THERMOCHEMISTRY, a branch of Energetics, treating of the thermal phenomena which are associated with chemical change.
§ i. That vigorous chemical action is accompanied by a brisk evolution of heat is evident from such familiar examples as the combustion of fuel or the explosion of gunpowder. The heat attendant on these actions, and on the vital processes of the animal organism, naturally first attracted attention. Robert Boyle, A. Crawford, A. L. Lavoisier and P. S. Laplace, P. L. Dulong, H. Davy, Count Rumford, all concerned themselves with thermochemical investigations of such processes. Their quantitative experiments were, however, too rough to permit of accurate generalization; and although Lavoisier and Laplace stated the principle that the same amount of heat must be supplied to decompose a compound as would be produced on its formation, the statement was not based on exact experiment, and only received experimental confirmation much later.
The beginnings of modern thermochemistry, though made independently of the doctrine of the conservation of energy, are practically contemporaneous with the recognition of that law, and without it the science could scarcely have reached the degree of development which it rapidly attained. Thomas Andrews and, especially, G. H. Hess (18 4 o) were the first who systematically investigated thermochemical effects in solution, and arrived at conclusions from their experimental data which still possess validity. Andrews, for example, found that when a series of acids were under similar conditions used to neutralize a given amount of a base, the quantity of heat evolved on the neutralization was the same in all cases. Hess, from his work, arrived at the converse conclusion, that when a series of bases were used to neutralize a given amount of an acid, the heat of neutralization was always the same. Both of these statements are correct when the powerful mineral acid and bases are considered, exceptions only arising when weak acids and bases are employed. Again, Andrews discovered that when one metal displaces another from solution of its salts (e.g. zinc with solutions of copper salts), the thermal effect is practically independent of the nature of the acid radical in the salt employed. Andrews likewise found that when the heat evolved on. the displacement from its salts of a metal M' by a metal M is added to the heat of displacement of another metal M" by M', the sum is equal to the heat which is evolved on the direct displacement of M" from its salts by M. This affords an example of a principle which had been stated by Hess in a very general form under the name of the Law of Constant Heat Sums - namely, that the thermal effect of a given chemical action is the same, independently of the character and number of the stages in which it takes place. Thus, in the above example, it is immaterial whether M displaces M" from its salt directly, or whether M first displaces M', which is then used to displace M". This important principle is a direct consequence of the law of the conservation of energy, but was discovered independently by Hess from accurate experiment.
Oxidation of Zn to ZnO .
„ S to SO 3
Dissolution of SO 3 in much water. .. .
ZnO in the resulting aqueous H2S04.
1585 o „
Deduct heat of dissolution of anhydrous ZnSO 4 .
11 93 „
Heat of formation of ZnSO 4 from Zn, S, and 40 =
Hess employed this principle to determine indirectly the heat of formation of compounds from their elements, when this magnitude, as is generally the case, was inaccessible to direct measurement. Thus the heat of formation of anhydrous zinc sulphate, ZnSO 4j which cannot be determined directly, may be arrived at by summation (in Hess's units) as follows: Heats of formation are still determined for the most part in a precisely similar manner.
Hess also stated another principle on empirical grounds, which, although admitting of many exceptions, is of considerable utility and significance. It had been known long before his time that when solutions of neutral salts were mixed, and no precipitate resulted, the mixed solution was also neutral. Hess now observed that in the process of mixing such neutral solutions no thermal effect was produced - that is, neutral salts in aqueous solution could apparently interchange their radicals without evolution or absorption of heat. These experimental results were generalized by him under the title of the Law of Thermoneutrality. After the investigations of Hess and Andrews, a great deal of excellent experimental work was performed by P. A. Favre and J. T. Silbermann, whose chief theoretical achievement was the recognition that the heat of neutralization of acids and bases was additively composed of two constants, one determined by the acid and the other by the base. This deduction harmonized the observations of Andrews and of Hess previously alluded to, and also accounted satisfactorily for the Law of Thermoneutrality.
Julius Thomsen was the first investigator who deliberately adopted the principle of the conservation of energy as the basis of a thermochemical system. His thermochemical work was begun in 1853, but most of his experiments were performed in the years 1869-82, the whole being published collectively, under the title Thermochemische Untersuchungen, in four volumes. Somewhat later than Thomsen, Marcellin P. E. Berthelot began (in 1873) a long series of thermochemical determinations. It is to these two investigators and their pupils that most of our exact thermochemical data are due.
Thomsen and Berthelot independently enunciated a generalization (commonly known as Berthelot's Third Principle, or Principle of Maximum Work), which may be stated in brief as follows: - Every pure chemical reaction is accompanied by evolution of heat. Whilst this principle is undoubtedly applicable to the great majority of chemical actions under ordinary conditions, it is subject to numerous exceptions, and cannot therefore be taken (as its authors originally intended) as a secure basis for theoretical reasoning on the connexion between thermal effect and chemical affinity. The existence of reactions which are reversible on slight alteration of conditions at once invalidates the principle, for if the action proceeding in one direction evolves heat, it must absorb heat when proceeding in the reverse direction. As the principle was abandoned even by its authors, it is now only of historical importance, although for many years it exerted considerable influence on thermochemical research.
§2. From the standpoint of the law of conservation of energy, the relation between chemical and thermochemical action bears the following aspect: - A given amount of any substance under given conditions possesses a perfectly definite amount of intrinsic energy, and, no matter what chemical and physical transformations the substance may undergo, it will, when it returns to its original state, possess the original amount of intrinsic energy. If we consider now the transformation of one system of chemical substances into another system under specified conditions, we shall find that in general the intrinsic energy of the second system is different from the intrinsic energy of the first. Let us assume, as is commonly the case, that the intrinsic energy of the initial system is greater than that of the final system. When the first system then is transformed into the second, the excess of energy which the former possesses must appear in the shape of heat, light, electrical energy, mechanical energy, &c. It is for the most part a simple matter to obtain the excess of energy entirely in the form of heat, the amount of which is easily susceptible of measurement, and thus the existence of thermochemistry as a practical science is rendered possible. Since the intrinsic energies of the two systems under given conditions are invariable, the difference between them is constant, so that the heat evolved when the first system is converted into the second is equal to that absorbed when the second system is re-transformed into the first (cf. Lavoisier and Laplace, ante, § 1). The total thermal effect, too, which is associated with the transformation, must be the same, whether the transformation is conducted directly or indirectly (Hess's Law of Constant Heat Sums), since the thermal effect depends only on the intrinsic energies of the initial and final systems.
Since the intrinsic energy of a substance varies with the conditions under which the substance exists, it is necessary, before proceeding to the practical application of any of the laws mentioned above, accurately to specify the conditions of the initial and final systems, or at least to secure that they shall not vary in the operations considered. It is also a necessary condition for the application of the preceding laws that no form of energy except heat and the intrinsic energy of the substances should be ultimately involved. For example, when metallic zinc is dissolved in dilute sulphuric acid with production of zinc sulphate (in solution) and hydrogen gas, a definite quantity of heat is produced for a given amount of zinc dissolved, provided that the excess of energy in the initial system appears entirely as heat. This provision may not always be fulfilled, since by placing the zinc in electrical contact with a piece of platinum, likewise immersed in the sulphuric acid, we can generate a current of electricity through the solution and the metallic part of the circuit. The reaction as before is completely expressed by the chemical equation Zn+H2S04 =ZnSO 4 H+ 2, the initial and final systems being exactly the same as in the first case; yet the amount of heat generated by the action is much smaller, a quantity of the intrinsic energy having been converted into electrical energy. This electrical energy, however, is equivalent to the heat which has disappeared, for it has been shown experimentally that if it is converted into heat and added to the heat actually evolved, the total quantity of heat obtained is exactly equal to that produced by the direct dissolution of the zinc in the absence of platinum.
§3. The following conditions have to be considered as affecting in a greater or less degree the intrinsic energy of the initial and final systems: (1) Dilution of solutions.
(2) Physical state.
(3) Change of volume.
(4) Allotropic modifications.
(i) Generally speaking, there is a considerable thermal effect when a substance is dissolved in water, and this effect varies in magnitude according to the amount of water employed. It is only, however, when we deal with comparatively concentrated solutions that the heat-effect of diluting the solutions is at all great, the heat-change on diluting an already dilute solution being for most practical purposes negligible. In dealing, therefore, with dilute solutions, it is only necessary to state that the solutions are dilute, the exact degree of dilution being unimportant. It occasionally happens that a change in dilution affects the chemical action that occurs. Thus if concentrated instead of dilute sulphuric acid acts upon zinc, the action takes place to a great extent not according to the equation given above, but according to the equation Zn +2H 2 SO 4 = ZnS04+S02+2 H20, sulphur dioxide and water being produced instead of hydrogen. Here we have a different final system with a different amount of intrinsic energy, so that the thermal effect of the action is altogether different.
(2) The physical state of the reacting substances must be considered, since comparatively large amounts of heat are absorbed on fusion and on vaporization. Thus the heat of fusion of ice (for H 2 O=18 g) is 1440 cal., and the heat of vaporization of water at 100°, for the same quantity, 9670 cal.
(3) The effect of change of volume against external pressure (due to production or consumption of mechanical energy) may be neglected in the case of solids, liquids or solutions, but must usually be taken into account when gases are dealt with. Each grammemolecule of a gas which appears under constant pressure during a chemical action (e.g. hydrogen during the action of zinc on dilute sulphuric acid) performs work equivalent to 580 cal. at the ordinary temperature, which must be allowed for in the thermochemical calculation. A similar correction, of opposite sign, must be made when a gramme-molecule of gas disappears during the chemical action.
(4) When a substance - e.g. carbon, phosphorus, sulphur - exists in allotropic forms, the particular variety employed should always be stated, as the conversion of one modification into another is frequently attended by a considerable thermal effect. Thus the conversion of yellow into red phosphorus evolves about one-sixth of the heat of combustion of the latter in oxygen, and so the knowledge of which variety of phosphorus has been employed is of essential importance in the thermochemistry of that element.
(5) The influence of temperature on the thermal effect of a chemical action is sometimes considerable, but. since the initial and final temperatures, which alone determine the variation in the thermal effect, are in almost all cases within the ordinary laboratory range of a few degrees, this influence may in general be neglected without serious error.
§ 4. Methods. - In order to estimate the thermal effect of any chemical process, use is made of the ordinary methods of calorimetry, the particular method being selected according to the nature of the chemical action involved. In almost every case the method of mixture (see Calorimetry) is employed, the method of fusion with Bunsen's ice-calorimeter being only used in special and rarely occurring circumstances.
As a very great number of important chemical actions take place on mixing solutions, the method for such cases has been thoroughly studied. When the solutions employed are dilute, no water is placed in the calorimeter, the temperature-change of the solutions themselves being used to estimate the thermal effect brought about by mixing them. Known quantities of the solutions are taken, and the temperature of each is accurately measured before mixing, the solutions having been allowed as far as possible to adjust themselves to the same temperature. The change of temperature of the solutions after the mixing has taken place is then observed with the usual precautions. It is of course in such a case necessary to know the specific heat of the liquid in the calorimeter. Thomsen by direct experiment found that the heat-capacity of a dilute aqueous solution diverged in general less than i per cent. from the heat-capacity of the water contained in it, the divergence being sometimes in one sense, sometimes in the other. He therefore abstained from determining for each case the specific heats of the solutions he employed, and contented himself with the above approximation. Berthelot, on the other hand, assumed that the heat-capacity of an aqueous solution is equal to that of an equal volume of water, and calculated his results on this assumption, which involves much the same uncertainty as that of Thomsen. Since thermochemical measurements of this type may be frequently performed with an error due to other causes of much less than i per cent., the error introduced by either of these assumptions is the chief cause of uncertainty in the method.
The calorimeter used for solutions is usually cylindrical, and made of glass or a metal which is not, attacked by the reacting substances. The total quantity of liquid employed need not in general exceed half a litre if a sufficiently delicate thermometer is available. The same type of calorimeter is used in determining the heat of solution of a solid or liquid in water.
Combustion calorimeters are employed for observing the heat generated by the brisk interaction of substances, one of which at least is gaseous. They are of two kinds. In the older type the combustion chamber (of metal or glass) is sunk in the calorimeter proper, tubes being provided for the entrance and exit of the gaseous substances involved in the action. These tubes are generally in the form of worms immersed in the water of the calorimeter. In the newer type (which was first proposed by Andrews for the combustion of gases) the chemical action takes place in a completely closed combustion chamber of sufficient strength to resist the pressure generated by the sudden action, which is often of explosive violence. The steel combustion chamber is of about 250 c.c. capacity, and is wholly immersed in the calorimeter. To withstand the chemical action of the gases, the " calorimetric bomb " is lined either with platinum, as in Berthelot's apparatus, or with porcelain, as in Mahler's. For ordinary combustions compressed oxygen is used, so that the combustible substance burns almost instantaneously, the action being induced by means of some electrical device which can be controlled from without the calorimeter. The accuracy of heats of combustion determined in the closed calorimeter is in favourable cases about one-half per cent. of the quantity estimated.
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The heat-units employed in thermochemistry have varied from time to time. The following are those which have been in most general use: Small calorie or gramme calorie. cal.
Large or kilogramme calorie. Cal.
Centuple or " rational " calorie. K.
The centuple calorie is the amount of heat required to raise 1 g. of water from o° C. to C., and is approximately equal to ioo cal. The large calorie is equal to 1000 cal. In view of the not very great accuracy of thermochemical measurements, the precise definition of the heat-unit employed is not a matter of special importance. It has been proposed to adopt the joule, with the symbol j, as thermochemical unit for small quantities of heat, large amounts being expressed in terms of the kilojoule, Kj =100o j. (For the exact relation between these heat-units, see Calorimetry.) For ordinary thermochemical work we may adopt the relation 1 cal. = 4.18 j, or 1 Cal. = 4.18 Kj.
Except for technological purposes, thermochemical data are not referred to unit quantity of matter, but to chemical quantities - i.e. to the gramme-equivalents or gramme-molecules of the reacting substances, or to some multiples of them. The notation which Julius Thomsen employed to express his thermochemical measurements is still extensively used, and is as follows: - The chemical symbols of the reacting substances are written in juxtaposition and separated by commas; the whole is then enclosed in brackets and connected by the sign of equality to the number expressing the thermal effect of the action. The chemical symbols stand for quantities measured in grammes, and heat-evolution is reckoned as positive, heat-absorption as negative. Thus [S, 20] =71100 cal.
indicates that 71100 calories are evolved when 32 grammes of sulphur react with 2 X 16 grammes of free oxygen to form sulphur dioxide. It is of course necessary in accurate work to state the conditions of the reaction. In the above instance the sulphur is supposed to be in the solid rhombic modification, the oxygen and sulphur dioxide being in the gaseous state, and the initial and final systems being at the ordinary temperature. Again, the equation [2N, 0] =-18500 cal.
indicates that if 28 grammes of nitrogen could be made to unite directly with 16 grammes of oxygen to form nitrous oxide, the union would cause the absorption of 18500 calories. When substances in solution are dealt with, Thomsen indicates their state by affixing Aq to their symbols. Thus [NaOH Aq, HNO 3 Aq] =13680 cal.
represents the heat of neutralization of one gramme-equivalent of caustic soda with nitric acid, each in dilute aqueous solution before being brought into contact. One drawback of Thomsen's notation is that the nature of the final system is not indicated, although this defect in general causes no ambiguity.
Berthelot's notation defines both initial and final systems by giving the chemical equation for the reaction considered, the thermal effect being appended, and the state of the various substances being affixed to their formulae after brackets. W. Ostwald has proposed a modification of Berthelot's method which has many advantages, and is now commonly in use. Like Berthelot, he writes the chemical equation of the reaction, but in addition he considers the chemical formula of each substance to express not only its material composition, but also the (unknown) value of its intrinsic energy. To the right-hand member of the equation he then adds the number expressing the thermal effect of the reaction, heat-evolution being as before counted positive, and heat-absorption negative. The mass-equation then becomes an energy-equation. He thus writes S+02=S02+7110o cal., which expresses the fact that the intrinsic energy of the quantities of sulphur and oxygen considered exceeds that of the sulphur dioxide derived from them by 71100 cal. when thermal units are employed. The equation H2+12=2HI-12200 cal.
expresses that under certain conditions the intrinsic energy of hydriodic acid is greater than the intrinsic energy of its component elements by 12200 cal., i.e. that hydriodic acid is formed from its elements with absorption of this amount of heat. Energyequations, such as the above, may be operated with precisely as if they were algebraic equations, a property which is of great advantage in calculation. Thus by transposition we may write the last equation as follows 2HI =H2+12+12200 cal., and thus express that hydriodic acid when decomposed into its elements evolves 12200 cal. for the quantity indicated by the equation.
Ostwald has made the further proposal that the formulae of solids should be printed in heavy type (or within square brackets), of liquids (solutions, &c.) in ordinary type, and of gases in italics (or within curved brackets), so that the physical state of the substances 'might be indicated by the equation itself. Thus the equation Cl 2 -1-2KI, Aq=2KC1, Aq+12+52400 cal., or (C12) +2KI, Aq =2KC1, Aq+-I-52400 cal., would express that when gaseous chlorine acts on a solution of potassium iodide, with separation of solid iodine, 52400 calories are evolved.
§ 6. Heat of Formation. - For thermochemical calculations it is of great importance to know the heat of formation of compounds from their elements, even when the combination cannot be brought about directly. As an example of the use of Ostwald's energy-equations for the indirect determination we may take the case of carbon monoxide.
The following equations give the result of direct experiment :- C +20 = CO 2+943 oo cal. CO+ O=CO 2 +68000 cal.
If now it is required to find the heat of formation of the compound CO, which cannot be directly ascertained, we have merely to subtract the second equation from the first, each symbol representing constant intrinsic energy, and thus we obtain C+0 - 00= 26300 cal., or C+0=C0+26300 cal., that is, the heat of formation of a gramme-molecule of carbon monoxide is 26300 cal.
As has already been stated, the heat of formation of a compound is the amount (expressed in thermal units) by which its intrinsic energy exceeds or falls short of that of the elements which enter into its composition. Now of the absolute values of intrinsic energy we know nothing; we can only estimate differences of intrinsic energy when one system is compared with another into which it may be directly or indirectly converted. But since the elements cannot be converted one into the other, we are absolutely without knowledge of the relative values of their intrinsic energy. This being the case, we are at liberty to make the assumption that the intrinsic energy of each element (under specified conditions) is zero, without thereby introducing any risk of self-contradiction in thermochemical calculations. This assumption has the great advantage, that the intrinsic energy of a compound relatively to its elements now appears as the heat of formation of the compound with its sign reversed. Thus if we consider the energyequation C +02 = CO 2+943 00 cal., and replace the symbols by the values of the intrinsic energy, viz. zero for carbon and oxygen, and x for carbon dioxide, we obtain the equation o+o=x+94300 cal.
or x = - 94300 cal.
With knowledge then of the heats of formation of the substances involved in any chemical action, we can at once calculate the thermal effect of the action, by placing for each compound in the energy-equation its heat of formation with the sign reversed, i.e. its heat of decomposition into its elements. Thus if we wish to ascertain the thermal effect of the action Mg+CaO =MgO+Ca, we may write, knowing the heats of formation of CaO and Mg0 to be 131000 and 146000 respectively, 0-131000 = 0-146000+x x =15000 cal.
Since heats of formation afford such convenient data for calculation on the above method, they have been ascertained for as many compounds as possible.
Substances with positive heats of formation are termed exothermic; those with negative heats of formation are termed endothermic. The latter, which are not very numerous, give out heat on decomposition into their elements, and are more or less unstable. Amongst endothermic compounds may be noted hydriodic acid, HI, acetylene, C 2 H 2, nitrous oxide, N 2 O, nitric oxide, NO, azoimide, N 3 H, nitrogen trichloride, NC1 3. Some of these pass into their elements with explosive violence, owing to the heat generated by their decomposition and the gaseous nature of the products.
The thermochemical magnitude which is universally determined for organic compounds is the heat of combustion, usually by means of the calorimetric bomb. The relation between the heat of combustion of a hydrocarbon and its heat of formation may be readily seen from the following example. The hydrocarbon methane, CH 4, when completely burned to carbon dioxide and water, generates 213800 cal. We may therefore write CH 4 +40 = C02+2H20+213800.
Now we know the heats of formation of carbon dioxide (from diamond) and of liquid water to be 94300 cal. and 68300 cal. respectively. The above equation may consequently be written, if x is the heat of formation of methane, -x+0 = -94300-(2 X 68300) +213800 x =17000 cal.
This heat of formation, like that of most hydrocarbons, is comparatively small: the heat of formation of saturated hydrocarbons is always positive, but the heat of formation of unsaturated hydrocarbons is frequently negative. For example, ethylene, C2H4 j is formed with absorption of 16200 cal., acetylene, C 2 H 2, with absorption of 59100 cal., and liquid benzene, C 6 H 6, with absorption of 9100 cal. Since the heat of combustion of a hydrocarbon is equal to the heat of combustion of the carbon and hydrogen it contains minus its heat of formation, those hydrocarbons with positive heat of formation generate less heat on burning than the elements from which they were formed, whilst those with a negative heat of formation generate more. Thus the heat generated by the combustion of acetylene, C 2 H 2, is 316000 cal., whereas the heat of combustion of the carbon and hydrogen composing it is only 256900 cal., the difference being equal to the negative heat of formation of the acetylene.
For substances consisting of carbon, hydrogen and oxygen, a rule was early devised for the purpose of roughly calculating their heat of combustion (J. J. Welter's rule). The oxygen contained in the compound was deducted, together with the equivalent amount of hydrogen, and the heat of combustion of the compound was then taken to be equal to the heats of combustion of the elements in the residue. That the rule is not very accurate may be seen from the following example. Cane-sugar has the formula C12H22011. According to Welter's rule, we deduct II 0 with the equivalent amount of hydrogen, namely, 22 H, and are left with the residue 12 C, the heat of combustion of which is 1131600 cal. The observed heat of combustion of sugar is, however, 1354000, so that the error of the rule is here 20 per cent. A much better approximation to the heat of combustion of such substances is obtained by deducting the oxygen together with the amount of carbon necessary to form C02, and then ascertaining the amount of heat produced by the residual carbon and hydrogen. In the above case we should deduct with II 0 the equivalent amount of carbon 5.5 C, thus obtaining the residue 6.5 C and 22 H. These when burnt would yield (6.5 X9430o)+(II X68300) =1364250 cal., an amount which is less than 1 per cent. different from the observed heat of combustion of sugar. Neither of the above rules can be applied to carbon compounds containing nitrogen.
It has already been stated that the heats of neutralization of acids and bases in aqueous solution are additively composed of two terms, one being constant for a given base, the other constant for a given acid. In addition to this, the further regularity has been observed that when the powerful monobasic acids are neutralized by the powerful monacid bases, the heat of neutralization is in all cases the same. The following table gives the heats of neutralization of the commoner strong monobasic acids with soda: - Hydrochloric acid Hydrobromic acid Hydriodic acid Nitric acid Chloric acid Bromic acid Within the error of experiment these numbers are identical.
It was at one time thought that the greater the heat of neutralization of an acid with a given base, the greater was the strength of the acid. It is now known, however, that when weak acids or bases are used, the heat of neutralization may be either greater or less than the normal value for powerful acids and bases, so that there is no proportionality, or even parallelism, between the strengths of acids and their heats of neutralization (see Solutions).
I. Sodium carbonate-
Na 2 CO 3.. .
Na 2 CO 3, H 2 O .
Heat of Solution.
Heat of Hydration.
+339 0 cal.
Na 2 CO 3, 2H 2 0 .
Na 2 CO 3, IoH 2 O .
- 16160 „
II. Sodium sulphate-
Na 2 SO 4
Na 2 SO 4, H 2 O .
- 1900 „
Na2S04, IoH 2 O .
- 18760 „
When substances readily combine with water to form hydrates, the heat of solution in water is usually positive; when, on the other hand, they do not readily form hydrates, or when they are already hydrated, the heat of solution is usually negative. The following examples show the effect of hydration on heat of solution in a large quantity of water: § io. Application of the Second Law of Thermodynamics to Thermochemistry. - What is commonly understood by thermochemistry is based entirely on the first law of thermodynamics, but of recent years great progress has been made in the study of chemical equilibrium by the application of the second law. For an account of work in this direction see Chemical Action.
BIBLIOGRAPHY. - Julius Thomsen, Thermochemische Untersuchungen (Leipzig, 1882-86); M. Berthelot, Essai de Mecanique Chimique fondee sur la Thermochimie (Paris, 1879); Thermochimie, donnees et lois numeriques (Paris, 1897); W. Ostwald, Lehrbuch der allgemeinen Chemie, 2nd ed., vol. ii. part I, pp. 1-517 (Leipzig, 1893); M. M. P. Muir and D. M. Wilson, Elements of Thermal Chemistry (London, 1885); P. Duhem, Traite de Mecanique Chimique (Paris, 18 97-99); J. J. van Laar, Lehrbuch der mathematischen Chemie (Leipzig, 1901). (J. WAL.)
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