Treatise on Thermodynamics (1903) — part 2 of 14

Qualitatively, all substances conform to these regularities, but the values of the constants differ widely.

§ 31. Kegarding the transition from the liquid to the solid state, the same considerations hold as for that from the gaseous to the liquid state. The system of isotherms might be drawn for this process, and it is probable that tlieoretical regions and a critical point would be verified here also, if the means of experimental investigation were adequate. A continuous passage from the liquid to the solid state would then become possible along a path inter- secting the critical isotherm on either side of the critical

• Obtained as follows : —

Re _ _o

^ ~ ,7^r«i e{v + h)- ^ -^

Kdvje (^v-af^eiv + hf-'' ■ ■ ■ ■ ^^) fd'P\ - 2Re 6c^ _

W>~("-")' eiv + by-'' ^ ^

From (2) ami (3), v = 3a +2b (-4)

Substituting (4) in (2) and reducing, we get

And substituting (4) and (5)i(^l) and reducing, we have

°^ , , (6) Tr.

216(a + by ^ '^

20 THERMODYNAMICS.

point. In fact, there are certain substances which under ordinary pressures pass without appreciable discontinuity from the solid to the liquid state (pitch, glass, etc.), while others possess for a definite temj)erature a definite pressure of liquefaction or pressure of solidification, at which the substance splits into two portions of different densities. The pressure of liquefaction, however, varies with temperature at a much greater rate than the pressure of the saturated vapour. This view is physically justified, in particular by the experiments of Barus and Spring, in which the pressures were varied within wide limits.

In its most complete form the characteristic equation would comprise the gaseous, liquid, and solid states simul- taneously. No formula of such generality, however, has as yet been established for any substance.

§ 32. Mixtures. — While, as shown in § 19, the charac- teristic equation of a mixture of perfect gases reduces in a simple manner to that of its components, no such simpli- fication takes place, in general, when substances of any kind are mixed. Only for gases and vapours does Dalton's law hold, at least with great approximation, that the total pressure of a mixture is the sum of the partial pressures which each gas would exert if it alone filled the total volume at the given temperature. This law enables us to establish the characteristic equation of any gas mixture, provided that of the constituent gases be known. It also decides the question, unanswered in § 17, whether to the individual gases of a mixture common pressure and different volumes, or common volume and different pressures, should be pscribed. From the consideration of a vapour differing widely from an ideal gas, it follows that the latter of these views is the only one admissible. Take, for instance, atmo- spheric air and water vapour at O'' C. under atmospheric pressure. Here the water vapour cannot be supposed to be subject to a pressure of 1 atm., since at 0^ C. no Mater vapour exists at this pressure. The only choice remaining is to assign to the air and water vapour a common volume

TEMPERATURE. 21

(that of the mixture) and different pressures (partial pressures).

For mixtures of solid and liquid substances no law of general validity has beeii found, that reduces the characteristic equation of the mixture to those of its constituents.

CHAPTER II.

MOLECULAR WEIGHT.

§ 33. In the preceding chapter only snch physical changes have been discussed as concern temperature, pressure, and density. The chemical constitution of the substance or mixture in question has been left untouched. Cases are frequent, however (much more so, in fact, than was formerly supposed) in which the chemical nature of a substance is altered by a change of temperature or pressure. The more recent development of thermodynamics has clearly brought out the necessity of establishing a fundamental difference between physical and chemical changes such as will exclude continuous transition from the one kind to the other {cf. § 42, et seq., and § 238). It has, however, as yet not been possible to establish a practical criterion for distinguishing them, applicable to all cases. However strikingly most chemical processes differ from physical ones in their violence, suddenness, develoj)ment of heat, changes of colour and other properties, yet there are, on the other hand, numerous changes of a chemical nature that take place with continuity and comparative slowness; for example, dissociation. One of the main tasks of physical chemistry in the near future will be the further elucidation of this essential difference.*

§ 34. Experience shows that all chemical reactions take place according to constant proportions by weight. A

• In a word, we may, in a certain sense, say, that physical changes take place continuously, chemical ones, on the otlicr hand, discontinuously. In consequence, the science of physics deals, primarily, with continuously vary- ing numbers, the science of chemistry, on the contrary, with whole, or witli simple rational numbers.

MOLECULAR WEIGHT. 23

certain weight (strictly speaking, a mass) may therefore be used as a characteristic expression for the nature of a given chemically homogeneous substance, whether an element or a compound. Such a weight is called an equiva- lent weight. It is arbitrarily fixed for one element — generally for hydrogen at 1 gr. — and then the equivalent weight of any other element {e.g. oxygen) is that weight which will combine with 1 gr. of hydrogen. The weight of the compound thus formed is, at the same time, its equiva- lent weight. By proceeding in this way, the equivalent weights of all chemically homogeneous substances may be found. The equivalent weights of elements that do not combine directly with hydrogen can easily be determined, since in every case a number of elements can be found that combine directly with the element in question and also with hydrogen.

The total weight of a body divided by its equivalent weight is called the number of equivalents contained in the body. Hence we may say that, in every chemical reaction, an equal number of equivalents of the different substances react with one another.

§ 35. There is, however, some ambiguity in the above definition, since two elements frequently combine in more ways than one. For such cases there would exist several values of the equivalent weight. Experience shows, how- ever, that the various possible values are always simple multiples or submultiples of any one of them. The ambiguity in the equivalent weight, therefore, reduces itself to multiplying or dividing that quantity by a simple integer. We must accordingly generalize the foregoing statement, that an equal number of equivalents react with one another, and say, that the number of equivalents that react with one another are in simple numerical proportions. Thus 16 parts by weight of oxygen combine with 28 parts by weight of nitrogen to form nitrous oxide, or with 14 parts to form nitric oxide, or with 9^ parts to form nitrous anhydride, or with 7 parts to form nitrogen tetroxide, or with 5§ parts to

24 THERMOD YNA MICS.

form nitric anhydride. Any one of these numbers may be assigned to nitrogen as its equivalent weight, if 16 be taken as that of oxygen. They are in simple rational proportions, since

28 : 14 : 91 : 7 : 55 = 60 : 30 : 20 : 15 : 12.

§ 36. The ambiguity in the definition of the equivalent weight of nitrogen, exemplified by the above series of numbers, is removed by selecting a particular one of them to denote the molecular weight of nitrogen. In the definition of the molecular weight as a quite definite quantity depend- ing only on the particular state of a substance, and independent of possible chemical reactions with other sub- stances, lies one of the most important and most fruitful achievements of theoretical chemistry. Its exact statement can at present be given only for special cases, viz. for perfect gases and dilute solutions. We need consider only the former of these, as we shall see from thermodynamics that the latter is also thereby determined.

The definition of the molecular weight for a chemically homogeneous perfect gas is rendered possible by the further empirical law, that gases combine, not only in simple multiples of their equivalents, but also, at the same tempera- ture and pressure, in simple volume proportions (Gay-Lussac). It immediately follows that the number of equivalents, con- tained in equal volumes of different gases, must bear simple ratios to one another. The values of these ratios, however, are subject to the above-mentioned ambiguity in the selection of the equivalent weight. The ambiguity is, however, removed by putting all these ratios = 1, i.e. by establishing the condition that equal volumes of different gases shall contain an equal number of equivalents. Thus a definite choice is made from the different possible values, and a definite equivalent weight obtained for the gas, which is henceforth denoted as the molecidar weiglit of the gas. At the same time the number of equivalents in a quantity of the gas, which may be found by dividing the total weight by the molecular weight, is defined as the nwmher of

MOLECULAR WEIGHT. 25

moleeuhs contained in that quantity. Hence, equal volumes of perfect gases at the same temperature and pressure contain an equal number of molecules (Avogadro's law). The mole- cular weights of chemically homogeneous gases are, therefore, directly proportional to the masses contained in equal volumes, i.e. to the densities. The ratio of the densities is equal to the ratio of the molecular weights.

§ 37. Putting the molecular weight of hydrogen = m^, that of any other chemically homogeneous gas must be equal to mo multiplied by its specific density relative to hydrogen (§ 11). The following table gives the specific densities relative to hydrogen, and the molecular weights of several gases : —

Specific Density. Molecular Weight.

Hydrogen 10 m^

Oxygen 160 .... 160 m^

Nitrogen 140 .... 140 «?<,

Water vapour .... 9-0 ... . 9-0 m^ Ammonia 8'5 85 m^

Now, since water vapour consists of 1 part by weight of hydrogen and 8 parts by weight of oxygen, the molecule of water vapour, 9 «io, must consist of Wo parts by weight of hydrogen and 8 mo parts by weight of oxygen — i.e., according to the above table, of one molecule of hydrogen and half a molecule of oxygen. In the same manner ammonia, accord- ing to analysis, consisting of 1 part by weight of hydrogen and 4| parts by weight of nitrogen, its molecule 8*5 mo must necessarily contain 1-5 mo parts by weight of hydrogen and 7 mo parts by weight of nitrogen — i.e., according to the table, 1^ molecules of hydrogen and ^ molecule of nitrogen. Thus Avogadro's law enables us to give in quite definite numbers the molecular quantities of each constituent present in the molecule of any chemically homogeneous gas, pro- vided we know its density and its chemical composition.

§ 38. The smallest weight of a chemical element entering into the molecules of its compounds is called an atom.

26 THERM OD YNA MFCS.

Hence half a molecule of hydrogen is called an atom of hydrogen, H ; similarly, half a molecule of oxygen an atom of oxygen, ; and half a molecule of nitrogen an atom of nitrogen, N. The diatomic molecules of these substances are represented by H2, O2, No. An atom of mercury, on the contrary, is equal to a whole molecule, because in the mole- cules of its compounds no fractions of the molecular weight of mercury vapour occurs. It is usual to put the atomic weight of hydrogen H = 1. Then its molecular weight becomes Hg = ;»o = 2, and the molecular weights of our table become :

Molecular Weight.

Hydrogen 2 = Hj

Oxygen 32 = Oj

Nitrogen 28 = N,

Water vapour 18 = H.O

Ammonia 17 = HjN

§ 39. In general, then, the molecular weight of a chemically homogeneous gas is twice its density relative to hydrogen. Conversely, the molecular weight, m, of a gas being known, its specific density, and consequently the constant C in the characteristic equation (.5), can be calcu- lated. Denoting all quantities referring to hydrogen by the suffix 0, we have, at any temperature and pressure, for hydrogen,

for any other gas at the same temperature and pres- sure,

.'. C : Co = — : - = mo : m,

Vo V

« = "^ on)

MOLECULAR WEIGHT, 27

Now Wo = 2, and Co is to be calculated from the density of hydrogen at 0° C. and atmospheric pressure (§ 11).

Since - = 0-00008988, « = 1013650, = 273,

. p _ moCo _ Wo |JVo 2 . 101 3650 _ 82600000

• m ~ m*0 ~m. 273. 0-00008988 ~ m

Putting, for shortness, 82600000 = R, the characteristic equation of a chemically homogeneous perfect gas of mole- cular weight ni becomes

f = , (14)

where R, being independent of the nature of the individual gas, is generally called the absolute gas constant. The molecular weight may be deduced directly from the charac- teristic equation by the aid of the constant R, since

m = ^ (15)

V

Since v = ^' we have

M

But — is the quantity defined above as the number of

. M

molecules in the eras, and, therefore, if — = «,

to ' ' m

V = — "n, p

which means that at a given temperature and pressure the volume of a quantity of gas depends only on the number of the molecules present, and not at all on the nature of the gas.

§ 40. In a mixture of chemically homogeneous gases of

28 THERM OD YNA MICS.

molecular weights iiii, m^, . . . the relation between the partial pressures is, according to (9),

Ih' Ih' ' ' ' = CiMi : C2M2 . . .

But in (15) we have Ci = — ; C2 = — ; . . .

Ml Ma

^ ^ mi 1112

i.e. the ratio of the partial pressures is also the ratio of the number of molecules of each gas present. Equation (10) gives for the total volume

^ ^ (CiMi + C2M2 + ...)e

p

^ m Ml m

p \mi m.2

•■)

= „ 0^1 +112+ ..

•)

= — n ....

P

(16)

The volume of the mixture is therefore determined by the total number of the molecules present, just as in the case of a chemically homogeneous gas.

§ 41. It is evident that we cannot speak of the molecular weight of a mixture. Its apparent molecular weight, how- ever, may be defined as the molecular weight which a chemically homogeneous gas would have if it contained in the same mass the same number of molecules as the mixture. If we denote the apparent molecular weight by m, we have

Ml + 312 + . . . ^ Ml _^ M2 ^

m mi TO2 * ' "

Ml + :\r2 + . . .

and m =

Ml M2 wii m^

MOLECULAR WEIGHT. 29

The apparent molecular weight of air may thus be calcu- lated. Since

wji = O2 = 32 ; ^2 = N2 = 28 ; Mi : Ma = 0-3

we have m = tt-s ^ = 28*8,

32 "^28

which is somewhat larger than the molecular weight of nitrogen.

§ 42. The characteristic equation of a perfect gas, whether chemically homogeneous or not, gives, according to (16), the total number of molecules, but yields no means of deciding whether or not these molecules are all of the same kind. In order to answer this question, other methods must be resorted to, none of which, however, is practically applicable to all cases. A decision is often reached by an observation of the process of diifusion through a porous or, better, a semi- permeable membrane. The individual gases of a mixture will separate from each other by virtue of" the differences in their velocities of diffusion, which may even sink to zero in the case of semi-permeable membranes, and thus disclose the inhomogeneity of the substance. The chemical consti- tution of a gas may often be inferred from the manner in which it originated. It is by means of the expression for the entropy (§ 237) that we first arrive at a fundamental definition for a chemically homogeneous gas.

§ 43. Should a gas or vapour not obey tbe laws of perfect gases, or, in other words, should its specific density depend on the temperature or the pressure, Avogadro's definition of molecular weight is nevertheless applicable. The number of molecules in this case, instead of being a constant, will be dependent upon the momentary physical condition of the substance. We may, in such cases, either assume the number of molecules to be variable, or refrain from applying Avoga- dro's definition of the number of molecules. In other words, the cause for the deviation from the ideal state may be

30 THE R MOD YNA MICS.

sought for either in the chemical or physical conditions. The latter view preserA-es the chemical nature of the gas. The molecules remain intact under changes of temperature and pressure, but the characteristic equation is more complicated than that of Boyle and Gay-Lussac — like that, for example, of van der AVaals or of Clausius. The other view differs essentially from this, in that it represents any gas, not obey- ing the laws of perfect gases, as a mixture of various kinds of molecules (in nitrogen peroxide N2O4 and NO^, in j)hos- phorus pentachloride PCI5, PCI3, and CI2). The volume of these is supposed to have at every moment the exact value theoretically required for the total number of molecules of the mixture of these gases. The volume, however, does not vary with temperature and pressure in the same way as that of a perfect gas, because chemical reactions take place between the different kinds of molecules, continuously alter- ing the number of each kind present, and thereby also the total number of molecules in the mixture. This hypothesis has proved fruitful in cases of great differences of density — • so-called abnormal vapour densities — especially where, be- yond a certain range of temperature or pressure, the specific density once more becomes constant. When this is the case, the chemical reaction has been completed, and for this reason the molecules henceforth remain unchanged. Hydro- bromamylene, for instance, acts like a perfect gas below 160"^ and above 360"^, but shows only half its former density at the latter temperature. The doubling of the number of molecules corresponds to the equation

CsHuBr = C5H10 + HBr.

Mere insignificant deviations from the laws of jierfect gases are generally attributed to physical causes — as, e.g., in water vapour and carbon dioxide — and are regarded as the forerunners of condensation. The separation of chemical from physical actions by a principle which would lead to a more perfect definition of molecular weight for variable vapour densities, cannot be accomplished at the present time. The increase in the specific density which many

MOLECULAR WEIGHT. 31

vapours exhibit near their poiut of condensation might just as well be attributed to such chemical phenomena as the formation of double or multiple molecules. In fact, diiferences of opinion exist in a number of such cases. The molecular weight of sulphur vapour below 800°, for instance, is generally assumed to be Se = 192 ; but some assume a mixture of molecules Sg = 256 and S2 = 64, and others still different mixtures. In doubtful cases it is safest, in general, to leave this question open, and to admit both chemical and physical changes as causes for the deviations from the laws of perfect gases. This much, however, may be affirmed, that for small densities the physical influences will be of far less moment than the chemical ones, for, according to experience, all gases approach the ideal condition as their densities decrease (§ 21). This is an important point, which we will make use of later.

CHxiPTER III.

QUANTITY OF HEAT.

§ 44. If we plunge a piece of iron and a piece of lead, both of equal weight and at the same temperature (100^ C), into two precisely similar vessels containing equal quantities of water at 0"" C, we find that, after thermal equilibrium has been established iu each case, the vessel containing the iron has increased in temperature much more than that contain- ing the lead. Conversely, a quantity of water at 100° is cooled to a much lower temperature by a piece of iron at 0", than by an equal weight of lead at the same temperature. This phenomenon leads to a distinction between temperature and quantity of heat. As a measure of the heat given out or received by a body, we take the increase or decrease of temperature which some normal substance {e.g. water) under- goes when it alone is in contact with the body, provided all other causes of change of temperature (as compression, etc.) are excluded. The quantity of heat given out by the body is assumed to be equal to that received by the normal sub- stance, and vice versa. The experiment described above proves, then, that a piece of iron in cooling through a given interval of temperature gives out more heat than an equal weight of lead (about four times as much), and conversely, that, in order to bring about a certain increase of tempera- ture, iron requires a correspondingly larger supply of heat than lead.

§ 45. It was, in general, customary to take as the unit of heat that quantity which must be added to 1 gr. of water to raise its temperature from 0^ C. to 1° C. (zero

QUANTITY OF HEAT. 33

calorie). This is almost equal to the quantity of heat which will raise 1 gr. of water V C. at auy temperature. The refinement of calorimetric measurements has since made it necessary to take account of the initial temperature of the water, and it is often found convenient to define the calorie as that quantity of heat which will raise 1 gr. of water of mean laboratory temperature (15' to 20") 1 degree of the Centigrade scale. This laboratory calorie is about

I'OOfi ^^ ^ ^^^° calorie. Finally, a mean calorie has been

introduced, namely, the hundredth part of the heat required to raise 1 gr. of water from 0° C. to 100° C. The mean calorie is about equal to the zero calorie. Besides these so-called small calories, there are a corresponding number of large or kilogram calories, which contain 1000 small calories.

§ 46. The ratio of Q, the quantity of heat each gram of a substance receives, to Ad, the corresponding increase of temperature, is called the mean specifie heat, or mean heat capacity of 1 gr. of the substance between the initial and final temperatures of the process —

Hence, the mean heat capacity of water between 0° and 1° is equal to one zero calorie.

Passing to infinitely small differences of temperature, the specific heat of a substance, at the temperature B, becomes

^-c

This, in general, varies with temperature, but very slowly for most substances. It is usually permissible to put the specific heat at a certain temperature equal to the mean specific heat of an adjoining interval of moderate size.

§ 47. The heat capacity of solids and liquids is very

D

34 THERMODYNAMICS,

nearly independent of any variations of external pressure that may take place during the process of heating. Hence the definition of the heat capacity is not, usually, encumbered with a condition regarding pressure. The specific heat of gases, however, is influenced considerably by the conditions of the heating process. In this case the definition of specific heat would, therefore, be incomplete without some statement as to the accompanying conditions. Neverthe- less, we speak of the specific heat of a gas, without further specification, when we mean its specific heat at constant (atmospheric) pressure, as this is the value most readily determined.

§ 48. That the heat capacities of different substances should be referred to unit mass is quite arbitrary. It arises from the fact that quantities of matter can be most easily compared by weighing them. Heat capacity might, quite as well, be referred to unit volume. It is more rational to compare masses which are proportional to the molecular and atomic weights of substances, for then certain regu- larities at once become manifest. The corresjjonding heat capacities are obtained by multiplying the specific heats (per unit mass) by the molecular or atomic weights. The values thus obtained are known as the molecular or atomic heats.

§ 49. The chemical elements, especially those of high atomic weight, are found to have nearly the constant atomic heat of 6*4 (Dulong and Petit). It cannot be claimed that this law is rigorously true, since the heat capacity depends on the molecular constitution, as in the case of carbon, and on the state of aggregation, as in the case of mercury, as well as on the temperature. The effect of temperature is especially marked in the elements, carbon, boron, and silicon, which show the largest deviations from Dulong and Petit's law. The conclusion is, however, justified, that Dulong and Petit's law is founded on some more general law of nature, which has not yet been formulated.

■ QUANTITY OF HEAT. 35

§ 50. Similar regularities, as appear in the atomic heats of elements, are also found in the molecular heats of compounds, especially with compounds of similar chemical constitution. According to F. Neumann's law, subsequently confirmed by Kegnault, compounds of similar constitution, when solid, have equal molecular heats. Joule and Woestyn further extended this law by showing that the molecular heat is merely the sum of the atomic heats, or that in any com- bination every element preserves its atomic heat, whether or not the latter be 6*4^ according to Dulong and Petit's law. This relation also is only approximately true.

§ 51. Since all calorimetric measurements, according to § 44, extend only to quantities of heat imparted to bodies or given out by them, they do not lead to any conclusion as to the total amount of heat contained in a body of given temperature. It would be absurd to define the heat con- tained in a body of given temperature, density, etc., as the number of calories absorbed by the body in its passage from some normal state into its present state, for the quantity thus defined would assume diflerent values according to the way in which the change was effected. A geis at 0"^ and atmospheric pressure can be brought to a state where its temperature is 100" and its pressure 10 atmospheres, either by heating to 100° under constant pressure, and then com- pressing at constant temperature; or by compressing isothermally to 10 atmospheres, and then heating isopie- stically to 100°; or, finally, by compressing and heating simultaneously or alternately in a variety of ways. The total number of calories absorbed would in each case be different (§ 77). It is seen, then, that it is useless to speak of a certain quantity of heat which must be applied to a body in a given state to bring it to some other state. If the " total heat contained in a body " is to be expressed numerically, as is done in the kinetic theory of heat, where the heat of a body is defined as the total energy of its internal motions, it must not be interpreted as the sum- total of the quantities of heat applied to the body. As we

36 THERM OD YNA MICS.

shall make no use of this quantity in our present work, no definition of it need be attempted.

§ 52. In contrast to the above representation of the facts, the older (Oarnot's) theory of heat, which started from the hypothesis that heat is an indestructible substance, neces- sarily reached the conclusion that the " heat contained in a body " depends solely on the number of calories absorbed or given out by it. The heating of a body by other means than direct application of heat, by compression or by friction for instance, according to that theory produces no change in the "total heat." To explain the rise of temperature which takes place notwithstanding, it was necessary to make the assumption that compression and friction so diminish the body's heat capacity, that the same amount of heat now produces a higher temperature, just as, for example, a moist sponge appears more moist if compressed, although the quantity of liquid in the sponge remains the same. In the meantime, Rumford and Davy proved by direct experi- ment that bodies, in which any amount of heat can be generated by an adequate expenditure of work, do not in the least alter their heat capacities with friction. Regnault, likewise, showed, by accurate measurements, that the lieat capacity of gases is independent of or only very slightly dependent on volume ; that it cannot, therefore, diminish, in consequence of compression, as much as Carnot's theory would require. Finally, AV. Thomson and Joule have demonstrated by careful experiments that a gas, when ex- panding without overcoming external pressure, undergoes no change of temperature, or an exceedingly small one (c/- § '^0)5 so that the cooling of gases generally observed when they expand is not due to the increase of volume per se, but to the work done in the expansion. Each one of these experimental results would by itseK be sufficient to disprove the hypothesis of the indestructibility of heat, and to overthrow the older theory.

§ 53. While, in general, the heat capacity varies con- tinuously with temperature, every substance possesses,

QUANTITY OF HEAT. 37

under certain external pressures, so-called singular values of temperature, for which the heat capacity, together with other properties, is discontinuous. At such tempera- tures the heat absorbed no longer affects the entire body, but only one of the parts into which it has split ; and it no longer serves to increase the temperature, but simply to alter the state of aggregation, i.e. to melt, evaporate, or • sublime. Only when the entire substance has again become homogeneous will the heat imparted produce a rise in temperature, and then the heat capacity becomes once mora capable of definition. The quantity of heat necessary to change 1 gram of a substance from one state of aggregation to another is called the latent heat, in particular, the heat of fusion, of vajwrization, or of suUimation. The same amount of heat is set free when the substance returns to its former state of aggregation. Latent heat, as in the case of specific heat, is best referred, not to unit mass, but to molecular or atomic weight. Its amount largely depends on the external conditions under which the process is carried out (§ 47), constant pressure being the most important condition.

§ 54. Like the changes of the state of aggregation, all processes involving mixture, or solution, and all chemical reactions are accompanied by an evolution of heat of greater or less amount, which varies according to the external con- ditions. This we shall henceforth designate as the heat effect (Warmetonung) of the process under consideration, in particular as the heat of mixture, of solution, of combina- tion, of dissociation, etc. It is reckoned i^ositive when heat is set free or developed, i.e. given out by the body (exo- thermal processes) ; negative, when heat is absorbed, or rendered latent, i.e. taken up by the body (endothermal processes).

.^^

PART II.

The First Fundamental Principle of Thermodynamics.

CHAPTEE T.

GENERAL EXPOSITION.

§ 55. The jirst law of thermodynamics is nothing more than the principle of the conservation of energy applied to phenomena involving the production or absorption of heat. Two ways lead to a deductive proof of this principle. AVe may take for granted the correctness of the mechanical view of nature, and assume that all changes in nature can be reduced to motions of material points between which there act forces which have a potential. Then the principle of energy is simply the well-known mechanical theorem of kinetic energy, generalized to include all natural processes. Or we may, as is done in this work, leave open the question concerning the possibility of reducing all natural processes to those of motion, and start from the fact which has been tested by centuries of human experience, and repeatedly verified^iz. that it is in no way possible, either hy mechanical, thermal, chemical, or other devices, to obtain perpetual motion, i.e. it is impossible to construct an engine which will work in a cycle and produce continuous work, or kinetic energy, from nothing. We shall not attempt to show how this single fact of experience, quite independent of the mechanical view of nature, serves to prove the principle of energy in its generality, mainly for the reason that the validity of the energy principle is. nowadays no longer disputed. It

GENERAL EXPOSITION. 39

will be different, however, in the case of tlie second law of thermodynamics, the proof of which, at the present stage of the development of our subject, cannot be too carefully- presented. The general validity of this law is still con- tested from time to time, and its significance variously interpreted, even by the adherents of the principle.

§ 56. The energy of a body, or system of bodies, is a magnitude depending on the momentary condition of the system. In order to arrive at a definite numerical expression for the energy of the system in a given state, it is necessary to fix upon a certain normal arbitrarily selected state (e.g. 0° C. and atmospheric pressure). The energy of the system in a given state, referred to the arbitrarily selected normal state, is then equal to the alge- hraie sum of the meehameal equivalents of all the effects produced outside the system when it passes in any way from the given to the normal state. The energy of a system is, therefore, sometimes briefly denoted as the faculty to produce external effects. Whether or not the energy of a system assumes different values according as the transition from the given to the normal state is accomplished in different ways is not implied in the above definition. It will be necessary, however, for the sake of completeness, to explain the term " mechanical equivalent of an external effect."

§ 57. Should the external effect be mechanical in nature — should it consist, e.g., in lifting a weight, overcoming atmospheric pressure, or producing kinetic energy — then its mechanical equivalent is simply equal to the mechanical work done by the system on the external body (weight, atmosphere, projectile). It is positive if the displacement take place in the direction of the force exercised by the system — when the weight is lifted, the atmosphere pushed back, the projectile discharged, — negative in the opposite sense.

But if the external effect be thermal in nature — if it consist, e.g., in heating surrounding bodies (the atmosphere,

40 THERMOD YNA MICS.

a calorimetric liquid, etc.) — then its mechanical equivalent is equal to the number of calories which will produce the same rise of temperature in the surrounding bodies multiplied by an absolute constant, which depends only on the units of heat and mechanical work, the so-called meclianical equivalent of heat. This proposition, which appears here only as a definition, receives through the principle of the conservation of energy a physical meaning, which may be put to experimental test.

§ 58. The Principle of the Conservation of Energy asserts, generally and exclusively, that the energy of a system in a given state,, referred to a fixed normal state, has a quite drfferenlr value ; in other words — substituting the definition given in § 56 — that the algebraic sum of the mechanical equivalents of the external effects produced outside the system, when it passes from the given to the normal state, is independent of the manner of the trans- formation. On passing into the normal state the system thus produces a definite total of effects, as measured in mechanical units, and it is this sum — the "work- value" of the external effects — that represents the energy of the system in the given state.

§ 59. The validity of the principle of the conservation of energy may be experimentally verified by transferring a system in various ways from a given state to a certain other state, which may here be designated as the normal state, and measuring the mechanical equivalents of all external effects in each case. Special care must be taken, however, that the initial state of the system is the same each time, and that none of the external effects is overlooked or taken into account more than once.

§ 60. As a first application we shall discuss Joule's famous experiments, in which the external effects produced by weights falling from a certain height were compared, first, when performing only mechanical work [e.g. lifting a load), and second, when by suitable contrivances generating heat

GENERAL EXPOSITION. ^\

by friction. The initial and final position of the weights may be taken as the two states of the system, the work or heat produced, as the external effects. The first case, where the weights produce only mechanical work, is simple, and requires no experiment. Its mechanical equivalent is the product of the sum of the weights, and the height through which they fall. The second case requires accurate measurement of the increase of temperature, which the surrounding substances (water, mercury) undergo in conse- quence of the friction, as well as of their heat capacities, for the determination of the number of calories which will produce in them the same rise of temperature. It is, of course, entirely immaterial what our views may be with regard to the details of the frictional generation of heat, or with regard to the ultimate form of the heat thus generated. The only point of importance is that the state produced in the liquid by friction is identical with a state produced by the absorption of a definite number of calories.

Joule, by equating the mechanical work, corresponding to the fall of the weights, to the mechanical equivalent of the heat produced by friction, showed that the mechanical equivalent of a gram-calorie is, under all circumstances, equal to the work done in lifting a weight of a gram through a height of 423-55 meters. That all his experiments with different weights, different calorimetric substances, and different temperatures, led to the same value, goes to prove the correctness of the principle of the conservation of energy.

§ 61. In order to determine the mechanical equivalent of heat in absolute units, we must bear in mind that Joule's result refers to laboratory calories (§ 45), and the readings of a mercury thermometer. At the temperature of the laboratory, Y of the mercury thermometer represents about

iWr *^^ ^° ^^ ^^^^ ^^^ thermometer. A calorie referred to the gas thermometer has, therefore, a mechanical equivalent of 423-55 X 1-007 = 427.

42

THE R MOD YNA MlCS.

The acceleration of gravity must also be considered, since raising a gram to a certain height represents, in general, different amounts of work in different latitudes. The absolute value of the work done is obtained by multi- plying the weight, i.e. the product of the mass and the acceleration of gravity, by the height of fall. The follow- ing table gives the mechanical equivalent of heat in tlie different calories : —

Unit of heat referred to gas thennometer.

Corresponding height in

meters to which 1 gr.

must be raised in places

of mean latitude.

Absolute value of the mechanical equi- valent (C.G.S. system, erg).

Laboratory calorie , Zero calorie . . .

427

430

419 X 10* 422 X 10*