Theory of Coal Pyrolysis W&
&tzh
dtQ. q. &+z&u#
The Pennsylvania State College, State College, Penna.
1. The rule of least molecular deformation states that “the decomposition by heat will follow that reaction which requires the lea& ossible deformation of the molecule” (IO). 2. Haier’s rule states that “the C-C linkage in the aromatic series is more stable than the C-H linkage, the reverse of which is true in the aliphatic series” (IO). This rule is not without exception.
The outline of a comprehensive theory of coal pyrolysis is presented. Coal is considered to be comprised of a variety of molecular species which are illustrated by suitable examples. It is derived from fundamental principles that, in the pyrolysis of coal, aliphatic carbon-carbon linkages are the first to break, that carbon-hydrogen linkages are severed next as the temperature of 600” C. is approached and exceeded, and that aromatic carboncarbon linkages do not break readily in the temperature range of coal carbonization, because these linkages have been stabilized by resonance. In the temperature range from 400” to 700” C., oxygen-containing complexes and other heterocyclic structures are breaking away from the coal (or semicoke) molecules; this process is induced by an activation energy of roughly 39,000 calories per formula weight of split product and results in a decrease of free energy. The liberated radicals transform into more stable products, especially aldehydes) these, in turn, decompose to nitrogen bases, phenols, hydrocarbons, carbon monoxide, water, and hydrogen. All these processes are illustrated b y suitable examples.
General Results of Pyrolysis Studies
Brooks (a) applied Haber’s rule to the formation of coke during cracking of petroleum and by carbonization of coal. He emphasized that “the paraffins can be cracked at moderate temperatures without forming coke, which is another way of saying that heat causes splitting of the carbon-to-carbon bond in the paraffin hydrocarbons, but in the aromatic series, in which the remarkably stable six-carbon ring structure is present, the result is chemical condensation, carbon-tocarbon combination to larger and larger molecules and eventually to coke, with the liberation of hydrogen and other simple split products such as methane, ethylene, ethene, etc.” The application of Haber’s rule to the formation of petroleum coke seems much better founded than the application to coke obtained by carbonization of coal. Coal cannot be classified as a strictly aromatic compound nor as a hydrocarbon. Coal contains oxygen, nitrogen, and sulfur; it contains naphthenic and heterocyclic units, and no general rule concerning pyrolysis of naphthenic and heterocyclic compounds has yet been formulated. These compounds decompose sometimes like aliphatics and sometimes like aromatics. Finally, in view of recent x-ray evidence (1) it seems doubtful whether we may speak of a coke molecule and whether such a molecule would be larger than a coal molecule. To explain the mechanism of pyrolysis, various theoretieal assumptions have been made. The Nef theory (10) emphasizes the importance of bivalent carbon; this theory is still useful though no longer in general favor. At present, explanations are preferred which assume “the temporary existence of fragments of molecules or free radicals (CHa for example) which are forbidden by classical rules of valence. Any hypothesis which involved the existence of such nonconformist units would have been condemned immediately a few years ago, but a t the present time the existence of free radicals is quite generally accepted” (6). “Electronically, the free radical is neutral but unstable because it has an odd electron” ( $ 1 ) . A typical example may be expressed by the symbol Ha :C. Radicals to be conceived of as possible intermediates in coal pyrolysis should, on the basis of a preliminary consideration, contain oxygen in carbonyl or heterocyclic linkages and be capable of breaking up into water, carbon monoxide, methane, perhaps hydrogen, and compounds of the type isolated by Pictet (16). The interconnection between a mixture of Pictet compounds (or low-temperature carbonization tar) and high-temperature tar is well established by the work of Schrader (19) and Morgan and Soule (15).
Investigators who have studied mainly the pyrolysis of pure compounds in the gas phase have derived certain useful empirical rules which are applicable t o more complicated cases. The two most widely applicable rules are as follows:
Pyrolytic reactions will proceed in such a manner as t o cause a decrease of free energy. Some general conclusions
I
P
YROLYSIS has been defined as “the decomposition of organic substances by heat” (8). The expression ‘%hermal decomposition” is also descriptive of the transformation of a substance into another substance or into other substances through the severance of chemical linkages under the influence of heat. The term “pyrolysis”, however, commends itself by brevity and is therefore exclusively used in this paper. By definition, pyrolysis is the fundamental principle underlying carbonization and proximate analysis. The general topic of pyrolysis has been the object of numerous publications; yet, the subject of coal pyrolysis has hardly been given a coherent treatment, although experimental data are available in abundance. This paper attempts to develop thermodynamic information pertaining to the topic, to discuss pertinent published experimental data, to present results of experimental work, and to correlate all this information by the use of thermodynamic and kinetic considerations.
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Application of Thermodynamics
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obtained by the application of this principle to the rule of least molecular deformation, to Haber's rule, and to the pyrolysis of carbon-hydrogen-oxygen complexes follow:
application of the Newton approximation method: 6-H 848; C-C, 6303; C-CHs, 337; and C=C, 2437' K.
1. The thermodynamic mechanism of complicated decompositions, such as simultaneous formation from coal of coke, tar, water, carbon monoxide, methane, hydrogen, may be conceived as the effect of a sequence of simple steps. These steps may be established experimentally, or they may be assumed on the basis of kinetic studies and quantum mechanical considerations. 2. The relative stren th of the carbon-hydrogen bond, the aliphatic carbon-carbon tond, and the aromatic carbon-carbon bond, may be compared by ascertaining the free energy changes that accompany making or breaking of these bonds. 3. "The fact that upon heating one coal eives more volatile split products than another is in all probability suEciently exglained by a larger oxygen content; structures comprising oxygen reak up more easily than hydrocarbon structures" ( 7 ) . This statement also requires a quantitative thermodynamic analysis.
TABLEI. FREEENERGY CHANGE,AF',
As Bruins and Czarnecki (9) showed: "(1) It is possible to correlate logically free energies of formation with the nature and number of atomic linkages within a molecule by means of bond equations. (2) Such equations are of the conventional form and have considerable predicting value.'' They state also that "it must not be assumed that bond equations will produce perfect results. There are two important obstacles: First, we are limited by the number of available molecular free energy equations, and secondly, by natural phenomena such as atomic interactions, steric hindrance, resonance, etc." Equations 1 to 4 are simplified (and in one case slightly corrected) versions of the more elaborate formulas of Bruins and Czarnecki for the respective bonds in the paraffin and olefin series: C-H:AF' C-C:AF" C==C:AF" C-CHt:AF'
= =
-3,845 +4,440
= +28,020 = -7,620
T,'X. 600 700 800 900 1000
+ + + +
6C-H: AF' = -23,070 18.OT In 3c-c: A F o = 13,320 1.25"In 3C=C: AF" = 84,060 8.4T In COHO: AF' = -49,910 - 8.OT In COH8:AF" = 24,400 19.6T In
T T T T T
- 0.012T2 - 82.22' 5) + 0.006T2 f 28.5T [6) - 0.003TZ- 74.7T (7) - 0.005Tz + 41.45" (8) - 0.014TZ- 87.02'
(9)
Practically, Equation 8 for AFR is obtained by subtracting the sum of bond Equations 5, 6, and 7 from Equation 9. Such computations were made for a number of aromatic compounds for which empirical free energy equations are available. Table I presents the free energy changes accompanying the severance of aliphatic bonds a t various temperatures; these values were obtained by solving Equations 1 to 4 for the temperatures listed and reversing the signs. The C-CH8 linkage is important in the transformation of cresols to phenol. The following temperatures were computed for AF" = 0 by
-- 12,395 13,904 15,459 ---18,703 17,058
1281 652 42 571 -1178
-
OF
BONDSEVERANCE C=C
C-CHa 8,477 -11,372 - 15,276 18,700 -22,120
-
-23,466 -22,940 -22,433 -21,942 -21,462
Table 11 lists values for AFR necessary t o accomplish the reversion of the resonance phenomenon for several compounds. Inspection of Table I shows that all aliphatic carbon-to-carbon linkages will break before the carbon-hydrogen linkage; and from Table 11, the aromatic bonds will be much more difficult to sever than the aliphatic carbon-carbon or carbon-hydrogen linkages. The reversion of the resonance in the aromatic compounds implies an expenditure of free energy which makes for a very stable aromatic carbon-carbon linkage. This is essentially Haber's rule. TABLE11. FREEENERGY CHANGE, . AF;,
OF
RESONANCE RE-
VERSION
T
d.
Benzene
Xylene
Toluene
Na hthaEne
600 700 800 900
55,770 57,620 59,570 61,610 64,720
78,780 84,000 89,280 94,570 99,820
64,300 66,480 69,000 71,460 73,830
110,760 114,750 118,880 123.150 127,540
1000
Anthraoene A =
Anthracene Bo
156,620 166,630 177,040 186,430 198,990
167 050 173:300 178,800 185.020 191,440
a The computations are based on structure A and B , respectively:
+ 3.02' In T - 0.002T2 - 13.7T (I) + 0.42' In T + 0.002T2 + 9.5T (2) + 2.8T In 5" - O.OOITZ- 24.9T (3) + 10.4Tln T - 0.006T2 - 36.1T (4)
I n a n attempt to obtain similar bond equations for aromatic carbon linkages, consideration was given to the achievements of quantum mechanics, specifically to the concept of quantum mechanical resonance and the relation between atomic distance and bond energy. It was concluded that the transformation of a compound composed exclusively of aliphatic linkages into an aromatic compound was accompanied by a change of free energy due t o the resonance phenomenon ( A F g ) which would have to be reversed before the severance of linkages would follow. Values for AFR were derived in a manner illustrated by the following example: The empirical free energy equation for the formation of benzene (16) is considered to be obtainable by summation of the free energies of 6C-H, 3C-C, and 3C=C linkages, plus the free energy change accompanying formation of the final resonating structure:
c-c
C-H
B
A
An average AFR of severance for single aromatic linkages may also be deduced from Table 11; however, it is doubtful whether, in connection with quantum mechanical resonance, single linkages should be contemplated. An example that may be elucidated by reference to bond equations refers to the pyrolysis of Boghead coals. Proximate analyses of these coals show considerably less coke and more volatile matter than coking coals of comparable carbon content. Stadnikoff, in a study of Siberian Boghead coals, found that a coal with, e. g., 81.1 per cent C, 8.3 H, 8.7 0, 1.1 N (ash- and moisture-free basis) gave, on lowtemperature carbonization, 39.4 per cent semicoke and 39.1 per cent tar. From Table I the conclusion may be drawn that, compared to regular coking coals, Bogheads contain a considerably higher percentage of aliphatic C-C linkages. Moore (12) says: "There seems little doubt that the organic matter of the Bogheads is a step nearer petroleum and natural gas than the ordinary coa1s." Stadnikoff established the participation of saturated and unsaturated fatty acids in the make-up of these coals, and in his words, "the composition of sapropelitic coals establishes their origin from fats beyond doubt" (20). A consideration of the free energy changes a t constant pressure due to the formation of low-molecular-weight split prodTABLE111. AF"
OF
LOW-MOLECVLAR SPLITPRODVCTS
T. K.
H.0
CO
CO,
CHI
600 700 800 900
-50,936 -49,675 -48,471 -47,061 -45,716
-39,041 -41,215 -43,384 -45,547 -47,702
-94,261 -94,258 -94,253 -94,249 -94,245
-5714 -3391 -1047 4-1298 4-3636
1000
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ucts in decomposing systems may well constitute an important link in developing a theory of coal pyrolysis. As pointed out (15), in chemical reactions involving the formation of water the tendency to form that compound supplies, over a wide range of temperature, the actual thermodynamic driving force. This is due to the fact that a large decrease of free energy is concomitant with the formation of water, and therefore, reactions resulting in such formation are favored. A preliminary survey of published data showed that a similar condition exists where formation of other low-molecular oxygencontaining split products, such as carbon monoxide or carbon dioxide, is possible. This is further illustrated by data which are descriptive of the free energies of formation of some lowmolecular split products in the temperature range of proximate analysis and carbonization (Table 111). These data were computed by the use of the free energy equations given by Randall (18). They show an unmistakably strong trend toward the formation of the oxygen-containing split products. The formation of water and carbon monoxide is, however, descriptive of only one of the aspects of coal pyrolysis. Up to 600" K. illuminants and methane are characteristic products of coal distillation. Up t o 850" K. the main products of coal pyrolysis are semi-coke, low-temperature tar, and gas rich in illuminants and methane but poor in hydrogen. The abundant formation of low-temDerature tar with its aliphatic, naphthenic, and oxygenated compounds points to the splitting of larger atomic arrangements into solid structures of preponderantly aromatic nature and volatile substances as mentioned; these processes require further annlysis. Above 850" K. the stability of the aromatic linkages and the instability of the carbonhydrogen linkages are betrayed in the increased evolution of hydrogen and the formation of hightemperature coke. I n view of the importance of low-niolecular split products in coal pyrolysis, it is pertinent to consider the relation between Droximate analvsis. temperature, and gas yield.* Several p u b h a tions (4, 11, 17) contain data descriptive of gas yields a t different temperatures. A study of this information revealed the existence of 8,straight-line relation between temperature and gas yield which had apparently escaped the attention of the authors. The results of our analysis are presented in Table IV. From Table IV the conclusion is drawn that below a certain limiting temperature there is no significant evolution of gas, TABLEIV. RELATION BETWEEN GASYIELDSIN CUBWFEET/ TON(y) AND CARBONIZATION TEMPERATURE (z) Coal Anthrsoite Semibituminous Bituminous Bitumjnous Bituminous Bituminous Bituminous
-
% Volatile Matter Dry- Value of by Least n (y O), Ash-kree Sauare &estment C. Citation 8.4 17.0 31.0 38.2 31.0 32.9
672 489 416 421 355 358 336
..
~~~
~
~~
TABLEV. GASYIELDOF B SEAMCOALS Lab. No.
TABLEVI. ACTIVATION ENERQIESIN SEAMCOALS Lab.
No. 50 51 52 63 54 65 56
Coal Volatile Matter, %
24.8 17.5 16.6 24.3 36.6 17.1 25.7
Average
Volatile Matter, ? 'Z
Gas Yield, Cu. 600Q
400' C. 1650 1330 1280 1780 1410 1490 1860 61
c.
2830 3000 3040 4210 3550 3610 4100
THE
PYROLYSIS OF B
Activation Energy, Small Cal./Gram/O C.
& 1849 1734 1085 1717 1586 1622 1768 1623
1574 1829 2139 1176 1372 1713 1592 1628
1561 1841 1502 1443 1602 1502 1561 1659
1661 1801 1575 1445 1387 1612 1640 1689
and this conclusion would still be justified even if the curves should slope off toward the origin below the lowest temperature recorded. Table V presents results of experimental studies on a number of coals from the lower Kittanning (or B) seam in Pennsylvania. I n these studies only insignificant gas yields were obtained below 400" and above 800' C. A statistical treatment of the data between 400" and 800' C. showed that, in spite of inbx can dividual differences, equations of the form y = a well be used to express the relation between gas yield and temperature, just as in the cases summarized in Table IV. It is not inconceivable that some coals may behave differently-e. g., because of a peculiar petrographic composition.
+
Figure
1. Example of
d
Coal Molecule
Application of Kinetics
In laboratory carbonizations the bulk of the evolved gas may be collected within an hour, although evolution of gas in small quantities may continue for a long time. Considering that in rate studies deviations of *20 per cent are within the limits of tolerance (5),gas yields obtained within an hour were taken to be sufficiently accurate measures for the purpose of comparing reaction rates a t different temperatures. It is indeed the ratio of reaction rates which is used in the application of the Arrhenius equation (14). Results of computations of activation energies are compiled in Table VI. For an understanding of the physical meaning of activation energies in coal pyrolysis, a consideration based upon the Boltzmann principle was found helpful. To quote from Hammett (9): The fundamental principle that describes the behavior of chemical and physical systems at equilibrium is the Boltzmann expression : 1.
Coal
569
N = Aps-E'KT
N is the number of molecules present at equilibrium in a state in which the energy per molecule is E ; A is a proportionality constant; p , the a priori probability, is a factor that expresses the number of fundamentally different ways in which the ener y c can be attained; K is the Boltzmann constant defined by if = R / N , and T is the absolute temperature. The principle may be derived from more abstract considerations of probabilities, but the justification for these as for the Boltzmann expression itself is the widespread applicability of the expression.
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methods of chemical research may be traced. Examples of coal molecules are given in Figures 1,2,and 3; these structures are compatible with the proximate and ultimate analyses of bituminous coals, with the results of group determinations, and with the experiences of oxidation, reduction, and thermal decomposition. These model molecules are probably yellow, or perhaps even colorless if pure; the dark color of coal is probCi35H97 09 N S ably the result of the presence of a 4 7 0 X 2 2 . 5 1 9 . 0 a'. 0. 7.5% PI- 1 . 5 % variety of species forming innumerS L 1.6% able interfaces, and other-reasonse. g., surface films and van der Figure 2. Example of Coal Pyrolysis Waals forces. These model molecules were also built up with the It is not even necessary to consider the structures underFisher-Hirschfelder atom models so that the molecular digoing decomposition as molecules. Given a number of strucmensions could be tentatively ascertained. Real structures tures in equilibrium a t temperature T,a certain reaction inwill differ from the representative molecules by isomery, and a new state of duced by raising the temperature to TI, homology, different skeletal structure, etc. it appears that a t every temperature TI, T,, equilibrium at TI, I n the pyrolysis of such structures we may expect to ob.. . T, a certain number of structures will collapse and give tain coke, tar, and gas. It is not probable that the variety of rise to volatile products. The temperature-yield curves, then, compounds making up gas and tar originate simultaneously, indicate the relation between N and 2'. or a t least it is not necessary to make such an assumption. In a number of unimolecular decomposition reactions, acIt seems to be more probable that these compounds which are tivation energies of roughly 32,000 calories per mole were esfound in the tar and gas have originated from the breakdown tablished (6). If it is assumed that the reaction mechanism of slightly larger units split off from structures such as that in coal pyrolysis is analogous to a unimolecular reaction, a shown in Figures 2 and 3. molecular weight of approximately 20 for the decomposing With this assumption in mind, it is comparatively simple structure follows. This is obviously the average molecular to account for primary split products which, when within a weight of the gas, and this result suggests the following certain temperature range, will decompose t o give low-temperamechanism: Within the arrangement of atoms characteristic ture tar and low-temperature gas, and when within a higher of bituminous coals there will be certain groups capable of temperature range will yield high-temperature tar and highbreaking away from the arrangement if provided with a temperature gas. The solid residues will lose first their oxycertain amount of energy. For conditioning these groups t o gen content and then their hydrogen content as the temperathe point where they may sever the union with other atoms ture rises through the low- and high-temperature carbonizain the arrangement and assume the nature of free-moving tion ranges. molecules, an amount of energy must be supplied which is Figure 2 shows how pyrolysis may give rise to heterocyclic found to be equal to the activation energy of mono-molecular complexes, two of which are illustrated. Such hydrogenated radicals give, in turn, hydrogenated aldehydes which then decompositions. It is not intended a t this time to develop more specific interpretations; however, it is emphasized that break up into bases, alcohols, phenols, and hydrocarbons, even this tentative correlation of data and Boltmann princisimultaneously yielding gaseous illuminants, methane, carbon monoxide, hydrogen, and water. The same general patple was found helpful in integrating apparently disconnected tern may be followed in the pyrolysis of the molecule picdetails into a single picture. tured in Figure 3. Attention should be called to the possiCod Structure and Pyrolysis bility (not directly shown in Figure 3) of deriving substituted hydrogenated naphthalenes or such complicated hydrocarAlthough coal is undoubtedly a heterogeneous system cornbons as dinaphthocoronene or coke from the structure, deprised of a multitude of chemical species, it is advantageous pending upon the rate of severance of the carbon-hydrogen for some purposes to work with an iverage or representative bonds in the large aromatic residual complex. coal molecule to which the experiences gathered from various On the assumption that a large number of these types of atomic arrangments are present in coal, quantitaH tive relations between coal constitution and constituents of both low- and high-temperature tar may be derived. Morgan and Soule (IS) presented a theory of coal 4 b carbonization with emphasis upon the mechanism of tar formation. They summarized the salient features of their theory in six points which are now expanded and modified as a result of the present investigation: +ot co t
-od 4 0 .3.
c70 "406 N 16.5 Y 19.5 X 6.0 A.'
Figure 3. Example of
Ne 1.42%
Coal Pyrolysis
1. The decom sition of coal substance when subjected to the action of g a t is a process of progressive, step-bystep decomposition, in which pyrogenetic syntheses play only a secondary part. 2 . Homocyclic and heterocyclic ring systems characterize coal as well as the entire series of its decomposition products. The decompositions during carbonization are essentially reactions effecting the eliminationof hetero-
cyclic complexes and progressive aromatization.
May, 1942
I N D U S T R I A L AND E N G I N E E R I N G CHEMISTRY
3. The average molecular weights of the volatile intermediate products constantly decrease as the temperature of carbonization rises. This decrease is marked by the evolution of water, carbon monoxide, hydrogen, methane, and other hydrocarbons. 4. The initial decom osition of coal, giving semicoke and low-temperature tar, is irought about by (a) splitting off of radicals containing heterocyclic and homocyclic rings, (b) opening of some heterocycles to intermediate aldehydes which give rise to phenols, amines, and naphtiienic compounds, (c) progressive dehydrogenation and splitting off of side chains by the action of the develo ing hydrogen. 5 . Final &compositions are a t a maximum between 600’ and 800” C. and are marked by (a)loss of hydrogen and other simple gases from semicoke and hydroaromatic volatile compounds, (b) hydrogenation of phenols to aromatic hydrocarbons with the formation of methane, ethane, and water, and ( c ) formation of higher aromatics from semicoke proper and by secondary pyrogenetic syntheses.
Literature Cited (1) Blayden, H. E.,Riley, H. L., and Taylor, A., J. Am. Chem. Soc., 62, 180 (1940). (2) Brooks, B. T., IND.ENQ.CHEM.,18, 621 (1926). (3) Bruins, P. F., and Czarnecki, J. D., Ibid., 33, 201 (1941). (4) Burgess, M. J., and Wheeler, R. V., Fuel, 4,208 (1925). ( 5 ) Daniels, F., “Chemical Kinetics”, pp. 14 and passim, 49, Ithaca, Cornell Univ. Press, 1938.
571
(6) Fowler, (7)
R. H., and Guggenheim, E. A., “Statistical Thermodynamics”, p. 528, Cambridge Univ. Press, 1939. Fuchs, W., Am. Inst. Mining Met. Engrs., Tech. Publ. 1333 (1941).
(8) Hackh, I. W. D., with Grant, J., Chemical Dictionary, 2nd ed., p. 770 (1937). (9) Hammett, L. P., “Physical Organic Chemistry”, p. 69 (1940). (10) Hurd, C. D., “Pyrolysis of Carbon Compounds”, A. C. S. Monograph 60, pp. 11, 16, New York, Chemical Catalog Co., 1929. (11) Marson, C. B., J. Armstrong College Min. Soc., 6,7 (1930). (12) Moore, E.S., “Coal”, 2nd ed., p. 188 (1940). (13) Morgan, J. J., and Soule, R. P., Chem. & Met. Eng., 26, 1025 I , Ann\
(lYL.6).
Noyes, A. A.,and Sherrill, M. S., “Course of Study in Chemical Principles”, p. 426 (1938). (16) Parks, G. S., and Huffman, H. M., “Free Energies of Some Organic Compounds,” pp. 93-4, New York, Chemical Catalog Co., 1932. (16) Pictet, A., in Fuchs’ “Die Chemie der Kohle”, pp. 310 & ff.
(14)
(1931). (17) Porter, H. C.,and Ovitz, F. K., U.S. Bur. Mines, Bull. L (1913). (18) Randall, Merle, in International Critical Tables, Vol. VII,pp. 224 & ff ., New York, McGraw-Hill Book Co., 1930. (19) Schrader, H., in Fuchs’ “Die Chemie der Kohle”, p. 364 (1931). (20) Stadnikoff, G. L., “Die Enstehung von Kohle und Erdol”, p 144 (1930). (21) Whitmore, F. C., “Advanced Organic Chemistry”, p. 841, New York. D. Van Nostrand Co., 1937.
of Solution of the System Trioxide-Water &.&!p& A table of partial molal heats can be used to calculate the heat involved in forming, mixing, and diluting solutions. Values are given for partial molal heats for the system sulfur trioxide-water which include all concentrations for sulfuric acid in water and oleums up to 100 per cent free sulfur trioxide. Similar data for systems in which crystallization occurs can be used to calculate the heat of crystallization.
P
ARTIAL molal quantities (3) have not received the widespread attention in engineering calculations that their value deserves. Any extensive propeey of a solution, G, has ita corresponding partial molal value Gl for the solvent and G, for the solute. The discussion is valid for any number of components but will be confined to the case of a two-component system. At any given temperature and pressure GI and G2are defined b y the equations:
One of the fundamental equations of calculus (4) gives the relations between partial and total derivatives as follows:
/$fv,
University of Florida, Gainesville, Fld.
(3)
whence i t follows that dG = Bldnl
+ &dnl
(4)
The use of these equations implies that all variables other than concentration remain constant. Since G1and G2 are always the same for a given concentration of a given solution and are independent of the quantity or past history of the solution, the partial molal values are intensive properties of the solution. A table of partial molal values can be used to calculate the corresponding extensive property of the solution. Partial M o l a l Heats of Solution
The partial molal heats of solution of the system sulfur trioxide-water are used to illustrate the usefulness of partial molal quantities. Thermodynamically the heat of formation of a solution can be handled in the same manner as the heat of any chemical reaction. If the standard states are defined as the pure materials in their most stable form a t the given temperature and pressure, then the heat of reaction for the formation of the solution is the sum of the partial molal heats of each constituent multiplied by the number of moles of each constituent involved. For example, if one mole of liquid sulfur trioxide is added to one mole of liquid water, the resulting product can be considered a solution of sulfur trioxide in water in which the mole fraction of sulfur trioxide, 22, is 0.5. As Table I shows, this value is the same as the heat of forma-
