrate a further use to which eq. 5 can be put. Experimental techniques used in the measurement of the velocity of sound in molten alkali halides involve a corrosion problemlo which could possibly vitiate the compressibility results. Uncertainty as to the reliability of the results could exist, therefore, particularly in the absence of other compressibility data for molten salts. Hon-ever, the correlation between observed and computed u and fl in Table T’I supports the validity of the sound velocity measurements of Bockris and Richards.lo Similarly, the good correlatioii with eq. 5 found for the inorganic chlorides in the second half of Table I1 supports the validity of the sound velocity techniques used for these inorganic liquids. The computed values of a, and up in Table VI are consistent with the measured11 interatomic distances for these halides in the gas phase. I n a previous papert5it was pointed out that the interatomic distances for the gas molecules were the best data available for the hardcore diameter parameter a. The observed interatomic distances, in units of 10-8 cm., for the gas are: NaC1, 2.36; NaBr, 2.50; NaI, 2.71; KI, 3.05. Examination of Table VI shows that a temperature could be selected, for each of the salts, a t which a, or up in the liquid would equal the observed interatomic distances in the gas. (10) J. O’M. Bockris a n d N. E. Richards, Proc. Rou SOC.(London). A241 44 (1957). (11) A. Honig, M. Mandel, M. L. Stitch, and C. H. Townes, Phw. Rev., 96, 629 (1954).
REALIZATION OF QUANTITATIVE DIFFERENTIAL THERMAL ANALYSIS. I HEATS AND RATES OF SOLID-LIQUID TRANSITIONS BY D. R4. SPEROS AND R. L. WOODHOUSE Lanap Research Laboratory, General Electric Company, Nela Park, Cleveland, Ohio Received March 8, 1963 A direct method for accurate quantitative differential thermal analysis has been developed. The method involves addition or withdrawal of electrical energy from a sample structure in such a manner as to maintain a constant differential temperature between sample and reference thermocouples. The theoretical conditions under which this electrical energy is equal t o the true energy of the process are determined. The results of the theoretical analysis a r e applied to the development of a working model. The instrument is used to redetermine the heats of fusion of Sn, iYaiTO,, Pb, Al, and Ag. The application of this procedure to the study of process kinetics is demonstrated. The rates of fusion of the above substances are examined.
I. Introduction Ever since its inception by LeChatelier in 1887,’ the inethod of differential thermal analysis (d.t.a.)Z has been used extensively for the qualitative study of processes involving changes ( A H ) in heat content. The method is simple in principle: a sample substance and a thermally inert reference substance are heated (1) H. LeChatelier, B~LZZ.SOC. franc. mineral., 10, 204 (1887). ( 2 ) W. J. Smothers a n d Y . Chiang, “Differential Thermal Analysis,” Chemical Publ. Co., New York, N. Y . , 1968.
a t a given rate by an external furnace. The junctions of a differential therniocouple are embedded in or near the sample and inert substances. The electromotive forces developed by these junctions are amplified and continuously compared by displaying on a recorder. Exothermic or endothermic changes that occur in the sample as the furnace temperature is raised are therefore recorded as deviations from a base line and appear as peaks on one side of it or the other. So by means of this method the following information is obtained:
Oct., 1963
QUANTITATIVE DIFPEREXTIAL THERMAL ANALYSIS
(1) the temperature range in which a thermal change takes place; ( 2 ) the net exothermic or endothermic nature of the process; ( 3 ) a very approximate magnitude of AH of the process. The reduction of c1.t.a. to a firm quantitative basis whereby AH could be determined with an accuracy in the vicinity of 1% has appeared of such potential usefulness that numerous attempts have been made to bring it alboutS2-* These attempts fall into two categories. 1. Computation Methods.-These methods attempt to measure A H by means of the characteristics (height, area, etc.) of the d.t.a. peaks. Although a number of them are highly d e ~ e l o p e d they , ~ ~ attempt in essence to accomplish what in classical calorimetry would be akin to computing the energy equivalent of a calorimeter. 2. Calibration Methods.-These have met with very limited success either because they involve calibrations peculiar to each substance and set of conditions as in the work of Wittigls or because, as in the case of Eyraud13the principles of thermal balance described below are not taken into account. In this work a solution to this problem has been found. This paper examines its applicability to fusion processes. Subsequent reports mill examine its applicability to other processes. This solution is based on the following two concepts: (a) the concept of temperature balance between the sample and reference structures and (b) the concept of thermal balance within the sample structure. (a) Temperature I3alance.--Bn auxiliary heater is inserted in the sample structure of an otherwise normal d.t.a. apparatus. As long as the sample undergoes n o change due to an endothermic or exothermic process, the heater power input is kept a t a given value. When such a change begins taking place, the input to the heater is increased or decreased. This is done in such a manner that the temperature difference between the sample and reference thermocouples is maintained a t the same value as before the change began. I n order to accomplish this temperature balance between sample and reference thermocouples, an amount of electrical energy AQ must be expended. It follows that AQ is related to A H . However, AQ mill equal AH only fortuitously unless the rate of heat gain or loss from or t o the surroundings is made the same during the process as it was beflore the process began. (b) Thermal Balance.-In order to obtain a general idea of what the condition AQ = AH entails, imagine the arrangement of Fig. 1 in which dimensionless heat source H, heal sink F1 (such as a melting sample), and thermocouple Tc are embedded in a solid which is immersed in a gas. The rate of heat loss from the heater will be given by d& = k(Th - T,) -k dL dt dt where T , = melting point, Ti, = heater temperature, and dL/dt is the rate of heat loss to the gaseous atmosphere. C . E y r a u d , Cornpt. rend., 238, 1611 (1954). H. J. Borchardt and F. Daniels, J . Am. Chem. SOC.,79, 41 (1957). E. Sturrn, J . P h y s . Cham., 65, 1935 (1961). (6) E . Deeg, Ber. deut. keram. Ges., 33, 321 (1956). (7) M. J. \'old, Anal. Chem,., 2 1 , 683 (1949). (8) F. E. Wittig, J . Elekhorhem., 54, 288 (1950).
Fig. 1.-Thermal
2165
DISTANCE. balance within imaginary sample structure.
The rate of heat absorption by the sample will be given by
where dG/dt is the rate of heat gain from the atmosphere. For thermal balance, namely for equality between (1) and ( 2 ) , the following condition must be satisfied dG - ---dL (3) dt dt The appropriate relationships of heat transfer areg
dG _ --- T- - T , - T,dt R rd dL _ - T,' - T dt R'
Th
Tm
- T,' r'd'
where T is the gas temperature, Tw and T,' are the wall temperatures a t the sample and heater sides of the solid, respectively, d and d' are distances of sample and heater, respectively, from the adjacent walls of the solid, and R, r, R', and r' are heat resistance constai~ts.~ The preceding relations yield dG dt
-
T - T, R rd
+
dL Th - T - -dt R' r'd'
+
Therefore for the condition expressed by eq. 3 to exist, we must have ~T - T m - R+rd = A (3) Th - T R' r'd'
+
We now consider the condition of compatibility between eq. 4 and the temperature balance condition, namely that the thermocouple temperature T , = TI the gas temperature. The thermal currents to and from the thermocouple a t steady state must be equal; i.e.
(9) W. H. AIcAdams, "Heat Transmission," MoGrawHill Book Co., Ino., New York, N. Y., 1942, p. 22.
D. -1.1.SPEROS AND R.L. WOOIEIOCSE
2 166
Tol. 67
tive of operating with a ki1.sw.n (dG/dt) - ( d L / d t ) # 0 found by calibrat’ioii is, as will be seen, normally unnecessary.
11. Experimental Design and Procedure *4pract,ical arrangement t,ha.t satisfies or approximates the
Fig. Z.-Vertical cross section of sample structure: 1, sample container (material compatible with sample); 2, heater core ( S i ) ; 3, ceramic-coated heater wire (TT7); 4, heater shield (Xi); 5, sample; 6, insulator (SiOz); i, thermocouple (Pt-Pt :13% R h ) ; 8, ceramic conduits and supports (porcelain); 9, current and potential leads (Pt).
where D and D‘ are distances from sample to thermocouple and heater to thermocouple, respectively, and To is the temperature of the thermocouple. Since, for temperature balance, T , = T , the above expression may be initten
T
T, rD - A Th - T T’D’ -
where A has already been defined by (-1). In the case of a homogeneous solid, r = I”,and
a = -D
D’
T’C’hen an exothermic process is anticipated, a coiistant power input is applied to H. This results in a constant difference between T , and 1’. By appropriately reducing this input to the heater during the exothermic process, this temperature difference is maiiitained constant. As a result, the rate of heat loss to the surroundings from the entire structure before, during, and after the process is held constant and the above concept still applies. Equations 4 and 5 indicate that, even for the idealized iiiodel of Fig. 1, the requiremeiits for equality betx-een electrical aiid process energies are strict. If, for example, rD/r’D’ # A, then AQ # AH even if the temperature balance is maintained. Howeyer, these equations indicate that, in principle, it is possible to adjust or balance the sample structure until dGldt = d L / d t by varying the parameters d , d’, D , and D‘ and, if necessary, R, R‘, r, and r‘. The apparent alterna-
general ideas conveyed by eq. 4 and 5 may possibly take any one of a number of forms. The approach preferred here was to fold the armngement of Fig. 1 upon itself in the sense of surrounding the sample by the heater as shown in Fig. 2. For an endothermic process, the temperature of the heater is kept at T as determined by comparison with the temperature of the reference thermocouple (not shown in Fig. 2) or made to oscillate equally above and below T , so that for the entire structure f dG E f dL 0. For an exothermic process, the surface temperature is maint,ained a t a temperature T‘ higher than T , or made to oscillate equally above and below T’, so t.hat for the entire structure dL/dt is constant before, during, and after the process. The above temperature balance is maintained by one of two methods. The first is manual switching of the heater on and off a t constant, power, the duration of each pulse being such (usually more than 1 sec.) that TCoscillites equally above and below T (or T’). The second is continuous adjust.ment of the power eit>her nianua.lly or automatically by use of an electronic control circuit. In the first method the oscillation amplitude is normally held below 0.03”, which is equivalent to a recorder pen travel of ’/I6 in. The total “power on” time is found by means of a stop watch or another recorder. The “distance” between the thermocouple and the sample cont,ainer (the concept conveyed by eq. 5) is adjusted until the correct heat of fusion of a substnnce such as Pb is obtained. In order to have a compact design, this distance is increased, not, by placing the sample f w from the thermocouple, but by inserting a Tafer of quartz 1.5 mm. in thickness between the two. The bearing of eq. 5 is illustrated by the fact t,h,rt if no insulator is used a t all, the error becomes +2%; if the thermocouple is in direct contact with the sample container, but not with the heater casing, the error amounts to + l o % or more depending on the intimacy of the contact; if the thickness of the quartz is increased, the error becomes negative, amount,ing to -8% for a quartz thickness of 10 mm. Complete lack of electrical interference between the thermocouple (microvolt) circuit, and the heater circuit, is, of course, necessary. Conductance through the heater insulation, for example, can cause effects varying from intermittent noise to results faulty by several per cent.
111. Experimental Results 1. Heats of Fusion.-Once the sample struct’urewas balanced using Pb, the correct heats of fusion were obtained without further adjustm,en,t of the sample structure for Sn, NaKOa,AI, and Ag, t’liuscovering a useful range of temperature (Table I). Origin of Samples.-The tin, lead, and silver samples were obtained from The Consolidated Mining aiid Smelting Co. of Canada, Ltd. Their purit,y was statsed by the manufacturer t o exceed 99.999 mole %. The aluminum sample was obtained from United Mineral and Chemical Corp. and likewise was st,ated t o exceed 99.999 mole % in purity. The sodium nitrate sample was of cert,ified reagent grade obtained from Fisher Scientific Co. The certificat’e of analysis for this material indicated a purit’yof 99.98 mole %. Materials for Sample Containers.-The choice of container materials was such as to eliminate iiit’eractmionbetween the molt’en samples aiid cont’ainers. This choice was dictated by available kno-vvledgclO,ll and tdheresults of this investigat,ion. Pb, Sli, AI, and Ag were melted in containers made of “spect’roscopic (10) L. R. Kelman, W,D. JVilkinson, a n d F. L. Yagee, “Resistance ot Materials t o Attack b y Liquid Metals,” U. S . Atomic Energy Comn~ission $NL-4417, U. S.Government Printing Office, V a s l ~ i n g t o n D. , C., 1950. (11) lU. Hansen and K. Anderko, ”Constitution of I3inary Alloys,” >IcGraw-Hill Rook Co., Inc., New York. K.Y., 1958.
Oct., 1963
&ITANTITATIVE
L)ETERMINATIOS O F
DIFFEREKTIAL THERMAL ANALYSIS
2167
TABLE I HEATSO F FCSION AXD COIY~PARISON WITH LITERATURE VALUES
Substance and sample container(8)"
Fusion temp., O C .
g.
Sn-(a)
231.8
0.3411-0.7671
Range of sample mass,
----Range Vatta
of heater input*--Seconds
Mean AHfusion a n d range. cal./mole
-"Best" critical valuesKubasohewski and Evans'* Kelley'a
1690 i 30 1720 1693 (1686-1702) SaKO8--( e ) 306.0 ,1861- ,2105 ,1196- .1879 193.5-270.4 3549 3490 (3492-3575) 1150 It 30 1140 127.7-293.5 1135 Pb-(a)( In) 327.4 ,9673-1.2783 ,0759- ,2303 (1129-1140) 2570 2566 2500 2~ 30 AI-(a) 182.2-637.0 ,0964-0.2165 ,1268- ,2102 658.8 (2652-2582) 2690 3~ 100 2856 145.1-268.0 2728 ,2454- .5229 Ag-( a ) 960.8 .4414- ,7535 (2595-2867) Range of heating rates provided by external a (a) Spectroscopically pure graphite; ( b ) S o . 446 stainless steel; (c) 99.999% pure Ag. furnace: 0.01-0.5O0/min.; range of pressures of argon gas: 378-1010 mm.; range of areas under normal (uncompensated) d.t.a. peaks: 0.8-1.5 cni.2/joule. 0.0563-0.1964
234.5-628.8
I
graphite" obtained from the National Carbon Co. In a 1iumb.r of experiments P b was also melted in containers made of KO. 446 stainless steel in order to compare directly the heat of fusion value for P b obtained in this work with that found by Douglas and Dever.l* The fusions of S a x 0 3 were performed in containers made of Ag of the same purity as that of the Ag samples. Geometry of Sample Containers.-In many instances the sample container cionsisted simply of a hollow cyliuder with a lid. Therefore, the shape of the sample in them was thLat of a cylinder the height of which depended on the amount of sample used. I n some instances, as for example during studies of the melting behavior of SaXOs, several silver spheres were inserted in the container together with the sample in order to investigate the effect of changes of over-all thermal conductivity 0111 the results. For most experiments the container was formed with a cylindrical center post (see Fig. 2 ) so as to obtain an annular or tubular sample configuration. This was found especially useful whenever increased resolution was desirable. Initial State of Samples.-Samples were handled, when necessary, in a nitrogen drybox using phosphorus pentoxide as the final desiccant. Samples and coiitainers were weighed on a semimicro analytical balance in the drybox. Pb and Sii were melted in the drybox and cast in the containers. I n other experiments these materials were introduced into the containers as small shot or scrapings. The A1 samples consisted of 1-mm. thick sheets bent into tubular forms fitting around the posts of containers such as shown in Fig. 2. The silver samples were introduced into the graphite containers as small spherical shot, although iii a number of experiments scrapings were added. The sodium nitrate sample was introduced into the silver container as a coarse Linground powder and pressed into it with a high-carbon steel plunger. A411the fusion exprriments were performed in an atmosphere of purified argon. Table I summarizes the results and compares them with critical literature values.13l 4 It shows that the agreement between the results of this work and the (12) T I3 Douglan and J I. D e i e r J Am Chem. Soe., 76, 4824 (1954) (13) 0 Kubaschexski a n d E. L E\ans, "Metallurgical Thermochemistrg," Pergamon Press, New York, K 1-, 1958 (14) K K Kellex, U Bureau of Mines Bulletin 584, U 9 Government P r i n t i n e Office, 1% ashiriptun E C , 1960
.
.
1
I 5
TIME IN MINUTES.
6
7 1 10
Fig. 3.-Fusion of tin: the rate of electrical energy input to the sample heater as a function of time. Points: experimental; full line: theoretical with slope equal to 2AH/to2.
TABLE I1 DETERMIXATIOS OF THE HEATOF FUSION OF ALUXINCM Mass, g.
0.0964 ,0964 .0964 ,0991 ,0991 ,2165
,----Heater Watts
0.1270 ,2102 ,2102 ,1268 ,1268 ,1354
input---Seconds
304.1 183.4 182.2 309.6 309. i
637.0
-\Hf"3iC'Il,
cal.imole
2582 2678 2561 2552 2553 2568 Average : 2566
critical values is as good as the agreement between the critical values. It is to be noted that the results were found to be independent of the following variables, a t least within the limits shown in Table I or indicated in the text: (a)
D. R f . SPEROS AND R. L. WOODHOUSE
2168 8
Table I1 gives an example of results of individual experiments. Due to supercooling all substances in Table I froze a t rates too high for manual compensation. This tendency was arrested in the case of Pb by using a container of pure S i . The heat of crystallization found for this sample was within 1% of its heat of fusion. For these exothermic experimeiits the initially constant input to the sample heater (see sections I and 11) was 0.1174 watt. The e.m.f. of the reference thermocouple was proportionately increased either by supplying an external constant e.m.f. or by utilizing a reference heater which is normally present since both sample and reference structures are used for measurements indiscriminately. 2. Rates of Fusion.-Kinetically, a fusion is a zeroorder process described by
7
6
-I . z
5
m W J
4
Vol. 67
4
..5
v
0 0 3
2
dn - 1 dH AH dt dt
I
where n = moles of molten substance and dH/dt is the rate a t which heat energy is actually absorbed by the substance. For a balanced sample structure dH/dt = k(Th - Tm). If a linear heating rate A is provided by the external furnace, T1, - TI, = At. Combining the last three equations results in dnldt = (kA/AH)t and dHldt = kAt. The last equation upon integration yields k = 2AH/Ato2or
0 TIME I N MINUTES.
Fig. 4.-Fusion of sodium nitrate: the rate of electrical energy input to the sample heater as a function of time. Points: experimental; dashed line: theoretical with slope equal to 2AH/ toa for zero-order kinetic process. 8
dH 2AH Xt dt to2
I ~
7
I 6
-
I
I 5 -
v)
w
-I 3
I I
TIME I N MINUTES.
Fig. 5.-Fusion of aluminum: the rate of electrical energy input to the sample heater as a function of time. Points: experimental; dashed line: theoretical with slope equal to 2AH/to2 for zero-order kinetic process.
amount of sample; (b) sample preparation and initial state; (e) material and internal geometry of container; (d) level of power input to the auxiliary heater; (e) pressure of the argon atmosphere; (f) heating rate provided by the external furnace.
where to is the total time required for the fusion. Accordingly, if dH/dt = dQ/dt, that is if the kinetics of the process can be determined by the quantitative d.t.a. procedure described here, a plot of the experimental dQ/dt as a function of time should prove to be a straight line with origin a t zero and slope equal to 2AH/to2. Figure 3 for Sa shows that this is so. The same result is obtained also with P b and Ag. The experimental rates of heat input dQ/dt into the auxiliary heater were determined by displaying, on a high-speed recorder, the on-off periods during manual switching. The energy Q for each 0.5-min. interval of the fusion period then was plotted appropriately as dQ/dt us. t to give the experimental points on the plots. The melting behavior of sodium nitrate and, surprisingly, aluminum (Fig. 4 and 5. respectively) is complex. In both instances the linear behavior is preceded by ai1 exponential dependence of dQ/dt on 2. I n past work16 this A-type melting behavior of XaSO, has been indicated and this work confirms those results. The energy involved in the A-type premelting phenomenon of aluminum amounts to 3.5% of the total. More work, however, will be necessary t o confirm the cause of this behavior. Acknowledgments,-The authors are indebted to Drs. J. 5. Saby, W. G. Segelken, and 31. A. Weinstein for continued interest and very helpful discussions. The work and suggestions of G. TT. hrmstrong on the construction and maintenance of this apparatus are gratefully aclinomledged. 115) A. XIustajoki, Ann. h a d . Sct Feiinlcae, Ser A , TI 5 (1957).
