Treatise on Thermodynamics (1903) — part 1 of 14

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TREATISE ON THERMODYNAMICS

TREATISE

ON

THERMODYNAMICS

BY

Dr. max PLANCK

PROFESSOR OF THEORETICAL PHYSICS IN THE UNIVERSITY OF BERLIN

TRANSLATED WITH THE AUTHORS SANCTION

ALEXANDER OGG, MA., B.Sc, Ph.D.

LATE 1851 EXHIBITION SCHOLAR AND UNIVERSITY ASSISTANT, ABERDEEN UNIVERSITY ASSISTANT .MASTER, ROYAL NAVAL ENGINEERING COLLEGE, DEVONPORT

LONGMANS, GREEN, AND CO.

39 PATERNOSTER ROW, LONDON

NEW YORK AND BOMBAY

1903

All rights resemed

QC.

311

TRANSLATOR'S NOTICE

The modern developments of Thermodynamics, and the applications to physical and chemical problems, have become so important, that I have ventured to translate Professor Planck's book, which presents the whole subject from a uniform point of view.

A few notes have been added to the present English edition by Professor Planck. He has not found it neces- sary to change the original text in any way.

To bring the notation into conformity with the usual English notation, several symbols have been changed. This has been done with the author's sanction. Here I have followed J. J. van Laar and taken ^ to signify what he calls the Planck' sches Potential, i.e. the thermodynamic potential of Gibbs and Duhem divided by — 6.

Professor Planck's recent paper, "tjber die Grundlage der Losungstheorie " {A7m. d. Phys. 10, p. 436, 1903), ought to be read in connection with his thermodynamical theory of solution.

I am indebted to Herren Veit & Co., Leipzig, for kindly supplying the blocks of the five figures in the text.

A. 0.

Devospokt,

June, 1903.

a 3

PREFACE

The oft-repeated requests either to publish my collected papers on Thermodynamics, or to work them up into a comprehensive treatise, first suggested the writing of this book. Although the first plan would have been the simpler, especially as I found no occasion to make any important changes in the line of thought of my original papers, yet I decided to rewrite the whole subject-matter, with the intention of giving at greater length, and with more detail, certain general considerations and demonstra- tions too concisely expressed in these papers. My chief reason, however, was that an opportunity was thus offered of presenting the entire field of Thermodynamics from a uniform point of view. This, to be sure, deprives the work of the character of an original contribution to science, and stamps it rather as an introductory text-book on Thermo- dynamics for students who have taken elementary courses in Physics and Chemistry, and are familiar with the elements of the Differential and Integral Calculus.

Still, I do not think that this book will entirely super- sede my former publications on the same subject. Apart from the fact that these contain, in a sense, a more original presentation, there may be found in them a number of details expanded at greater length than seemed advisable in the more comprehensive treatment here required. To

viii PREFACE.

enable the reader to revert in particular cases to the original form for comparison, a list of my publications on Thermodynamics ha^ been appended, with a reference in each case to the section of the book which deals with the same point.

The numerical values in the examples, which have been worked as applications of the theory, have, almost all of them, been taken from the original papers ; only a few, that have been determined by frequent measurement, have been taken from the tables in Kohlrausch's " Leitfaden der praktischen Physik." It should be emphasized, however, that the numbers used, notwithstanding the care taken, have not undergone the same amount of critical sifting: as the more general propositions and deductions.

Three distinct methods of investigation may be clearly recognized in the previous development of Thermodynamics. The first penetrates deepest into the nature of the processes considered, and, were it possible to carry it ont exactly, would be designated as the most perfect. Heat, according to it, is due to the definite motions of the chemical molecules and atoms considered as distinct masses, which in the case of gases possess comparatively simple properties, but in the case of solids and liquids can Ije only very roughly sketched. This kinetic theory, founded by Joule, Waterston, Kronig and Clausius, has been greatly extended mainly by Maxwell and r»oltzmann. Obstacles, at present unsurmountable, however, seem to stand in the way of its further progress. These are due not only to the highly complicated mathematical treatment, but principally to essential difficulties, not to be discussed here, in the mechanical interpretation of the fundamental principles of Thermodynamics.

PREFACE. ix

Such difficulties are avoided by the second method, developed by Helmholtz. It confines itself to the most important hypothesis of the mechanical theory of heat, that heat is due to motion, but refuses on principle to specialize as to the character of this motion. This is a safer point of view than the first, and philosophically quite as satisfactory as the mechanical interpretation of nature in general, but it does not as yet offer a foundation of sufficient breadth upon which to build a detailed theory. Starting from this point of view, all that can be obtained is the verification of some general laws which have already l)een deduced in other ways direct from experience.

A third treatment of Thermodynamics has hitherto proved the most fruitful. This method is distinct from the other two, in that it does not advance the mechanical theory of heat, but, keeping aloof from definite assump- tions as to its nature, starts direct from a few very general empirical facts, mainly the two fundamental principles of Thermodynamics. From these, by pure logical reasoning, a large number of new physical and chemical laws are deduced, which are capable of extensive application, and have hitherto stood the test without exception.

This last, more inductive, treatment, which is used ex- clusively in this book, corresponds best to the present state of the science. It cannot be considered as final, however, but may have in time to yield to a mechanical, or perhaps an electro-magnetic theory. Although it may be of advan- tage for a time to consider the activities of nature — Heat, Motion, Electricity, etc. — as different in quality, and to suppress the question as to their common nature, still our aspiration after a uniform theory of nature, on a mechanical basis or otherwise, which has derived such powerful en- couragement from the discovery of the principle of the

X PREFACE.

conservation of energy, can never be permanently repressed. Even at the present day, a recession from the assumption that all pliysical phenomena are of a common nature would be tantamount to renouncing the comprehension of a number of recognized laws of interaction between different spheres of natural phenomena. Of course, even then, the results we have deduced from the two laws of Thermo- dynamics would not be invalidated, but these two laws would not be introduced as independent, but would be deduced from other more general propositions. At present, however, no probable limit can be set to the time which it will take to reach this goal.

THE AUTHOE.

I^KKLIN,

Aiyt'il, 1807.

CONTENTS

PART I. Fundamental Facts and Definitions

OIIAITER PAGK

I. Tkmpek.vturk 1

II. Molecular Weight 22

III. Quantity of Heat 32

PART II. The First Fundamental Principle of Thermodynamics

1. General Exposition 38

II. Applications to Homogeneous Systems 46

III. Applications to Nox- Homogeneous Systems 67

PART III. The Second Fundamental Principle of Thermodynamics

I. Introduction 77

II. Proof 86

III. General Deductions 105

CONTENTS.

PART IV.

Applications to Special States of Equilibrium

CHAPTKR PAGR

I. HOMOUKNKOUS SvSTKMS 119

II. Systkm is Differext States of Agorkgatiox .... 132

III. System of axv Nt^mbkr ot Isdepexdext Coxstitiexts . 17.3

lY. Gaseous System 207

V. Dilute Solltioxs 223

Catalogue of the Author's Papers ox Thermodyxamrs 2tU

Index 207

ERRATA

The reader is requested kindly to make the following corrections in the pages -where they occur : —

Page 26, line 8, for " occurs " read '* occur."

„ 30, „ 13 from bottom, for " Hydrobromamylene " read " Amy- lene hydrobromide."

„ 40, „ 4 in § 58, for " dififerent " read " definite."

„ 78, „ 5, for " restablishment " read " re-establishment."

„ 78, „ 14, /or "stakes" read "states."

„ 79, „ 9 in § 108, /or "Occasionally," etc., read "That attempts are still made to represent this law as contained in the principle of energy may be seen from the fact that the too restricted term ' Energetics ' is sometimes applied to all investigations on these questions."

„ 87, „ 2 in § 118, /or " heat " read " work."

„ 104, „ 10 from end, for " metaphysicists " read " metaphysicians."

„ 149, „ 3, for " V " read " v^."

„ 152, lines 3 and 1 from end, read ( ^) and ( ^ ) throughout.

„ 176, line 8, for « AM, " read " AM,'."

„ 177, equation (149), for " M^" " read " ^M^"."

„ 180, line 22, for " quintiple " read " quintuple."

„ 185, „ 2, for " dMj " read " dM,'."

„ 186, equation 153, /or dMj5^^^, read dM,'S^^,

„ 191, line 3, left-hand side of equation, for " d log p " read " d log p^."

„ 202, lines 6 and 8, for " (f> " read " <p."

„ 214, line 15, for " ni(c,2 " read " ?i,(c„,."

„ 229, „ 14, for " are " read " is."

„ 230, „ 14, for second minus read plug.

„ 232, „ 2 from end, for " equations " read " equation,"

„ 241, equation (225), for " 0^ " read " d-."

„ 241, line 5 from end, /or " carbonic " read " carbon."

„ 242, „ 8,/or "^2" reod"eV

„ 243, „ 11 from end, /or "molecule " read "gram molecule."

PLA.NCK'S THERMODY.NAMICS.

TREATISE

ON

THERMODYNAMICS

PART I.

Fundamental Facts and Definitions.

CHAPTER I.

TEMPERATURE.

\ 1. The conception of " heat " arises from that particular sensation of warmth or coldness which is immediately experienced on touching a body. This direct sensation, however, furnishes no quantitative scientiijc measure of a body's state with regard to heat ; it yields only qualitative results, which vary according to external circumstances. For quantitative purposes we utilize the change of volume which takes place in all bodies when heated under constant pressure, for this admits of exact measurement. Heating produces in most substances an increase of volume, and thus we can tell whether a body gets hotter or colder, not merely by the sense of touch, but also by a purely mechanical observation affording a much greater degree of accuracy. We can also tell accurately when a body assumes a former state of heat.

§ 2. If two bodies, one of which feels warmer than the other, be brought together (for example, a piece of heated metal and cold water), it is invariably found that the hotter body is cooled, and the colder one is heated up to a certain

B

2 THERMOD YNA MICS.

point, and then all change ceases. The two bodies are then said to be in thermal equilibrium. Experience shows that such a state of equilibrium finally sets in, not only when two, but also when any number of differently heated bodies are brought into mutual contact. From this follows the important proposition : If a body, A, be in thermal equili- brium with two other bodies, B and C, then B and C are in thermal equilibrium with one another* For, if we bring A, B, and C together so that each touches the other two, then, according to our supposition, there will be equilibrium at the points of contact AB and AC, and, therefore, also at the contact BC. If it were not so, no general thermal equili- brium would be possible, which is contrary to experience.

§ 3. These facts enable us to compare the degree of heat of two bodies, B and C, without bringing them into contact with one another; namely, by bringing each body into contact with an arbitrarily selected standard body, A (for example, a mass of mercury enclosed in a vessel terminating in a fine capillary tube). By observing the volume of A in each case, it is possible to tell whether B and C are in thermal equilibrium or not. If they are not in thermal equilibrium, we can tell which of the two is the hotter. The degree of heat of A, or of any body in thermal equilibrium with A, can thus be very simply defined by the volume of A, or, as is usual, by the difference between the volume of A and its volume when in thermal equilibrium with melting ice under atmospheric pressure. This volumetric difference, which, by an appropriate choice of unit, is made to read 100 when A is in contact with steam under atmospheric pressure, is called the temperature in degrees Centigrade with regard to A as thermometric substance. Two bodies of equal temperature are, therefore, in thermal equilibriujii, and vice versa.

• As is well known, there exists no corresponding proposition for electrical equilibrium. For if we join together the substances Cu | CuSO, aq. j ZnSO, aq. | Zn to form a conducting ring, no electrical equilibrium is |H)ssible.

TEMPERA TURE. 3

§ 4. The temperature readings of no two thermometric substances agree, in general, except at 0' and 100°. The definition of temperature is therefore somewhat arbitrary. This we may remedy to a certain extent by taking gases, in particular those hard to condense, such as hydrogen, oxygen, nitrogen, and carbon monoxide, as thermometric substances. They agree almost completely within a considerable range of temperature, and their readings are sufficiently in accord- ance for most purposes. Besides, the coefficient of expansion of these different gases is the same, inasmuch as equal volumes of them expand under constant pressure by the same amount — about ^fg of their volume — when heated from 0^ C. to 1° C. Since, also, the influence of the external pressure on the volume of these gases can be represented by a very simple law, we are led to the conclusion that these regularities are based on a remarkable simplicity in their constitution, and that, therefore, it is reasonable to define the common temperature given by them simply as tempera- ture. We must consequently reduce the readings of other thermometers to those of the gas thermometer, and prefer- ably to those of the hydrogen thermometer.

§ 5. The definition of temperature remains arbitrary in cases where the requirements of accuracy cannot be satisfied by the agreement between the readings of the different gas thermometers, for there is no sufficient reason for the preference of any one of these gases. A definition of tem- perature completely independent of the properties of any individual substance, and applicable to all stages of heat and cold, becomes first possible on the basis of the second latv of thermodynamics (§ 160, etc.). In the mean time, only such temperatures will be considered as are defined with sufficient accuracy by the gas thermometer.

§ 6. In the following we shall deal chiefly with homo- geneous, isotropic bodies of any form, possessing throughout their substance the same temperature and density, and subject to a uniform pressure acting everywhere perpen- dicular to the surface. They, therefore, also exert the same

4 THERMODYNAMICS

pressure outwards. Surface phenomena are thereby dis- regarded. The condition of such a body is determined by its chemical nature ; its mass, M ; its volume, V ; and its temperature, t. On these must depend, in a definite manner, all other properties of the particular state of the body, especially the pressure, which is uniform throughout, in- ternally and externally. The pressure, jp, is measured by the force acting on the unit of area — in the C.Gr.S. system, in dynes per square centimeter, a d^ijne being the force which imparts to a mass of one gramme in one second a velocity of one centimeter per second.

§ 7. As the pressure is generally given in atmospheres, the value of an atmosphere in absolute C.G.S. units is here calculated. The pressure of an atmosphere is the weight of a column of mercury at 0° C, 76 cm. high, and 1 sq. cm. in cross-section, when placed in mean geographical latitude. This latter condition must be added, because the weight, i.e. the force of the. earth's attraction, varies with the locality. The volume of the column of mercury is 76 c.c. ; and since the density of mercury at 0' C. is 13'596, the mass is 76 x 13'596. Multiplying the mass by the acceleration of gravity in mean latitude, we find the pressure of one atmosphere in absolute units to be

76 X 13-596 X 981 = 1,013,650 ^^ or — ^^-o-

cm.'' cm.-sec.'*

This, then, is the factor for converting atmospheres into absolute units. If, as was formerly the custom in mechanics, we use as the unit of force the weight of a gramme in mean geographical latitude instead of the dyne, the pressure of an atmosphere would be 76 X 13596 = 1033*3 grms. per square centimeter.

§ 8. Since the pressure in a given substance is evidently controlled by its internal physical condition only, and not by its form or mass, it follows that }> depends only on tlie temperature and the ratio of the mass M to the volume A'

TEMPERA TURE. 5

[i.e. the density), or on the reciprocal of the density, the volume of unit mass —

which is called the specific volume of the substance. For every substance, then, there exists a characteristic relation —

V = fM,

which is called the characterisiie equation of the substance. For gases, the function / is invariably positive ; for liquids and solids, however, it may have also negative values under certain circumstances.

§ 9. Perfect Gases. — The characteristic equation as- sumes its simplest form for the substances which we used in § 4 for the definition of temperature. If the temperature be kept constant, then, according to the Boyle-Mariotte law, the product of the pressure and the specific volume remains constant for gases —

P^ = T,. (1)

where T, for a given gas, depends only on the tempera- ture.

But if the pressure be kept constant, then, according to § 3, the temperature is proportional to the difierence between the present volume v and the volume Vq at 0° ; i.e. —

t={v-v,)V, (2)

where P depends only on the pressure p. Equation (1) for Vq becomes

VV, = To, (3)

where To is the value of the function T, when ^ = 0" C.

Finally, as has already been mentioned in § 4, the expansion of all permanent gases on heating from 0° 0. to 1^ C. is the same fraction a (about 273) of their volume at

6 THERMODYNAMICS.

0^ (Gay-Lussac's law). Patting i = 1, we have v - v^ - uVq, and equation (2) becomes

\ = av,V (4)

By eliminating P, Vo> and v from (1), (2), (3), (4), we obtain the temperature function of the gas —

T = To(l + at),

which is seen to be a linear function of t. The characteristic equation (1) becomes

§ 10. The form of this equation is considerably simplified by shifting the zero of temperature, arbitrarily fixed in § 3,

by - degrees, and calling the melting point of ice, not 0^ C,

but i' C. {Le. about 273° C). For, putting i + i =

(absolute temperature), and the constant aTo = C, the characteristic equation becomes

This introduction of ahsoJide temperature is evidently tanta- mount to measuring temperature no longer, as in § 3, by a change of volume, but by the volume itself.

§ 11. The constant C, which is characteristic for the perfect gas under consideration, can be calculated, if the specific volume f be known for any pair of values of and jj (e.g. 0" and 1 atmosphere). For different gases, taken at the same temperature and pressure, the constants C evidently vary directly as the specific volumes, or inversely as the

densities -. It may be affirmed, then, that, taken at the

same temperature and pressure, the densities of all perfect gases bear a constant ratio to one another. A gas is, therefore, often characterized by the constant ratio which its

TEMPERA TURE. 7

density bears to that of a normal gas at the same tempera- ture and pressure {specific density relative to air or hydrogen). Ki (f G. {B = 273°) and 1 atmosphere pressure, the densities of the following gases are :

Hydrogen 0-00008988-^

Oxygen 00014291

Nitrogen 0-0012507

Atmospheric nitrogen 0-0012571

Air 00012930

whence the corresponding values of C in absolute units can be readily calculated.

All questions with regard to the behaviour of a substance when subjected to changes of temperature, volume, and pressure are completely answered by the characteristic equation of the substance.

§ 12. Behaviour under Constant Pressure (Isopiestio or Isobaric Changes). — Coefficient of expansion is the name given to the ratio of the increase of volume for a rise of temperature of 1° C. to the volume at 0° C. This increase

CM

for a perfect gas is, according to (5), . The same equa- tion (5) gives the volume of the gas at 0° C. as x 273,

hence the ratio of the two quantities, or the coefficient of expansion, is ._, ] .; = a.

§ 13. Behaviour at Constant Volume (Isochoric or Isopycnic Changes). — The pressure coefficient is the ratio of the increase of pressure for a rise of temperature of 1" to the pressure at 0° C. For a perfect gas, this increase, accord-

c]\r CM

ing -to equation (5), is -^. The pressure at 0° C. is — y- x 273,

whence the required ratio, i.e. the pressure coefficient, is ^^j-o, therefore equal to the coefficient of expansion a.

§ 14. Behaviour at Constant Temperature (Isother- mal Changes). — Coefficient of elasticitij is the ratio of an

8 THERMO D YNA MICS.

infinitely small increase of pressure to the resulting con- traction of unit volume of the substance. In a perfect gas, according to equation (5), the contraction of volume V, in consequence of an increase of pressure dp, is

„, CM0, V,

— rf V = — s- dp = — dp. p^ -^ p "^

The contraction of unit volume is therefore

_dY _dp Y ~ p'

and the coefficient of elasticity of the gas is

dp dl=i''

V that is, equal to the pressure.

The reciprocal of the coefficient of elasticity, i.e. the ratio of an infinitely small contraction of unit volume to the corresponding increase of pressure, is called the coefficient of compressibility.

§ 15. The three coefficients which characterize the be- haviour of a substance subject to isopiestic, isopycnic, and isothermal changes are not independent of one another, but are in every case connected by a definite relation. The general characteristic equation, on being diflerentiated, gives

where the suffixes indicate the variables to be kept constant while performing the diiferentiation. By putting dp = Q we impose the condition of an isopiestic change, and obtain the relation between dv and dO in isopiestic processes : —

\dvX

TEMPERA TURE.

For every state of a substance, one of the three coefficients, viz. of expansion, of pressure, or of compres- sibility, may therefore be calculated from the other two.

Take, for example, mercury at 0° C. and under atmo- spheric pressure. Its coefficient of expansion is (§ 12)

i^-^, . i = 0-00018,

its coefficient of compressibility in atmospheres (§ 14) is

• (P) ■ - = 0-000003, therefore its pressure coefficient in atmospheres (§ 13) is

(^P\ = - (^P\ (^^\ - - ^Ep - M^OIS _ fio \de), \dv),'\W~ (dv\ ~ 0-000003 ~^'

i;

,^p/

This means that an increase of pressure of 60 atmospheres is required to keep the volume of mercury constant when heated from 0° C. to 1° C.

§ 16. Mixture of Perfect Gases. — If any quantities of the same gas at the same temperatures and pressures be at first separated by partitions, and then allowed to come suddenly in contact with another by the removal of these partitions, it is evident that the volume of the entire system will remain the same and be equal to the sum-total of the partial volumes. Starting with quantities of different gases, experience still shows that, when pressure and temperature are maintained uniform and constant, the total volume continues equal to the sum of the volumes of the con- stituents, notwithstanding the slow process of intermingling — diffusion — which takes place in this case. Diffusion goes on until the mixture has become at every point of precisely the same composition, i.e. physically homogeneous.

§ 17. Two views regarding the constitution of mixtures thus formed present themselves. Either we might assume

lo THERMODYNAMICS.

that the individual gases, while mixing, split into a large number of small portions, all retaining their original volumes and pressures, and that these small portions of the difterent gases, without penetrating each other, distribute themselves evenly throughout the entire space. In the end each gas would still retain its original volume (partial volume), and all the gases would have the same common pressure. Or, we might suppose — and this view will be shown below (§ 32) to be the correct one — that the individual gases change and interpenetrate in every infinitesimal portion of the volume, and that after diffusion each individual gas, in so far as one may speak of such, fills the total volume, and is consequently under a lower pressure than before diffusion. This so-called partial pressure of a constituent of a gas mixture can easily be calculated.

§ 18. Denoting the quantities referring to the individual gases by suffixes — and ^J requiring no special designation, as they are supposed to be the same for all the gases, — the characteristic equation (5) gives for each gas before diftusion

CiMiO C2M20

The total volume,

V = Vi + V2 + .

remains constant during diffusion. After difi'usion we as- cribe to each gas the total volume, and hence the partial pressures become

CiMifl Vi C2M20 V2

i^i = y- = yv\ vi = — -^ = v^'; • • . (0

and by addition

This is Ualtou's law, that in a homogeneous mixture of

TEMPERATURE. 1 1

gases the pressure is equal to the sum of the partial pressures of the gases. It is also evident that

IH -.p,: . . . = Vi : A^2 : . . . =CxMi : C^M^ . (9)

i.e. the partial pressures are proportional to the volumes of the gases before diffusion, or to the partial volumes which the gases would have according to the first view of diffusion given above.

§ 19. The characteristic equation of the mixture, ac- cording to (7) and (8), is

p = (CiMi + CM, + . . .)|

^/Ci Mi + CaMa -t- .- AMg (10)

which corresponds to the characteristic equation of a perfect gas with the following characteristic constant : —

^ CiMi + C.2M 2 + . . • n 1 ^

Hence the question as to whether a perfect gas is a chemically simple one, or a mixture of chemically different y gases, cannot in any case be settled by the investigation of the characteristic equation.

§ 20. The composition of a gas mixture is defined, either by the ratios of the masses. Mi, M2, ... or by the ratios of the partial pressures 2h, p-i, • • . or the partial volumes Vi, V2, ... of the individual gases. Accordingly we speak of per cent, by weight or by volume. Let us take for example atmospheric air, which is a mixture of oxygen (1) and " atmospheric " nitrogen (2).

The ratio of the densities of oxygen, "atmospheric" nitrogen and air is, according to § 11,

0-0014291 : 0012571 : 0-0012930 = ^ • q • ^

12 THERMODYNAMICS.

Taking into consideration, the relation (11)-

C =

CiMi + C,M, Ml + M2

we find the ratio Mi : Ma = 02998, i.e. 23-1 per cent, by weight of oxygen and 769 per cent, of nitrogen. Furthermore,

CiMi : C2M2 = i^x : i)2 = Vi : V2 = 0-2637

i.e. 209 per cent, by volume of oxygen and 79*1 per cent, of nitrogen.

§ 21. Characteristic Equation of Other Substances. — The characteristic equation of perfect gases, even in the case of the substances hitherto discussed, is only an approxi- mation, though a close one, to the actual facts. A still further deviation from the behaviour of perfect gases is shown by the other gaseous bodies, especially by those easily condensed, which for this reason were formerly classed as vaj)ours. For these a modification in the form of the characteristic equation is necessary. It is worthy of notice, however, that the more rarefied the state in which we observe these gases, the less does their behaviour deviate from that of perfect gases, so that all gaseous substances, when suffi- ciently rarefied, may be said in general to act like j^erfect gases. The general characteristic equation of gases and vapours, for very large values of v, will pass over, therefore, into the special form for perfect gases.

§ 22. We may obtain by various graphical methods an idea of the character and magnitude of the deviations from the ideal geiseous state. An isothermal curve may, e.g., be drawn, taking v and ^ for some given temperature as the abscissa and ordinate, respectively, of a j)oint in a plane. The entire system of isotherms gives us a complete repre- sentation of the characteristic equation. The more the behaviour of the vapour in question approaches that of a perfect gas, the closer do the isotherms approach those of equilateral hyperbolie having the rectangular co-ordinate

TEMPERATURE. 13

axes for asymptotes, for j)v = const, is the equation of an isotherm of a perfect gas. The deviation from the hyper- bolic form yields at the same time a measure of the departure from the ideal state.

§ 23. The deviations become still more apparent when the isotherms are drawn taking the product j^v (instead of p) as the ordinate and say p as the abscissa. Here a perfect gas has evidently for its isotherms straight lines parallel to the axis of abscissae. In the case of actual gases, however, the isotherms slope gently towards a minimum value of pv, the position of which depends on the temperature and the nature of the gas. For lower pressures {i.e. to the left of the minimum), the volume decreases at a more rapid rate, with increasing pressure, than in the case of perfect gases ; for higher pressures (to the right of the minimum), at a slower rate. At the minimum point the compressibility coincides with that of a perfect gas. In the case of hydrogen the minimum lies far to the left, and it has hitherto been possible to observe it only at very low temperatures.

§ 24. To van der Waals is due the first analytical formula for the general characteristic equation, applicable also to the liquid state. He also explained physically, on the basis of the kinetic theory of gases, the deviations from the behaviour of perfect gases. As we do not wish to introduce here the hypothesis of the kinetic theory, we consider van der Waals' equation merely as an approximate expression of the facts. His equation is

• ^^ _ «_

V — b v^'

where R, a, and h are constants which depend on the nature of the substance. For large values of v, the equation, as required, passes into that of a perfect gas ; for small values of V and corresponding values of 9, it represents the charac- teristic equation of a liquid.

Expressing p in atmospheres and calling the specific

14 THERMODYNAMICS.

volume V uuity for = 273 and ji = .> van der Waals' constants for carbon dioxide are

n = 0-00369 ; a = 0-00874 ; I = 0-0023.

As the volume of 1 gr. of carbon dioxide at 0° C. and atmospheric pressure is 505 c.c, the values of v cal- culated from the formula must be multiplied by 505 to obtain the specific volumes in absolute units.

§ 25. Van der Waals' equation not being sufficiently accurate, Clausius supplemented it by the introduction of an additional constant. Clausius' equation is

For large values of v, this too approaches the ideal characteristic equation. In the same units as above, Clausius' constants for carbon dioxide are :

B = 0-003688 ; a = 0000843 ; h = 0-000977 ; e = 2-0935.

Andrews' observations on the compressibility of gaseous and liquid carbon dioxide are satisfactorily represented by Clausius' equation.

§ 26. If we draw the system of isotherms with the aid of Clausius' equation, employing the graphical method de- scribed in § 22, the characteristic graphs for carbon dioxide — Fig. 1 — are obtained.* For high temperatures the isotherms approach equilateral hyperbolae, as may be seen from equation (12). In general, however, the isotherm is a curve of the third degree, three values of v corresponding to one of J). Hence, in general, a straight line parallel to the axis of abscissae intersects an isotherm in three points, of which two, as actually happens for large values of d, may be imaginary. At high temperatures there is, consequently,

♦ For the calculation and construction of the curves, I am indebted to Dr. Richard Apt.

\

\

^

\

\

\

Jso'herm'i Oj

' Carbon

Dioxide

^

^K

"^

\ '

N

\

\

a,

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Fig. 1.

JO

i6 THERM OD YNA MICS.

only one real volume corresponding to a given pressure, while at lower temperatures, there are three real values of the volume for a given pressure. Of these three values (indicated on the figure by a, )3, 7, for instance) only the smallest (o) and the largest (y) represent practically realiz- able states, for at the middle point (/3) the pressure along the isotherm would increase with increasing volume, and the comjDressibility would accordingly be negative. Such a state has, therefore, only a theoretical signification.

§ 27. The point a corresponds to liquid carbon dioxide, and 7 to the gaseous condition at the temperature of the isotherm passing through the points and under the pressure measured by the ordinates of the line a/By. In general only one of these states is stable (in the figure, the liquid gtate at a). For, if we compress gaseous carbon dioxide, enclosed in a cylinder with a movable piston, at constant temperature, e.g. at 20° C, the gas assumes at first states corresponding to consecutive points on the 20° isotherm to the extreme right. The point representative of the physical state of the gas, then moves farther and farther to the left until it reaches a certain place C. After this, further com- pression does not move the point beyond (7, .but there now takes place a partial condensation of the substance — a split- ting into a liquid and a gaseous portion. Both parts, of course, possess common pressure and temperature. The state of the gaseous portion continues to be characterized by the point G, that of the liquid portion by the point A of the same isotherm. C is called the saturation point of carbon dioxide gas for the particular temperature considered. Isothermal compression beyond C merely results in precipitating more of the vapour in liquid form. During this part of the isothermal compression no change takes place but the con- densation of more and more vapour ; the internal conditions (pressure, temperature, specific volume) of both parts of the substance are always represented by the two points A and C. At last, when all the vapour has been condensed, the whole substance is in the liquid condition A, and again behaves

TEMPERATURE. J7

as a homogeneous substance, so that further compression gives an increase of density and pressure along the isotherm. The substance will now pass through the point a of the figure. On this side, as may be seen from the figure, the isotherm is much steeper than on the other, i.e. the compressi- bility is much smaller. At times, it is possible to follow tlje isotherm beyond the point G towards the point -y, and to prepare a so-called supersaturated vapour. Then only a more or less unstable condition of equilibrium is obtained, as may be seen from the fact that the smallest disturbance of the equilibrium is sufficient to cause an immediate con- densation. The substance passes by a jump into the stable condition. Nevertheless, by the study of supersaturated vapours, the theoretical part of the curve also receives a direct meaning.

§ 28. On any isotherm, which for certain values of jp admits of three real values of v, there are, therefore, two definite points, A and G, corresponding to the state of saturation. The position of these points is not immediately deducible from the graph of the isotherm. The propositions of thermodynamics, however, lead to a simple way of finding these points, as will be seen in § 172. The higher the tem- perature, the smaller becomes the region in which lines drawn parallel to the axis of abscissae intersect the isotherm in three real points, and the closer will these three points approach one another. The transition to the hyperbola-like isotherms, which any parallel to the axis of abscissoe cuts in one point only, is formed by that particular isotherm on which the three points of intersection coalesce into one, giving a point of inflection. The tangent to the curve at this point is parallel to the axis of abscissae. It is called the critical point (K of Fig. 1) of the substance, and its position indicates the critical temperature, the critical specific volume, and the critical pressure of the substance. Here there is no longer any difference between the saturated vapour and its liquid precipitate. Above the critical tempe- rature and critical pressure, condensation does not exist,

c

1 8 THERMODYNAMICS.

as the diagram plainly shows. Hence all attempts to condense hydrogen, oxygen, and nitrogen necessarily failed as long as the temperature had not been reduced below the critical temperature, which is very low^ for these gases.

^ § 29. It further appears from our figure that there is no definite boundary between the gaseous and liquid states, since from tl^^^gion of purely gaseous states, as at C, tha^»of purJJPJfquid Jfties, as at A, may be reached on a circuitous path that nowhere passes through a state of saturation — on a curve, for instance, drawn around the critical point. Thus a vapour may be heated at constant volume above the critical temperature, then compressed at constant temperature below the critical volume, and finally cooled under constant pressure below the critical temperature. Condensation nowhere occurs in this process, which leads, nevertheless, to a region of purely liquid states. The earlier fundamental distinction between liquids, vapours, and gases should therefore be dropped as no longer tenable. A more modern proposal to denote as gaseous all states above the critical temperature, and as vaporous or liquid all others according as they lie to the right or left of the theoretical regions (Fig. 1), has also this disadvantage, that thereby a boundary is drawn between liquid and gas on the one hand, and vapour and gas on the other hand, which has no physical meaning. The crossing of the critical temperature at a pressure other than the critical pressure differs in no way from the crossing of any other temperature.

§ 30. The position of the critical point may be readily calculated from the general characteristic equation. Accord- ing to § 28 we have

The first of these means that the tangent to the isotherm at K is parallel to the axis of abscissai ; and the second, that

TEMPERATURE. 19

the isotherm has a point of iuflection at K. On the basis of Clausius' form of the characteristic equation (12), we obtain for the critical point

«' = 27(^)R' ^' = 21(i(f+tf " = 3a + 26.'

These equations give for carbon dioxide from the above data

= 304 = 273" + 31°, w = 77 atm., v = 227 — '.

gr-