J. SCHUBERT, A. LINDENBAUM AND W. WESTFALL
390
calculated from zero and first-order rate laws are virmole fraction hr.-l tually identical, 1.67 X and 1.70 X lo6 hr.-I, respectively. The break toward a lower rate in the 890" curve may represent an approach to equilibrium, Le., the effect of the reverse reaction Na AgCl+ NaCl Ag. Since both zero- and first-order treatment yield nearly the same rate constants the same activational energies will apply in either case. The 890 and 970" data do not give very straight lines. However, from the straight-line portion of the 1080" and the 200-300 hour section of the 860" run (lc = 8.9 X hr.-I) we obtain as an estimate for the activational enthalpy AH* = 9.7 kcal. The entropy of activation, AS*, can be calculated from the theory of absolute reaction rates
+
+
Vol. 62
larly exchange reactions in which the chemical species are indistinguishable, Le., AFO = 0. Nevertheless, there is a certain correspondence between the reactions treated by Marcus, Zwolinski and Eyring and the present one. If it is assumed that AS* = R In ke and that the reaction is homogeneous (the initial step being the solution of silver in NaCl as atoms) then the tunnel distance Tab at which electron-exchange occurs can be calculated from the expression c
where m = mass of the electron, Z* = the positive charge = 1, ro = nx2u0,n* = effective quantum number for Na = 1.63, uo = 0.528 X lo-* cm., f ( n ) is a function of charge = 1 for the charge transfer between an atom and a univalent ion, D = dik~ = 'kT iE- eAS*/Re- AH*/RT (1) electric constant. If y e let D = 3, customary for Since k~ is in hr.-l, k T / h is in molecules-l hr.-1 = fused salts, Tab = 35 A.,which seems unreasonably 8.35 X Solving the equation we get AS* = large since it is difficult to visualize a mechanism by -99.0 e.u. The free energy of activation is which an electron can be transferred over such a large distance. However, electron transfer during AF* = AH* - TAS* (2) atom-ion collision in the melt cannot be ruled out. At 861" we get AI"* = 109 kcal. The present data insufficient evidence is available are not sufficient to establish a mechanism for the to Alternatively, establish a heterogeneous mechanism which reaction. Broadly it may be described as an elec- would have as its initial step the adsorption of sotron exchange reaction between atoms and ions dium.ions on the metallic silver surface. characterized by a large negative entropy of activaCertainly more work on this reaction, including tion; this appears to be primarily responsible for the effect of surface area and stirring speed on rate, the slow rate of reaction, since AH* is relatively as well as studies on other reactions of this type, is small. For reactions of this type in aqueous solu- needed. tion these characteristics have been as~ociatedl~,~3 Acknowledgment.-I wish to thank Mr. Thomas with a very low electronic transmission coefficient Ice = eAS*IR. Unfortunately, the equations have Wilson, a summer research assistant from El been derived only for aqueous solutions, particu- Dorado High School, for doing the "closed system" kinetics; Dr. R. W. Fink for his help and advice (12) R. J. Marcus, B. J. Zwolinski and H. Eyring, THISJOURNAL, with the radiochemical experiment, and for the use 68, 432 (1954). of his counting equipment; and Mr. K. Okada for (13) B. J. Zwolinski, R. J. Marcus and H. Eyring, Chem. Reus., 66, the actual counting. 157 (1955).
ION-EXCHANGE SEPARATION OF BERYLLTUM BY ELUTION WITH SALICYLATE ANALOGS BYJACK SCHUBERT, ARTHURLINDENBAUM AND WILLIAMWESTFALL Division of Biological and Medical Research, Argonne National Laboratory, Lemont, Illinois Received June 6, 1067
Rapid quantitative separations of trace uantities of Be from macro-concentrations of salts including those of Cu, U 'and Ca were devised. The method makes use the fact that Be forms a relatively strong complex with salicylate analogs in the region p H 3-4.5 whereas most cations of the alkaline earths do not react significantly with salicylates, and cations such as Cu++ and UO?++react weakly or not at all in the same p H region. The separation of Be from foreign cations is made by means of a cation-exchange resin. Be is eluted selectively with 0.02-0.10 M sulfosalic lie acid (SSA) a t p H 3.5-4.5. Neither CU++,UOz+'+nor Ca++are removed under these conditions. Uranyl ion is eluted by i S A at p H 4.5-4.7. The foreign cations can be removed subsequently by a variety of eluting agents such as HCl and HzS04. At p H > 6 and in the presence of SSA Be is strongly absorbed by an anion-exchange resin. Evidence is given to show that in the acid regions Be forms an uncharged complex with SSA and a negatively charged complex in neutral and oZkaline regions.
3
,
The sensitivity of analytical procedures for beryllium is severely limited by the presence of interfering substances. 2a-c This problem becomes (1) Work performed under the auspices of the U. S. Atomic Energy Commission. (2) (a) T. Y. Toribara and R. E. Sherman, Anal. Chem., 26, 1594 (1953): (b) J. R. Arnold and H. A. Al-Salih, Science, 121, 451 (1955); (0) A. J. Cruikshank, G. Cowper and W. E. Grummitt, Can. J . Chem., 34, 214 (1956).
especially acute when it is necessary to isolate micro-quantities of Be from biological and geological materials. Usual methods of extraction and precipitation in such cases are not only laborious and time-consuming but often give low or erratic results. Since the discovery that the radioactive nuclides, Be7 and Belo, are produced in the atmosphere by cosmic rays,2bit is especially worth-
a
April, 1958
ION-EXCHANGE SEPARATION OF BERYLLIUM
while t o be able to isolate quantitatively trace amounts of Be. I n the separation of a micro-component from large amounts of impurities it is usually desirable to remove the micro-component first. It was decided to utilize the selective metal complexing properties of salicylic acid derivatives in combination with ion exchange. Salicylic acid and derivatives, such as sulfosalicylic and gentisic acids, react negligibly with most alkaline earth cations, e.g., Ca, yielding formation constants with values of log K1 of 0.14-0.6 for ionic strengths in the region of 0.14.16.3-6 On the other hand, the formation constants for complexes of Be++ and UOz++ and for transitiorl elements such as Cu++, Ni++ and Fe+++ fall in the range log KI = 2-5.Va-f Since the ratio of the formation constants of Be to that of the other alkaline earths is approximately lo6, it is possible to devise rapid and efficient schemes for separation of Be from large amounts of mineral substances such as rock and bone without the necessity of resorting to relatively slow chromatographic techniques. Experimental
391
Results
1. Batch Tests. (a) Effects of pH and Sulfo-
salicylic Acid Concentrations.-The system consisted of 100 mg. of air-dried “Dowex 50” in the Hform containing absorbed Be7, 100 ml. of solution 0.16 N in NaC1, pH 7, and containing concentrations of sulfosalicylic acid (SSA) varying from 0.01 to 0.08 M . With 0.01 M SSA only 61% of Be7was eluted, but a t a concentration of SSA of 0.02 M or greater, elution was complete. Next the concentration of SSA was kept constant a t 0.02 M while the pH was varied. It was found that a t pH 2.5 and below absorption of Be7 by Dowex 50 was nearly complete while at pH 3.2 and greater the absorption dropped sharply to zero, Le., complete elution. The above results showed that Be formed a sufficiently strong complex ion with SSA at concentrations of a t least 0.02 M SSA a t pH > 2.5, so that rapid elution from a cation-exchange resin could be expected. The observed pH of 3.2 in 0.02 M SSA a t which complex formation became effective was in agreement with the observations of Meek and Banks* regarding the stability of the Be-SSA comThe resin used was 8% cross-linked, 60-100 mesh “Dowex plex as a function of pH. Further information on the nature of the com50,” a sulfonated polystyrene cation exchanger (Dow Chemical Company, Midland, Michigan). A 5 N HC1 plex ion formed between Be and SSA was obtained solution was passed through a column of the resin until all from determination of the uptake of Be from SSA traces of iron were removed. The resin was then alternately solutions by a strongly basic anion resin, “Amberconverted several times to the sodium and hydrogen forms by use of excess 2N NaOH and 2 N HC1. The H-form of lite IRA-410,” a quaternary amine exchanger the resin subsequently was washed with ethanol to remove (Rohm and Haas Company, Philadelphia, Pennsylsoluble organic material. Finally, the “fines” were re- vania). Up to a pH of 4.5 the absorption of Be moved by decantation and the resin was air-dried and by this resin was only a few per cent. In terms of stored in an air-tight bottle. In general, a small volume of a Be-containing solution the distribution coefficient, Kd, where
was absorbed a t the top of the resin column. Subsequently the Be was eluted with either sulfosalicylic acid or gentisic acid at a given pH. When all the Be had been eluted, the remaining bulk cations were usually eluted with either HCl or HzSOc. I n most instances the effluent was collected by means of an automatic fraction collector. Radiochemical analyses for Be, Ca and U were made, when specified, by use of the radioactive isotopes Be’, Ca46 and U233. The Cu concentration was determined spectrophotometrically by measurement of the optical density of 3 M HzSOlsolutions a t 455 mp. Calcium assays were made by precipitation of calcium oxalate from an effluent with 0.01 N sodium oxalate and subsequent titration with 0.01 N KMnO4. Semi-quantitative t,ests for Ca were made by adding brom thymol blue to 5-ml. volumes of effluent and then adding NH4OH dropwise until a blue color persisted in the solution. Upon the addition of 1 ml. of saturated ammonium oxalate a cloudiness proportional to the concentration of precipitated calcium oxalate was observed. The method was sensitive t o 0.1 mg. of Caper 5 ml. In some experiments a synthetic bone ash solution “ s n i k d ” with Be7 was used. A 20.00-ml. solution consisted of 0.0233 g. of Mg3(P0&4HIO, 0.1000 g. of CaCOi, 0.8500 g. of Cas(PO&, 0.0106 g. of NaCl and 0.0025 g. of KC1, all dissolved in 1.O N HCl. Several batch experiments were carried out. I n general, a weighed amount of air-dried resin was mixed with a given volume of solution and the mixture shaken mechanically for 2-3 hours. The supernatant solution was assayed for the element, usually Be7, and the fraction absorbed by the resin was calculated. (3) C. R. Davies, J . Chsm. Soc., 277 (1938). (4) N. R. Joseph, J . Eiol. Chem., 164,529 (1946). (5) R. P. Bell and C. M. Waind, J . Chem. Soc., 2357 (1951). (6) J. Schubert, J . A m . Chem. Sac., 7 6 , 3442 (1954). (7) (a) H.V. Meek and C. V. Banks, ibid., 73,4108(1951); (b) R. T. Foley and R. C. Anderson, ibid., 70, 1195 (1948); (c) 71, 909 (1949); (d) 8. E. Turner and R. C. Anderson, {bid.,71, 912 (1949); (e) R. T. Foley and R. C. Anderson, ibid., 72, 5609 (1950); (f) M. B. Lasater and R. C. Anderson, ibid., 1 4 , 1429 (1952).
Kd
=
70
cation in resin
vol. of soln. (v)
2,
% cation in soln. mass of resin (m) G the values of K d (when v was expressed in ml., m in grams) varied from 1 to 4. However, a t pH 6 and above the uptake rapidly increased to values of &I several hundred-fold higher. In the acid regions SSA forms a 1 : l complex with Ni++, Cu++, Fe++, UOz++ and A l + + + . y b - f The proton binding constant for the carboxyl group in SSA has a pK of about 2.86,7cwhile that of the phenolic-OH group is about 11.7.9 The phenolic hydroxide group participates in the complex formation even in the p H region 3-5 but without the loss of a proton. The -SO3 group gives only a minimum contribution. It is also known that Be forms at best only very weak complexes with aliphatic -OH groups and aliphatic carboxyl-hydroxyl compounds.l0 From such considerations and from the above ion-exchange data we can postulate that in the acid regions (pH 3.6) the uncharged complex
r
coo, \Be
+
OH’
is the predominant species. (b) Separation in Absence of Salicylates.-Be(8) H. V. Meek and C. V. Banks, Anal. Chsm.,22, 1512 (1950). (9) A. Agren, Acta Chsm. Scand., 9, 49 (1954). (10) (a) J. Schubert and A. Lindenbaum, J . Biol. Chem., 208, 359 (1954); (b) A. Lindenbaum, M. R. White and J. Schubert, Arch. Biochsm. Biophys., 62, 110 (1954).
392
J. SCHUBERT, A. LINDENBAUM AND W. WESTFALL
Vol. 62
and Ca-Containing Solutions.-Two
ml. of a solution consisting of carrier-free Be7 and 20 mg. of Ca3(P0& labeled with Ca45,and 0.6 M in HC1, was added to the top of a Dowex 50 column in the Hform. Sulfosalicylic acid, adjusted to p H 4.5 with i 100NH40H,14 was passed through the column. Additional experimental data and results are shown in 3 5 75 Fig. 1. No Be was eluted until the p H of the ef\ fluent rose to approximately 3. At that point the 50tBe was eluted from the column in a sharp band z 3 with no detectable Ca4Scontaminant. When all 25the Be had been eluted the Ca was eluted readily O L with 6 M HC1. Within experimental error all the 275 0 25 50 Be and Ca were recovered: 1922 counts/min. of VOLUME O F ELUANT [ m l . l . Fig. 1.-Ion-exchange separation of 20 mg. of calcium Be7 added, recovered 1937; 3298 counts/min. of phosphate from tracer quantities of Be7 (200-300 mesh Ca45added, recovered 3274. “Dowex 50,” 1.18 cm.l by 14 cm. deep, hydrogen form; The effectiveness of salicylate analogs in the elutriant, as shown, passed at 1 ml./min.). Be7 did not separation of trace amounts of Be from synthetic appear in effluent until pH 3-3.5 was reached. bone ash (see Experimental section) was demonstrated by use of 0.1 M gentisic acid (GA) adjusted to a p H of 6.0 with NaOH. Pertinent column data -0.02 M S U L F O S A L I C Y L I C ACID 3 M ties04 include: 15 g. of air-dried “Dowex 50,” H-form, p H * 3.5 I-‘ 2 60-100 mesh; column dimensions 12.5 cm. high X = l 3 1.0 cm. diameter. Thirty and a half ml. of the 3 w J 25 synthetic bone ash solution in 0.7 N HC1 was -E passed through the column at a flow rate of 1 ml./ 400 min. followed by a wash solution of 0.01 N HCl to c W 3 remove residual interstitial ions. Elution with f 300 gentisic acid was begun at a Bow rate of 1 ml./min. \ Effluent samples of 5-ml. volumes were taken and 5 200 analyzed for Be7 and Ca. When the pH of the 3 effluent reached 1.90 (at this point the total ef0 0 fluent volume including holdup was 190 ml.) the IO0 Be came off the column in a very sharp band which I I I 1 reached its peak a t p H 2.74. Within a total volume c 0 10 20 30 40 50 60 70 80 90 of 110 ml., or about 10 column volumes, the elution V O L U M E OF E L U A N T (mi.). of Be7 was quantitatively complete while Ca was Fig. 2.-Ion-exchange separation of 155 micrograms Cu quantitatively retained on the column. At the from tracer quantities of Be7 (same resin bed and flow rftte end of the elution the p H of the efluent was 5.60. as in Fig. 1). Another approach has the advantage that larger cause Be has an extremely small ionic radius and volumes of solution containing Be and foreign ions hence a large hydrated radius, its affinity for a can be handled. In this procedure, after the cation exchanger in acid media is considerably less Be and foreign salts were dissolved, SSA was added than for the other alkaline earth cations.ll Thus to the solution, the p H was adjusted, and the entire an additional means of separation is possible. I n solution passed through a resin column previously the range of 0.1 to 2 M HC1 it is possible to sepa- conditioned with the identical concentration and rate Be from the other alkaline earths by elution pH of NH4SSA. With this pretreatment Be broke with HC1 or simple salts. However, the separation through immediately and passed through quanfactors are still several orders of magnitude below titatively without the necessity for additional SSA the values obtained by the use of salicylates. elutriant. The results of one such separation are The uptake of Be++ by “Dowex 50” compared shown in Table I, where 150 ml. of a solution conto that of Srff ‘was measured in HC1 solutions. taining a mixture of 1.101 g. of CaC12and 5 mg. of In 0.1 M HC1 the Kd for Sr++ (where m, the mass of Be as BeSOc was completely resolved. resin, is given in milligrams) was 41 while that for Separation of Be and Cu.-At all pH regions Be++ was 2.1-a ratio of Ka(Sr)/Kd(Be) = 20. Cu(b) forms a complex with SSA that is only oneWith increasing HC1 concentration the ratio dimin- tenth as strong as the corresponding Be complexes ished to only G a t 2 M HCI. At concentration of with SSA7a.d; consequently it is possible to effect HC13 to 8 M the ratio of uptake of Sr++and Re++, rapid quantitative separations of trace amounts of rather than decreasing, remained roughly constant Be from copper salts. In Fig. 2 is shown a typical a t -0.25, a reversal of affinities that is in general separation accomplished by the use of 0.02 M SSA agreement with previous observations. w13 a t a pK of 3.5. The position of the Cu band was 2. Column Separations. (a) Separation of Be- unaffected by the SSA; Cu was removed later with (11) T. R. E. Kressman and J. A. Kitchener, J. Chsm. SOC.(Lon3 M H2SO4.
1
W
‘ v)
don), 1201 (1949). (12) K. A. Kraus, F. Nelson and G. W. Smith, THISJOURNAL, 58, 11 (1954).
(13) R. M. Diamond, J . Am. Chem. SOC.,77, 2978 (1985).
__
(14) The ammonium salts generally are preferable, as they allow a reduction in the bulk of ash to be used for subsequent radiochemical or spectrographic assay of Be.
c
ION-EXCHANGE SE~PARATION OF BERYLLIUM
April, 1958
(c) Separation of Be and U(VI).-Uranyl ion, TJ02++, forms a 1:l-complex with SSA a t pH 4.54.7.7c Above and below this pH-region the stability of the complex decreases markedly. Inasmuch as Be is eluted readily with SSA or GA in the general pH region 3 to 4, rapid and complete separations of Be and U(V1) are effected easily. In one run a column of “Dowex 50,” 12.5 cm. high X 1.0 em. in diameter, 60 to 100 mesh, was used. The elutriant was 0.02 M SSA a t a pH of 4.2. Two ml. of a solution consisting of 100 mg. of uranyl nitrate and 5 pg. (labeled with 42,000 counts/min. of P3) of Be (labeled with 6,990 counts/min. of Be7) as BeSO., were placed on the top of the column. Be began t o break through a t pH 1.8 and reached a peak at p H 3.68. At the point where all the Be had been eluted, pH 4.3, no U could be detected. When the concentration of SSA was increased t o 0.1 M and the p H to 6, the U was rapidly and quantitatively removed. The peak of the elution curve came in the p H region 4.54.7, in agreement with the pH region of maximum stability of the complex. When the pH of the elutriant varied in either direction from the optimum by more than about 0.5 p H unit, considerable tailing of the U elution curve was observed. For optimum Be-U separation it is suggested that 0.1 M SSA be used a t pH 3.5 to 3.8. When all the Be has been eluted, the U(V1) can be removed with 0.1 M SSA at pH 4.6-4.7. TABLE I BE FROM C A Solution: 150 ml. containing 1.101 g. of CaC12,5 micrograms Be as BeSO4, 156,300 counts/min. tracer Be?, all 0.1 M in sulfosalicylic acid (SSA) adjusted to pH 4.5 with “‘OH. Column: “Dowex 50” resin, GO-100 mesh, equilibrated with 0.1 M SSA adjusted to pH 4.5 with “,OH; 15.0 g. of resin, 39 cm. in height X 1.0 cm. diameter, flow rate, 2-3 ml./min. SEPARATION O F
Fraction collected (ml.)
Vol. (ml.)
e10 10 10-81 71 81-151.5 70.5 151.5-233 (includes 70 ml. SSA wash) 82 233-500 (3 N HC1) 26.7
% Total Be
Ca (mg.)
% Total Ca
47.1 45.6
0 0 0
0 0 0
6.0 0
0 295
0 99
0.96
Discussion When solid materials containing Be are put into solution it is sometimes found that the uptake of Be by a cation exchanger, or its extractability by acetylacetone, is nil or erratic. The difficulty seems to be caused by the conversion of Be into an insoluble oxide when the temperature of dry ashing exceeds 500°.2.15However, it has been found that the treatment recommended by Toribara and Chen,I6 namely, t80 heat the dry ashed residues strongly with concentrated sulfuric acid for several minutes, renders the Be absorbable if the ash has not been lost, wholly or in part, by fusion to the surface of the container. For samples containing very large amounts of calcium it is possible t o use a column large enough to absorb all the polyvalent cations. In the case of “Dowex 50” this would require a minimum of roughly 10 g. of resin per g. of calcium. Calcium (15) T.Y.Toribara and P. S. Chen, Jr., Anal. Chem., 24,639 (1953).
393
may also be removed prior to passage through the column by precipitation of the sulfate from an acidic solution without loss of Be.15 Other methods2.’a.*,l6 for minimizing interferences by Ca provide for the addition of EDTA or one of its analogs t o the Be-containing solution before passage through the cation exchanger. For example, from a solution at , p H 5 and containing 2 g. of the tetrasodium salt of EDTA per liter, Be was quantitatively absorbed by the “Iform of “Dowex 50,”16while those cations which form chelates with EDTA, including Ca, Fe, Mg, Ni and Al, etc., passed through. I n many cases, however, as with copper and uranyl ion, the differences in stability of the salicylic acid complexes of Be and many of the transition elements are great enough that the use of a more selective complexing agent such as EDTA is unnecessary. Iron was eluted along with Be by salicylates a t the usual pH range employed, namely, 3.5 to 4.5. However, since Fe(II1) formed a fairly strong complex with salicylates a t pH’s below 3, it was possible to elute the Fe prior to Be elution. (The polyamino chelating agents also can be used.) I n one experiment, for example, from a mixture of Fe, Be and Ca the Fe was eluted with 0.1 M gentisic acid at a p H of 2.1, then the pH was raised to elute the Be, while the Ca was eluted later with HC1. Another procedure for rapidly removing many interfering elements from Be would be to pass a strong HC1 solution of the Be through an anion exchanger first. In such solutions Be is unabsorbed while many elements, including iron and uranium, are strongly ab~0rbed.l~ Finally there is the problem of how to deal with potential aluminum interference. Since AI is eluted by salicylates in the same pH range as Be, the AI must be separated by other means. Kakiham’s was able to separate Be from A1 by passage of a dilute (-0.01 N ) solution of their salts through a cation exchanger in the calcium form. The Be ran through while A1 was retained. Residual Be was eluted free of AI by the use of 0.1 N CaC12. To carry this procedure one step further, the Be could now be reabsorbed by a cation exchanger and eluted with SSA. Other workerslghave passed the oxalates of Be, A1 and Fe through a cation exchanger. The oxalates of A1 and Fe are stable a t pH 4.4 to 4.5 and pass through, while Be is retained on the column. EDTA forms a chelate with AI under conditions allowing little or no complex formation with Be.* Thus, it should be possible to effect a separation of AI from Be either by adding EDTA to the solution a t p H -4 before passage through a cation exchanger, or by eluting the AI with EDTA prior (16) (a) J. Hure, M. Kremer and F. LeBerquier, Anal. Chem. Acta, 7 , 37 (1952); (b) M. S. Das and V. T. Athavale, ibid., 12, 6 (1955); (0) J. Kinnunen and B. Wennerstrand, Chemist-Analyst, 44, 51 (1955). (17) K. A. Kraus and F. Nelson, “Anion Exchange Studies of the Fission Products,” Proceedings of the International Conference on the Peaceful Uses of Atomic Energy, Vol. 7, p. 113. Session 9B.1, P/837, United Nations (1956). (IS) H.ICakihana, J. Chem. Soe. Japan, 7 3 , 200 (1951). (19) D.I. Ryabchikov and V. E. Bukhtiarov, Z h w . Anal. Khim., 9, 196 (1954); C . A., 48, 12610h (1954).
394
ELI S. FREEMAN AND BENJAMIN CARROLL
to Be elution with salicylates. Acknowledgments.-We are indebted to Mr.
VOl. 62
Roman V. Lesko for technical assistance in the early stages of this investigation.
THE APPLICATION OF THERMOANALYTICAL TECHNIQUES TO REACTION KINETICS.' THE THERMOGRAVIMETRIC EVALUATION O F THE KINETICS O F THE DECOMPOSITION OF CALCIUM OXALATE MONOHYDRATE BY ELI S. FREE MAN^ AND BENJAMIN CARROLL Chemistry Department, Rutgers University, Newark 2, N . J., and The Pyrotechnics Chemical Research Laboratory, Picatinny Arsenal, Dover, N . J . Received July $8, 1967
The a plication of thermoanalytical techniques to the investigation of rate processes is discussed. Equations have been derived for non-reversing reactions, which may be used to calculate energy of activation and order of reaction from thermogravimetric and volumetric curves. An equation, recently presented in the literature, for evaluating these parameters by the technique of differential thermal analysis has also been considered, so as to eliminate the trial and error procedure. The thermal decomposition of calcium oxalate monohydrate which involves dehydration, decomposition of calcium oxalate and calcium carbonate, is used to illustrate the applicabdity of the derived relationships.
Introduction Thermoanalytical methods, such as thermogravimetry, thermovolumetry and differential thermal analysis, are being employed increasingly in the investigation of chemicaI reactions in the Iiquid and solid states a t elevated temperatures. These techniques involve the continuous measurement of a change in a physical property such as, weight, volume, heat capacity, etc., as sample temperature is increased, usually at a predetermined rate. In this article, equations are derived for non-reversing reactions so that rate dependent parameters such as energy of activation and order of reaction may be calculated from a single experimental curve. For this purpose a relationship between specific rate and temperature is assumed = ze-E*/RT
A general derivation is presented and applied to thermogravimetry. For the method of differential thermal analysis, the derivation of Borchardt and Daniels3 has been expanded upon, so that the trial and error procedure now required for evaluating order of reaction and activation energy, may be replaced by a graphical or analytical solution. It should be kept in mind that the treatment may be applied to the measurement of any physical property which is unaffected by sample temperature. The advantages of evaluating reaction kinetics by a continuous increase in sample temperature are that considerably less experimental data are required than in the isothermal method, and the kinetics can be probed over an entire temperature range in a continuous manner without any gaps. In addition, where a sample undergoes considerable reaction in being raised to the tem-
perature of interest, the results obtained by an isothermal method of investigation are often questionable. Theory and Derivation Consider a reaction, in the liquid or solid states, where one of the products B is volatile, all other substances being in the condensed state. a A = bB(g) + CC The rate expression for the disappearance of reactant A from the mixture is where X
=
concn., mole fraction or amount of reactant, A
k = specific rate x = order of reaction with respect to A
It is assumed that the specific rate may be expressed as k = Ze-E*/RT
(2)
Solving for k in (1) and substituting (2) for k gives Ze-E*/RT
=:
- (dX/dt) X"
(3)
where 2 = frequency factor E* = energy of activation
R = gas constant T = absolute temperature
The logarithmic form of equation 3 is differentiated with respect to, dX/dt, X and T, resulting in equation 4. E*dt = d In (-dX/dt) R T
- x d In X
(4)
(1) Thin paper has been presented i n p a r t st t h e North Jersey Meeting i n Miniature of the A.C.S. i n Jan., 1957. and before t h e Division
Integrating the above relationship gives
of Physical a n d Inorganic Chemistry at the National Meeting of the A.C.S. i n April, 1957. (2) Pyrotechnics Chemical Research Laboratory, Bldg. 1512, Picstinny Arsenal, Dover, New Jersey. (3) H. J. Borchardt and E'. D. Daniels, J . A m . Chem. Soc., 79, 41 (1957).
Dividing (4)and (5) by d In X and A In X , respectively, one obtains equations 6 and 7.
