ENERGETICS  . The most fundamental result attained by the progress of See also:

• PHYSICAL

physical See also:

• SCIENCE (Lat. scientia, from scire; to learn, know)

science in the 19th See also:

• CENTURY (from Lat. centuria, a division of a hundred men)

century was the definite enunciation and development of the See also:

• DOCTRINE

doctrine of See also:

• ENERGY (from the Gr. ivipyela; iv, in, ipyov, work)

energy, which is now See also:

• PARAMOUNT (Anglo-Fr. paramont, up above, par el mont, up or on top of the mountain)

paramount both in See also:

• MECHANICS

mechanics and in See also:

• THERMODYNAMICS (from Gr. Bepµos, hot, Siva trs, force)

thermodynamics . For a discussion of the elementary ideas underlying this conception see the See also:

• SEPARATE

separate heading ENERGY . Ever since physical See also:

• SPECULATION

speculation began in the atomic theories of the Greeks, its See also:

• MAIN (from the Aryan root which appears in " may " and " might," and Lat. magnus, great)
• MAIN (Lat. Moenus)

main problem has been that of unravelling the nature of the underlying correlation which binds together the various natural agencies . But it is only in See also:

• RECENT

recent times that scientific investigation has definitely established that there is a quantitative relation of See also:

• SIMPLE

simple equivalence between them, whereby each is expressible in terms of See also:

• HEAT (0. E. haktu, which like " hot," Old Eng. hat, is from the Teutonic type haita, hit, to be hot; cf. Ger. hitze, heiss; Dutch, hitte, heet, &c.)

heat or See also:

• MECHANICAL

mechanical See also:

• POWER [WILLIAM GRATTAN] TYRONE (1797-1841)

power; that there is a certain measurable quantity associated with each type of physical activity which is always numerically identical with a corresponding quantity belonging to the new type into which it is transformed, so that the energy, as it is called, is conserved in unaltered amount . The main obstacle in the way of an earlier recognition and development of this principle had been the doctrine of caloric, which was suggested by the principles and practice of See also:

• CALORIMETRY

calorimetry, and taught that heat is a substance that can be transferred from one See also:

• BODY

body to another, but cannot be created or destroyed, though it may become latent . So See also:

• LONG, GEORGE (1800-1879)
• LONG, JOHN DAVIS (1838– )

long as this See also:

• IDEA (Gr. Ibia, connected with i&eiv, to see; cf. Lat. species from specere, to look at)

idea maintained itself, there was no possible See also:

• COMPENSATION (from Lat. compensare, to weigh one thing against another)

compensation for the destruction of mechanical power by See also:

• FRICTION (from Lat. fricare, to rub)

friction; it appeared that mechanical effect had there definitely been lost . The idea that heat is itself convertible into power, and is in fact energy of See also:

• MOTION (Lat. motio, from movere, to move)
• MOTION, LAWS OF

motion of the See also:

• MINUTE
• MINUTE (Lat. minutes, small; minuere, to make less)

minute invisible parts of bodies, had been held by See also:

• NEWTON
• NEWTON, ALFRED (1829–1907)
• NEWTON, JOHN (1725-1807)
• NEWTON, JOHN (1823-1895)
• NEWTON, SIR CHARLES THOMAS (1816-1894)
• NEWTON, SIR ISAAC (1642-1727)

Newton and in a vaguer sense by See also:

• BACON
• BACON (through the O. Fr. bacon, Low Lat. baco, from a Teutonic word cognate with " back," e.g. O. H. Ger. pacho, M. H. Ger. backe, buttock, flitch of bacon)
• BACON, FRANCIS (BARON VERULAM, VISCOUNT ST ALBANS) (1561-1626)
• BACON, JOHN (1740–1799)
• BACON, LEONARD (1802–1881)
• BACON, ROGER (c. 1214-c. 1294)
• BACON, SIR NICHOLAS (1509-1579)

Bacon, and indeed long before their See also:

• TIME (0. Eng. Lima, cf. Icel. timi, Swed. timme, hour, Dan. time; from the root also seen in " tide," properly the time of between the flow and ebb of the sea, cf. O. Eng. getidan, to happen, " even-tide," &c.; it is not directly related to Lat. tempus)
• TIME, MEASUREMENT OF
• TIME, STANDARD

time; but it dropped out of the See also:

• ORDINARY (med. Lat. ordinarius, Fr. ordinaire)

ordinary creed of science in the following century . It held a See also:

• PLACE (through Fr. from Lat. platea, street; Gr. IrAar6s, wide)

place, like many other anticipations of subsequent See also:

• DISCOVERY

discovery, in the See also:

• SYSTEM

system of Natural See also:

• PHILOSOPHY (Gr. gthos, fond of, and vo4 (a, wisdom)

Philosophy of See also:

• THOMAS
• THOMAS (c. 1654-1720)
• THOMAS (d. 110o)
• THOMAS, ARTHUR GORING (1850-1892)
• THOMAS, CHARLES LOUIS AMBROISE (1811-1896)
• THOMAS, GEORGE (c. 1756-1802)
• THOMAS, GEORGE HENRY (1816-187o)
• THOMAS, ISAIAH (1749-1831)
• THOMAS, PIERRE (1634-1698)
• THOMAS, SIDNEY GILCHRIST (1850-1885)
• THOMAS, ST
• THOMAS, THEODORE (1835-1905)
• THOMAS, WILLIAM (d. 1554)

Thomas See also:

• YOUNG
• YOUNG, A
• YOUNG, ARTHUR (1741-1820)
• YOUNG, BRIGHAM (1801-1877)
• YOUNG, CHARLES MAYNE (1777–1856)
• YOUNG, EDWARD (1683–1765)
• YOUNG, JAMES (1811-1883)
• YOUNG, THOMAS (1773-1829)

Young (1804); and the discrepancies attending current explanations on the caloric theory were insisted on, about the same time, by See also:

• COUNT (Lat. comes, gen. comitis, Fr. comte, Ital. conte, Span. conde)

Count See also:

• RUMFORD, BENJAMIN THOMPSON, COUNT (1753-1814)

Rumford and See also:

• SIR

Sir H . See also:

• DAVY, SIR HUMPHRY

Davy . But it was not till the actual experiments of See also:

• JOULE, JAMES PRESCOTT (1818–1889)

Joule verified the same exact equivalence between heat produced and mechanical energy destroyed, by whatever See also:

• PROCESS

process that was accomplished, that the idea of caloric had to be definitely abandoned . Some time previously R .

See also:

• MAYER, JOHANN TOBIAS (1723-1762)
• MAYER, JULIUS ROBERT (1814-1878)

Mayer, physician, of See also:

• HEILBRONN

Heilbronn, had founded a weighty theoretical See also:

• ARGUMENT

argument on the See also:

• PRODUCTION (Lat. productionem, from producere, to pro-duce)

production of mechanical ENERGETICS power in the See also:

• ANIMAL
• ANIMAL (Lat. animalis, from anima, breath, soul)

animal system from the See also:

• FOOD
• FOOD (like the verb " to feed," from a Teutonic root, whence O. Eng. foda; cf. " fodder "; connected with Gr. lrareicOae, to feed)

food consumed; he had, moreover, even calculated the value of a unit of heat, in terms of its See also:

• EQUIVALENT

equivalent in power, from the data afforded by See also:

• REGNAULT, HENRI (1843-1871)
• REGNAULT, HENRI VICTOR (1810-1878)
• REGNAULT, JEAN BAPTISTE (1754-1829)

Regnault's determinations of the specific heats of See also:

• AIR (from an Indo-European root meaning " breathe," " blow ")
• AIR, or ASBEN

air at See also:

• CONSTANT, BENJAMIN (1845-1902)

constant pressure and at constant See also:

• VOLUME

volume, the former being the greater on Mayer's See also:

• HYPOTHESIS (from Gr. inrorcOivat, to put under; cf. Lat. suppositio, from sub-ponere)

hypothesis (of which his calculation in fact constituted the verification) solely on See also:

• ACCOUNT
• ACCOUNT (through O. Fr. acont, Late Lat. comptum, cornputare, to calculate)

account of the power required for the See also:

• WORK

work of expansion of the See also:

• GAS

gas against the surrounding constant pressure . About the same time See also:

• HELMHOLTZ, HERMANN LUDWIG FERDINAND VON (1821-1894)

Helmholtz, in his See also:

• EARLY
• EARLY, JUBAL ANDERSON (1816-1894)

early memoir on the Conservation of Energy, constructed a cumulative argument by tracing the ramifications of the principle of conservation of energy throughout the whole range of physical science . Mechanical and Thermal Energy.—The amount of energy, defined in this sense by convertibility with mechanical work, which is contained in a material system, must be a See also:

• FUNCTION

function of its physical See also:

• STATE
• STATE, GREAT OFFICERS OF

state and chemical constitution and of its temperature . The See also:

• CHANGE (derived through the Fr. from the Late Lat. cambium, cambiare, to barter; the ultimate derivation is probably from the root which appears in the Gr. «a urmu', to bend)

change in this amount, arising from a given transformation in the system, is usually measured by degrading the energy that leaves the system into heat; for it is always possible to do this, while the See also:

• CONVERSION (Lat. conversio, from convertere, to turn or ,change)

conversion of heat back again into other forms of energy is impossible without assistance, taking the See also:

• FORM (Lat. forma)

form of compensating degradation elsewhere . We may adopt the provisional view which is the basis of abstract physics, that all these other . foxms of energy are in their essence mechanical, that is, arise from the motion or See also:

• STRAIN (through O. Fr. straindre, estraindre, mod. i treindre, from Lat. stringere, to draw tight, related to stress, stretch, string, &c.)

strain of material or ethereal See also:

• MEDIA

media; then their distinction from heat will See also:

• LIE, JONAS LAURITZ EDEMIL (1833—1908)
• LIE, MARIUS SOPHUS (1842–1899)

lie in the fact that these-motions or strains are simply co-ordinated, so that they can be traced and controlled or manipulated in detail; while the thermal energy subsists in irregular motions of the molecules or smallest portions of See also:

• MATTER

matter, which we cannot trace on account of the bluntness of our sensual perceptions, but can only measure as regards See also:

• TOTAL

total amount . See also:

• HISTORICAL

Historical: Abstract See also:

• DYNAMICS
• DYNAMICS (from Gr. bbvayts, strength)

Dynamics.—Even in the See also:

• CASE
• CASE, JOHN (d. 1600)

case of a purely mechanical system, capable only of a finite number of definite types of disturbance, the principle of the conservation of energy is very far from giving a See also:

• COMPLETE

complete account of its motions; it forms only one among the equations that are required to determine their course . In its application to the See also:

• KINETICS (from Gr. taveiv, to move)

kinetics of invariable systems, after the time of Newton, the principle was emphasized as fundamental by See also:

• LEIBNITZ (LEIBNIZ), GOTTFRIED WILHELM

Leibnitz, was then improved and generalized by the Bernoullis and by See also:

• EULER, LEONHARD (1707-1783)

Euler, and was ultimately expressed in its widest form by See also:

• LAGRANGE, JOSEPH LOUIS (1736-1813)

Lagrange . It is recorded by Helmholtz that it was largely his acquaintance in early years with the See also:

• WORKS

works of those mathematical physicists of the previous century, who had formulated and generalized the principle as a help towards the theoretical dynamics of complex systems of masses, that started him on the track of extending the principle throughout the whole range of natural phenomena . On the other See also:

• HAND
• HAND (a word common to Teutonic languages; cf. Ger. Hand, Goth. handus)
• HAND, FERDINAND GOTTHELF (1786-185r)

hand, the ascertained validity of this See also:

• EXTENSION (Lat. ex, out ; tendere, to stretch)

extension to new types of phenomena, such as those of electrodynamics, now forms a main See also:

• FOUNDATION (Lat. fundatio, from fundare, to found)

foundation of our belief in a mechanical basis for these sciences . In the hands of Lagrange the mathematical expression for the manner in which the energy is connected with the geometrical constitution of the material system became a sufficient basis for a complete knowledge of its dynamical phenomena . So far as See also:

• STATICS (from Gr. root am-, stand, or cause to stand)

statics was concerned, this doctrine took its rise as far back as Galileo, who recognized in the simpler cases that the work expended in the steady See also:

• DRIVING (from " to drive," i.e. generally to propel, force along or in, a word common in various forms to the Teutonic languages)

driving of a frictionless mechanical system is equal to its output . The expression of this fact was generalized in a brief statement by Newton in the Principia, and more in detail by the Bernoullis, until, in the See also:

• ANALYTICAL

analytical See also:

• GUISE
• GUISE, HOUSE OF

guise of the so-called principle of " virtual velocities " or virtual work, it finally became the basis of Lagrange's See also:

• GENERAL
• GENERAL (Lat. generalis, of or relating to a genus, kind or class)

general formulation of dynamics .

In its application to kinetics a purely physical principle, also indicated by Newton, but See also:

• DEVELOPED

developed long after with masterly applications by d'See also:

• ALEMBERT, JEAN LE ROND

Alembert, that the reactions of the infinitesimal parts of the system against the accelerations of their motions staticallyequilibrate the forces applied to the system as a whole, was required in See also:

• ORDER
• ORDER (through Fr. ordre, for earlier ordene, from Lat. ordo, ordinis, rank, service, arrangement; the ultimate source is generally taken to be the root seen in Lat. oriri, rise, arise, begin; cf. " origin ")
• ORDER, HOLY

order to form a sufficient basis, and one which Lagrange soon afterwards condensed into the single relation of Least See also:

• ACTION

Action . As a matter of See also:

• HISTORY

history, however, the complete formulation of the subject of abstract dynamics actually See also:

• ENDYMION

ENDYMION arose (in 1758) from Lagrange's precise demonstration of the principle of Least Action for a particle, and its immediate ex-tension, on the basis of his new Calculus of See also:

• VARIATIONS
• VARIATIONS, CALCULUS
• VARIATIONS, CALCULUS OF

Variations, to a system of connected particles such as might be taken as a See also:

• REPRESENTATION

representation of any material system; but here too the same physical as distinct from mechanical considerations come into See also:

• PLAY

play as in d'Alembert's principle., (See DYNAMICS: Analytical.) It is in the cases of systems whose state is changing so slowly that reactions, arising from changing motions can be neglected, that the conditions are by far the simplest . In such systems, whether stationary or in a state of steady motion, the energy depends on the configuration alone, and its mathematical expression can be determined from measurement of the work required for a sufficient number of simple transformations; once it is thus found, all the statical relations of the system are implicitly determined along with it, and the results of all other transformations can be predicted . The general development of such relations is conveniently classed as a separate See also:

• BRANCH (from the Fr. branche, late Lat. branca, an animal's paw)

branch of physics under the name Energetics, first invented by W . J . M . See also:

• RANKINE, WILLIAM JOHN MACQUORN (182o-1872)

Rankine; but the essential limitations of this method have not always been observed . As regards statical change, the complete See also:

• SPECIFICATION (from Med. Lat. specificatio, specificare, to enumerate or mention in detail)

specification of a mechanical system is involved in its geometrical configuration and the function expressing its mechanical energy in terms thereof . Systems which have statical energy-functions of the same analytical form behave in corresponding ways, and can serve as See also:

• MODELS

models or representations of one another . Extension to Thermal and Chemical Systems.—This dominant position of the principle of energy, in ordinary statical problems, has in recent times been extended to transformations involving change of physical state or chemical constitution as well as change of geometrical configuration . In this wider See also:

• FIELD (a word common to many West German languages, cf. Ger. Feld, Dutch veld, possibly cognate with O.E. f olde, the earth, and ultimately with root of the Gr. irAaror, broad)
• FIELD, CYRUS WEST (1819-1892)
• FIELD, DAVID DUDLEY (18o5-1894)
• FIELD, EUGENE (1850-1895)
• FIELD, FREDERICK (18o1—1885)
• FIELD, HENRY MARTYN (1822-1907)
• FIELD, JOHN (1782—1837)
• FIELD, MARSHALL (183 1906)
• FIELD, NATHAN (1587—1633)
• FIELD, STEPHEN JOHNSON (1816-1899)
• FIELD, WILLIAM VENTRIS FIELD, BARON (1813-1907)

field we cannot assert that mechanical (or available) energy is never lost, for it may be degraded into thermal energy; but we can use the principle that on the other hand it can never spontaneously increase . If this were not so, cyclic processes might theoretically be arranged which would continue to See also:

• SUPPLY (through Fr. from Lat. supplere, to fill up)

supply mechanical power so long as energy of any See also:

• KIND (O. E. ge-cynde, from the same root as is seen in " kin," supra)

kind remained in the system; whereas the irregular and uncontrollable See also:

• CHARACTER (Gr. xapareri7p, from xap&crew, to scratch)

character of the molecular motions and strains which constitute thermal energy, in See also:

• COMBINATION (Lat. combinare, to combine)

combination with the vast number of the molecules, must place an effectual See also:

• BAR
• BAR (0. Fr. barre, Late Lat. barra, origin unknown)
• BAR, CONFEDERATION OF
• BAR, FRANCOIS DE (1538-1606)
• BAR, THE

bar on their unlimited co-ordination .

To establish a doctrine of energetics that shall form a sufficient foundation for a theory of the trend of chemical and physical change, we have, there-fore, to impart precision to this motion of available energy . See also:

• CARNOT, LAZARE EIPPOLYTE (18ot-1888)
• CARNOT, MARIE FRANCOIS SADI (1837-1894)
• CARNOT, SADI NICOLAS LEONHARD (1796-1832)

Carnot's Principle: Entropy.—The whole subject is involved in the new principle contributed to theoretical physics by Sadi Carnot in 1824, in which the far-reaching See also:

• MODERN

modern conception of cyclic processes was first scientifically developed . It was shown by Carnot, on the basis of certain axioms, whose theoretical See also:

• FOUNDATIONS

foundations were subsequently corrected and strengthened by See also:

• CLAUSIUS, RUDOLF JULIUS EMMANUEL (1822-1888)

Clausius and See also:

• LORD
• LORD (O. Eng. hldford, i.e. hldfweard, the warder or keeper of bread, hlIf, loaf; the word is not represented in any other Teutonic language)
• LORD, JOHN (1810-1894)

Lord See also:

• KELVIN, WILLIAM THOMSON, BARON (1824-1907)

Kelvin, that a reversible mechanical process, working in a See also:

• CYCLE
• CYCLE (Gr. KUK)^os, a circle)

cycle by means of thermal transfers, which takes heat, say HI, into the material system at a given temperature T1, and delivers the See also:

• PART

part of it not utilized, say 112, at a See also:

• LOWER

lower given temperature T2, is more efficient, considered as a working See also:

• ENGINE
• ENGINE (Lat. ingenium)

engine, than any other such process, operating between the same two temperatures but not reversible, could be . This relation of inequality involves a definite See also:

• LAW
• LAW (0. Eng. lagu, M. Eng. lawe; from an old Teutonic root lag, " lie," what lies fixed or evenly; cf. Lat. lex, Fr. loi)
• LAW, JOHN (1671—1729)
• LAW, WILLIAM (1686-1761)

law of equality, that the mechanical efficiencies of all reversible cyclic processes are the same, whatever be the nature of their operation or the material substances involved in them; that in fact the efficiency is a function solely of the two temperatures at which the cyclically working system takes in and gives out heat . These considerations constitute a fundamental general principle to which all possible slow reversible processes, so far as they concern matter in bulk, must conform in all their stages; its application is almost coextensive with the See also:

• SCOPE (through Ital. scopo, aim, purpose, intent, from Gr. o'KOaos, mark to shoot at, aim, o ic07reiv, to see, whence the termination in telescope, microscope, &c.)

scope of general physics, the See also:

• SPECIAL

special kinetic theories in which inertia is involved, being excepted . (See THERMODYNAMICS.) If the working system is an ideal gas-engine, in which a perfect gas (known from experience to be a possible state of matter) is passed through the cycle, and if temperature is measured from the See also:

• ABSOLUTE (Lat. absolvere, to loose, set free)

absolute zero by the expansion of this gas, then simple directcalculation on the basis of the See also:

• LAWS

laws of ideal gases shows that HI/TI=H1/T2; and as by the conservation of energy the work done is H1-H2, it follows that the efficiency, measured as the ratio of the work done to the supply of heat, is 1— T2/Tl . If we change the sign of HI and thus consider heat as See also:

• POSITIVE (or PORTABLE) ORGAN

positive when it is restored to the system as is 112, the fundamental See also:

• EQUATION (from Lat. aequatio, aequare, to equalize)

equation becomes HI/TI+H2/T2=o; and as any complex reversible working system may be considered as compounded in various ways of chains of elementary systems of this type, whose effects are additive, the general proposition follows, that in any reversible complete cyclic change which involves the taking in of heat by the system of which the amount is SH, when its temperature ranges between Tr and Tr+ST, the equation 1SHr/Tr o holds See also:

• GOOD, JOHN MASON (1764-1827)

good . Moreover, if the changes are not reversible, the proportion of the heat supply that is utilized for mechanical work will be smaller, so that more heat will be restored to the system, and ZSHr/T, or, as it may be expressed, fdH/T, must have a larger value, and must thus be positive . The first statement involves further, that for all reversible paths of change of the system from one state C to another state D, the value of fdH/T must be the same, because any one of these paths and any other one reversed would form a cycle; whereas for any irreversible path of change between the same states this integral must have a greater value (and so exceed the difference of entropies at the ends of the path) . The definite quantity represented by this integral for a reversible path was introduced by Clausius in 1854 (also adumbrated by Kelvin's investigations about the same time), and was named afterwards by him the increase of the entropy of the system in passing from the state C to the state D . This increase, being thus the same for the unlimited number of possible reversible paths involving See also:

• INDEPENDENT

independent variation of all its finite co-ordinates, along which the system can pass, can depend only on the terminal states . The entropy belonging to a given state is therefore a function of that state alone, irrespective of the manner in which it has been reached; and this is the See also:

• JUSTIFICATION

justification of the See also:

• ASSIGNMENT, ASSIGNATION, ASSIGNEE

assignment to it of a special name, connoting a See also:

• PROPERTY

property of the system depending on its actual See also:

• CONDITION (Lat. condicio, from condicere, to agree upon, arrange; not connected with conditio, from condere, conditum, to put together)

condition and not on its previous history .

Every reversible change in an isolated system thus maintains the entropy of that system unaltered; no possible spontaneous change can involve decrease of the entropy; while any defect of reversibility, arising from See also:

• DIFFUSION (from the Lat. diffundere; dis-, asunder, and fundere, to pour out)

diffusion of matter or motion in the system, necessarily leads to increase of entropy . For a physical or chemical system only those changes are spontaneously possible which would See also:

• LEAD
• LEAD (pronounced iced)

lead to increase of the entropy; if the entropy is already a maximum for the given total energy, and so incapable of further continuous increase under the conditions imposed upon the system, there must be See also:

• STABLE

stable See also:

• EQUILIBRIUM (from the Lat. aequus, equal, and libra, a balance)

equilibrium . This definite quantity belonging to a material system, its entropy 0, is thus concomitant with its energy E, which is also a definite function of its actual state by the law of conservation of energy; these, along with its temperature T, and the various co-ordinates expressing its geometrical configuration and its physical and chemical constitution, are the quantities with which the thermodynamics of the system deals . That branch of science develops the consequences involved in just two principles: (i.) that the energy of every isolated system is constant, and (ii.) that its entropy can never diminish; any complication that may be involved arises from complexity in the systems to which these two laws have to be applied . The General Thermodynamic Equation.—When any physical or chemical system undergoes an infinitesimal change of state, we have SE =SH+SU, where SH is the energy that has been acquired as heat from See also:

• SOURCES

sources extraneous to the system during the change, and SU is the energy that has been imparted by reversible agencies such as mechanical or electric work . It is, however, not usually possible to discriminate permanently between heat acquiredand work imparted, for(unless for isothermal transformations) neither 311 nor SU is the exact See also:

• DIFFERENTIAL

differential of a function of the constitution of the system and so independent of its previous history, although their sum SE is such; but we can utilize the fact that 311 is equal to TSB where 30 is such, as has just been seen . Thus E and cf, represent properties of the system which, along with temperature, pressure and other independent data specifying its constitution, must form the variables of an analytical exposition . We have, therefore, to substitute TS4 for SH; also the change of See also:

• INTERNAL

internal energy is determined by the change of constitution, involving a differential relation of type SU = —pSu+SW++s 26mz+ . • • • +l1nSm,,., when the system consists of an intimate mixture (See also:

• SOLUTION (from Lat. solvere, to loosen, dissolve)

solution) of masses ml, m2i ... in,, of given constituents, which differ physically or chemically but may be partially transformable into each other by chemical or physical action during the changes under See also:

• CONSIDERATION (from Lat. considerare, to look at closely, examine, generally taken to be from con-, and the base seen in sidus, sideris, a star, the word being supposed to be originally an astrological or astronomical term)

consideration, the whole being of volume v and under extraneous pressure p, while W is potential energy arising from physical forces such as those of gravity, capillarity, &c . The variables m,, m2, • • • • .. mn may not be all independent; for example, if the system were chloride of ammonium gas existing along with its gaseous products of See also:

• DISSOCIATION

dissociation, hydrochloric See also:

• ACID (from the Lat. root ac-, sharp; acere, to be sour)

acid and See also:

• AMMONIA (NH3)

ammonia, only one of the three masses would be independently variable . The sufficient number of these variables (independent components) together with two other variables, which may be v and T, or v and 0, specifies and determines the state of the system, considered as matter in bulk, at each instant . It is usual to include SW in µ,Sm, + .

. . ; in all cases where this is possible the single equation ENERGETICS SE =TSB—pbv+µ,Sm,+µ2&m2+• • . •+e &ni (i) thus expresses the complete variation of the energy-function E arising from change of state; and when the part involving the n constitutive differentials has been expressed in terms of the number of them that are really independent, this equation by itself becomes the unique expression of all the thermodynamic relations of the system . These are in fact the various relations ensuring that the right-hand See also:

• SIDE
• SIDE (mod. Eski Adalia)

side is an exact differential, and are of the type of reciprocal relations such as dp.r/d4,=dT/dm, .. The condition that the state of the system be one of stable equilibrium is that See also:

• ORB

Orb, the variation of entropy, be negative for all formally imaginable infinitesimal transformations which make SE vanish; for as S4, cannot actually be negative for any spontaneous variation, none of these transformations can then occur . From the form of the equation, this condition is the same as that SE—TS¢ must be positive for all possible variations of state of the system as above defined in terms of co-ordinates representing its constitution in bulk, without restriction . We can change one of the independent variables expressing the state of the system from 4, to T by subtracting S(OT) from both sides of the equation of variation: then S(E —See also:

• TOP (cf. Dan. top, Ger. Topf, also meaning pot)

Top) = — 03T — PSv+µ,&m1 + .... +µ„Sm,, . It follows that for isothermal changes, i.e. those for which ST is maintained null by an environment at constant temperature, the condition of stable equilibrium is that the function E—TO shall be a minimum . If the system is subject to an See also:

• EXTERNAL

external pressure p, which as well as the temperature is imposed constant from without and thus incapable of variation through internal changes, the condition of stable equilibrium is similarly that E—T¢+pv shall be a minimum . A chemical system maintained at constant temperature by communication of heat from its environment may thus have several states of stable equilibrium corresponding to different minima of the function here considered, just as there may be several minima of See also:

• ELEVATION

elevation on a landscape, one at the bottom of each depression; in fact, this See also:

• ANALOGY (Gr. avaXo-yLa, proportion)

analogy, when extended to space of ii dimensions, exactly fits the case . If the system is sufficiently disturbed, for example, by electric See also:

• SHOCK, or COLLAPSE

shock, it may pass over (explosively) from a higher to a lower minimum, but never (without compensation from outside) in the opposite direction .

The former passage, moreover, is often effected by introducing a new substance into the system; sometimes that substance is recovered unaltered at the end of the process, and then its action is said to be purely catalytic; its presence modifies the form of the function E— Tq5 so as to obliterate the See also:

• RIDGE (a word. common to many Teutonic languages, meaning " back," whether of a man or an animal, cf. German Rucke)
• RIDGE, WILLIAM PETT (1864- )

ridge between the two equilibrium states :n the graphical representation . There are systems in which the equilibrium states are but very slightly dependent on temperature and pressure within wide limits, outside which reaction takes place . Thus while there arecases in which a state of See also:

• MOBILE

mobile dissociation exists in the system which changes continuously as a function of these variables, there are others in which change does not sensibly.occur at all until a certain temperature of reaction is attained, after which it proceeds very rapidly owing to the heat developed, and the system soon becomes sensibly permanent in a transformed phase by completion of the reaction . In some cases of this latter type the cause of the delay in starting lies possibly in passive resistance to change, of the nature of viscosity or friction, which is competent to convert an unstable mechanical equilibrium into a moderately stable one; but in most such reactions there seems to be no exact equilibrium at any temperature, See also:

• SHORT, FRANCIS JOB (1857– )

short of the ultimate state of dissipated energy in which the reaction is completed, although the velocity of reaction is f ound to diminish exponentially with change of temperature, and thus becomes insignificant at a small See also:

• INTERVAL

interval from the temperature of pronounced activity . See also:

• FREE

Free Energy.—The quantity E — TO thus plays the same fundamental part in the thermal statics of general chemical systems at See also:

• UNIFORM

uniform temperature that the potential energy plays in the statics of mechanical systems of unchanging constitution . It is a function of the geometrical co-ordinates, the physical and chemical constitution, and the temperature of the system, which determines the conditions of stable equilibrium at each temperature; it is, in fact, the potential energy generalized so as to include temperature, and thus be a single function See also:

• RELATING

relating to each temperature but at the same time affording a basis of connexion between the properties of the system at different temperatures . It has been called the free energy of the system by Helmholtz, for it is the part of the energy whose variation is connected with changes in the bodily structure of the system represented by the variables m,, m2, . in,,, and not with the irregular molecular motions represented by heat, so that it can take part freely in physical transformations . Yet this holds good only subject to the condition that the temperature is not varied; it has been seen above that for the more general variation neither SH nor SU is an exact differential, and no See also:

• LINE

line of separation can be See also:

• DRAWN

drawn between thermal and mechanical energies . The study of the See also:

• EVOLUTION

evolution of ideas in this, the most abstract branch of modern mathematical physics, is rendered difficult in the manner of most purely philosophical subjects by the variety of terminology, much of it only partially appropriate, that has been employed to See also:

• EXPRESS (through the French from the past participle of the Lat. exprimere, to press out, by transference used of representing objects in painting or sculpture, or of thoughts, &c. in words)

express the fundamental principles by different investigators and at different stages of the development . Attentive examination will show, what is indeed hardly surprising, that the principles of the theory of free energy of See also:

• GIBBS, JOSIAH WILLARD (1839-1903)
• GIBBS, OLIVER WOLCOTT (1822-1908)

Gibbs and Helmholtz had been already grasped and exemplified by Lord Kelvin in the very early days of the subject (see the See also:

• PAPER
• PAPER (Fr. papier, from Lat. papyrus)

paper " On the Thermoelastic and Thermomagnetic Properties of Matter, Part I." Quarterly See also:

• JOURNAL
• JOURNAL (through Fr. from late Lat. diurnalis, daily)

Journal of See also:

• MATHEMATICS (Gr. /saBnµar1Kil, Sc. vOcvn or E7rlQTt'µn; from p iN.La, "learning" or "science ")

Mathematics, No. i, See also:

• APRIL

April 1855; reprinted in Phil . Mag., See also:

• JANUARY

January 1878, and in Math. and Phys . Papers, vol. i. pp .

291, seq.) . Thus the striking new advance contained in the more modern work of J . See also:

• WILLARD, FRANCES ELIZABETH (1839–1898)

Willard Gibbs (1875-1877) and of Helmholtz (1882) was rather the sustained general application of these ideas to chemical systems, such as the galvanic See also:

• CELL (from Lat. cella, probably from an Indo-European kal —seen in Lat. celare, to hide; another suggestion connects the word with Lat. cera, wax, taking the original meaning to refer to the honeycomb)

cell and dissociating gaseous systems, and in general See also:

• FASHION (adapted from Fr. facon, agon, Lat. factio, making, facere, to do or make)

fashion to heterogeneous concomitant phases . The fundamental paper of Kelvih connecting the electromotive force of the cell with the energy of chemical transformation is of date 1851, some years before the distinction between free energy and total energy had definitely crystallized out; and, possibly satisfied with the approximate exactness of his imperfect See also:

• FORMULA
• FORMULA (Lat. diminutive of forma, shape, pattern, &c., especially used of rules of judicial procedure)

formula when applied to a See also:

• DANIELL, JOHN FREDERIC (1790-1845)
• DANIELL, THOMAS (1749–1840)

Daniell's cell (infra), and deterred by See also:

• ABSENCE (Lat. absentia)

absence of experimental data, he did not return to the subject . In 1852 he briefly announced (Proc . See also:

• ROY, WILLIAM (1726-1790)

Roy . See also:

• SOC

Soc . Edin.) the principle of the dissipation of mechanical (or available) energy, including the See also:

• NECESSITY (Lat. necessitas)

necessity of compensation elsewhere when restoration occurs, in the form that " any restoration of mechanical energy, without more than an equivalent of dissipation, is impossible "—probably even in vital activity; but a sufficient specification of available energy (cf. infra) was not then developed . In the paper above referred to, where this was done, and illustrated by full application to solid elastic systems, the total energy is represented bye and is named "the See also:

• INTRINSIC (through Fr. intrinsique, from Lat. intrinsecus, inwardly; inter, within, secus, following, from root of sequi, to follow)

intrinsic energy," the energy taken in during an isothermal transformation is represented bye, of which H is taken in as heat, while the See also:

• REMAINDER, REVERSION

remainder, the change of free (or mechanical or available) energy of the system is the unnamed quantity denoted by the See also:

• SYMBOL (Gr. o-uµ(3oXov, a sign)

symbol w, which is " the work done by the applied forces " at uniform temperature . It is pointed out that it is w and not e that is the potential energy-function for isothermal change, of which the form can be determined directly by dynamical and physical experiment, and from which alone the criteria of equilibrium and stress are to be derived--simply for the See also:

• REASON (Lat. ratio, through French raison)

reason that for all reversible paths at constant temperature between the same terminal configurations, there must, by Carnot's principle, be the same gain or loss of heat . And a system of formulae are See also:

• DW(i+at)X587

dw given (5) to (11)—Ex. gr. e = w —t dt + J sdt for finding the total energy e for any temperature t when w and the thermal capacity s of the system, in a See also:

• STANDARD
• STANDARD, BATTLE OF THE

standard state, have thus been ascertained, and another for establishing connexion between the form of w for one temperature and its form for adjacent temperatures—which are identical with those developed by Helmholtz long afterwards, in 1882, except that the entropy appears only as an unnamed integral . The progress of physical science is formally identified with the exploration of this function w for physical systems, with continually increasing exactness and range—except where pure kinetic considerations prevail, in which cases the wider Hamiltonian dynamical formulation is fundamental .

Another aspect of the matter will be developed below . A somewhat different See also:

• PROCEDURE (Fr. procedure, from Lat. procedere, to go for-ward)

procedure, in terms of entropy as fundamental, has been adopted and developed by See also:

• PLANCK, GOTTLIEB JAKOB (1751–1833)
• PLANCK, KARL CHRISTIAN (1819-188o)

Planck . In an isolated system the trend of change must be in the direction which increases the entropy 4), by Clausius' form of the principle . But in experiment it is a system at constant temperature rather than an adiabatic one that usually is involved; this can be attained formally by including in the isolated system (cf. infra) a source of heat at that temperature and of unlimited capacity, when the energy of the See also:

• ORIGINAL

original system increases by&E this source must give up heat of amount SE, and its entropy therefore diminishes SE/T . Thus for the original system maintained at constant temperature T it is 3¢—SE/T that must always be positive in spontaneous change, which is the same criterion as was reached above . Reference may also be made to H . A . Lorentz's Collected Scientific Papers, part i . A striking anticipation, almost contemporaneous, of Gibbs's thermodynamic potential theory (infra) was made by Clerk See also:

• MAXWELL
• MAXWELL, JAMES CLERK (1831–1879)

Maxwell in connexion with the discussion of See also:

• ANDREWS, JAMES PETTIT (c. 1737–1797)
• ANDREWS, THOMAS (1813–1885)

Andrews's experiments on the See also:

• CRITICAL (in alphabetical order of authors)

critical temperature of mixed gases, in a See also:

• LETTER (through Fr. lettre from Lat. littera or litera, letter of the alphabet; the origin of the Latin word is obscure; it has probably no connexion with the root of linere, to smear, i.e. with wax, for an inscription with a stilus)

letter published in Sir G . G . See also:

• STOKES, SIR G
• STOKES, SIR GEORGE GABRIEL, BART
• STOKES, WHITLEY (1830-1909)

Stokes's Scientific See also:

• CORRESPONDENCE
• CORRESPONDENCE (from med. scholastic Lat. correspondentia, corrcspondere, compounded of Lat. cum, with, and respond ere, to answer; cf. Fr. correspondance)

Correspondence (vol. ii . P .

34) . Available Energy.—The same quantity 4), which Clausius named the entropy, arose in various ways in the early development of the subject, in the See also:

• TRAIN (M. Eng. trayn or trayne, derived through Fr. from Late Lat. trahinare, to drag, draw, Lat. trahere, cf. trail, trace, ultimately from the same source)

train of ideas of Rankine and Kelvin relating to the expression of the available energy A of the material system . Suppose there were accessible an See also:

• AUXILIARY (from Lat. auxilium, help)

auxiliary system containing an unlimited quantity of heat at absolute temperature To, forming a See also:

• CONDENSER

condenser into which heat can be discharged from the working system, or from which it may be recovered at that temperature: we proceed to find how much of the heat of our system is available for transformation into mechanical work, in a process which reduces the whole system to the temperature of this condenser . Provided the process of reduction is performed reversibly, it is immaterial, by Carnot's principle, in what manner it is effected: thus in following it out in detail we can consider each elementary quantity of heat 3H removed from the system as set aside at its actual temperature between T and T+ST for the production of mechanical work SW and the See also:

• RESIDUE (through the French, from the Lat. residuum, a remainder, from residere, to remain)

residue of it Rio as directly discharged into the condenser at To . The principle of Carnot gives SH/T=SHo/To, so that the portion of the heat SH that is not available for work is SHo, equal to To6H/T . In the whole process the part not available in connexion with the condenser at To is therefore Tof dH/T . This quantity must be the same whatever reversible process is employed: thus, for example, we may first transform the system reversiblyfrom the state C to the state D, and then from the state D to the final state of uniform temperature To . It follows that the value of TofdH/T, representing the heat degraded, is the same along all reversible paths of transformation from the state C to the state D; so that the function fdH/T is the excess of a definite quantity 4) connected with the system in the former state as compared with the latter . It is usual to change the law of sign of SH so that gain of heat by the system is reckoned positive; then, relative to a condenser of unlimited capacity at To, the state C contains more mechanic-ally available energy than the state D by the amount Ec—ED+TofdH/T, that is, by Ec-ED-To(4)c-4)o) . In this way the existence of an entropy function with a definite value for each state of the system is again seen to be the See also:

• DIRECT

direct analytical equivalent of Carnot's See also:

• AXIOM (Gr. &iwµa)

axiom that no process can be more efficient than a reversible process between the same initial and final states . The name motivity of a system was proposed by Lord Kelvin in 1879 for this conception of available energy . It is here specified as relative to a condenser of unlimited capacity at an assigned temperature To: some such specification is necessary to the See also:

• DEFINITION (Lat. definitio, from de-finire, to set limits to, describe)

definition; in fact, if To were the absolute zero, all the energy would be mechanically available .

But we can obtain an intrinsically different and self-contained comparison of the available energies in a system in two different states at different temperatures, by ascertaining how much energy would be dissipated in each in a reduction to the same standard state of the system itself, at a standard temperature To . We have only to See also:

• REVERSE

reverse the operation, and change back this standard state to each of the others in turn . This will involve abstractions of heat 8Ho from the various portions of the system in the standard state, and returns of SH to the state at To ; if this return were SHOT/To instead of OH, there would be no loss of availability in the .direct process; hence there is actual dissipation 8H—SHoT/To, that is T (34)--34o) . On passing from state r to state 2 through this standard state o the difference of these dissipations will represent the energy of the system that has become unavailable . Thus in this sense E —T4)+T4)o+const. represents for each state the amount of energy that is available; but instead of implying an unlimited source of heat at the standard temperature To, it implies that there is no extraneous source . The available energy thus defined differs from E—T4), the free energy of Helmholtz, or the work function of the applied forces of Kelvin, which involves no reference to any standard state, by a simple linear function of the temperature alonewhich is immaterial as regards its applications . The determination of the available mechanical energy arising from See also:

• DIFFERENCES, CALCULUS OF (Theory of Finite Differences)

differences of temperature between the parts of the same system is a more complex problem, because it involves a determination of the See also:

• COMMON

common temperature to which reversible processes will ultimately reduce them; for the simple case in which no changes of state occur the solution was given by Lord Kelvin in 1853, in connexion with the above train of ideas (cf . See also:

• TAIT, ARCHIBALD CAMPBELL (1811-1882)
• TAIT, ARTHUR FITZWILLIAM (1819-1905)
• TAIT, PETER GUTHRIE (1831-1901)

Tait's Thermodynamics, §179) . In the See also:

• PRESENT

present exposition the system is sensibly in equilibrium at each See also:

• STAGE (Fr. 6/age; from Lat. stare, to stand)

stage, so that its temperature T is always uniform throughout; isolated portions at different temperatures would be treated as different systems . Thermodynamic Potentials.—We have now to develop the relations involved in the general equation (1) of thermodynamics . Suppose the material system includes two coexistent states or phases, with opportunity for free interchange of constituents—for example, a See also:

• SALT (a common Teutonic word, cf. Dutch zout, Ger. Salz, Scand. salt; cognate with Gr. &Xs, Lat. sal)
• SALT, SIR TITUS, BART

salt solution and the aqueous vapour in equilibrium with it . Then in equilibrium a slight See also:

• TRANSFER (from Lat. transferre, to bear across, carry over)

transfer Sm of the See also:

• WATER

water-substance of See also:

• MASS (O.E. maesse; Fr. messe; Ger. Messe; Ital. messa; from eccl. Lat. missa)
• MASS, IN

mass m,. constituting the vapour, into the water-substance of mass m, existing in the solution, should not produce any alteration of the first order in SE—T34); therefore µr must be equal to µ, .

The quantity µr is called by Willard Gibbs the potential of the corresponding substance of mass m,.; it may be defined as its marginal available energy per unit mass at the given temperature . If then a system involves in this way coexistent phases which remain permanently separate, the potentials of any constituent must be the same in all of them in which that constituent exists, for otherwise it would tend to pass from the phases in which its potential is higher to those in which it is lower . If the constituent is non-existent in any phase, its potential when in that phase would have to be higher than in the others in which it is actually present; but as the potential increases logarithmically when the See also:

• DENSITY (Lat. densus, thick)

density of the constituent is indefinitely diminished, this condition is automatically satisfied —or, more strictly, the constitutent cannot be entirely absent, but the presence of the merest trace will suffice to satisfy the condition of equality of potential . When the action of the force of gravity is taken into account, the potential of each constituent must include the gravitational potential gh; in the equilibrium state the total potential of each constituent, including this part, must be the same throughout all parts of the system into which it is freely mobile . An example is See also:

• DALTON
• DALTON, JOHN (1766-1844)

Dalton's law of the independent distributions of the gases in the See also:

• ATMOSPHERE (Gr. fir µ6r, vapour; orkaipa, a sphere)

atmosphere, if it were in a state of See also:

• REST (O. Eng. rest, reste, bed, cognate with other Teutonic forms, e.g. Ger. Rast, Riiste, rest, and probably Gothic Rasta, league, i.e. resting or stopping place)

rest . A similar statement applies to other forms of mechanical potential energy arising from actions at a distance . When a slight constitutive change occurs in a galvanic See also:

• ELEMENT (Lat. elementum)

element at given temperature, producing available energy of electric current, in a reversible manner and isothermally, at the expense of chemical energy, it is the free energy of the system E— TO, not its total intrinsic energy, whose value must be conserved during the process . Thus the electromotive force is equal to the change of this free energy per electrochemical equivalent of reaction in the cell . This proposition, developed by Gibbs and later by Helmholtz, modifies the earlier one of Kelvin—which tacitly assumed all the energy of reaction to be available—except in the cases such as that of a Daniell's cell, in which the magnitude of the electromotive force does not depend sensibly on the temperature . The effects produced on electromotive forces by difference of concentrations in dilute solutions can thus be accounted for and traced out, from the knowledge of the form of the free energy for such cases; as also the effects of pressure in the case of gas batteries . The free energy does not sensibly depend on whether the substance is solid or fused—for the two states are in equilibrium at the temperature of See also:

• FUSION

fusion—though the total energy differs in these two cases by the heat of fusion; for this reason, as Gibbs pointed out, voltaic potential-differences are the same for the fused as for the solid state of the substances concerned . Relations involving Constitution only.—The potential of a component in a given solution can depend only on the temperature and pressure of the solution, and the densities of the various components, including itself; as no distance-actions are usually involved in chemical physics, it will not depend on the aggregate masses present .

The example above mentioned, of two coexistent phases liquid and vapour, indicates that there may thus be relations between the constitutions of the phases present in a chemical system which do not involve their total masses . These are developed in a very direct manner in Willard Gibbs's original procedure . In so far as attractions at a distance (a uniform force such as gravity being excepted) and capillary actions at the interfaces between the phases are inoperative, the fundamental equation (r) can be integrated . Increasing the volume k times, and all the masses to the same extent—in fact, placing alongside each other k identical systems at the same temperature and pressure—will increase el) and E in the same ratio k; thus E must be a homogeneous function of the first degree of the independent variables di, v, ml, . .., zn,,, and therefore by Euler's theorem relating to such functions E=Tpv+µimi+ .... +g,,mn• This integral equation merely expresses the additive character of the energies and entropies of adjacent portions of the system at uniform temperature, and thus depends only on the absence of sensible physical action directly across finite distances . If we form from it the expression for the complete differential BE, and subtract (I), there remains the relation o=SST—vap+m,Sµi+ .... } mnSµn . (2) This implies that in each phase the change of pressure depends on and is determined by the changes in T, µl, . . . µ„ alone; as we know beforehand that a physical property like pressure is an analytical function of the state of the system, it is therefore a function of these n+ i quantities . When they are all independently variable, the densities of the various constituents and of the entropy in the phase are expressed by the partial fluxions of p with respect to them: thus _—dp, mr_dp .

v dT v —dµ, But when, as in the case above referred to of chloride of ammonium gas existing partially dissociated along with its constituents, the masses are not independent, necessary linear relations, furnished by the laws of definite combining proportions, subsist between the partial fluxions, and the form of the function which expresses p is thus restricted, in a manner which is easily expressible in each special case . This proposition that the pressure in any phase is a function of the temperature and of the potentials of the independent constituents, thus appears as a consequence of Carnot's axiom combined with the energy principle and the absence of effective actions at a distance . It shows that at a given temperature and pressure the potentials are not all independent, that there is a necessary relation connecting them . This is the equation of stale or constitution of the phase, whose existence forms one mode of expression of Carnot's principle, and in which all the properties of the phase are involved and can thence be derived by simple differentiation . The Phase See also:

• RULE

Rule.—When the material system contains only a single phase, the number of independent variations, in addition to change of temperature and pressure, that can spontaneously occur in its constitution is thus one less than the number of its independent constituents . But where several phases coexist in contact in the same system, the number of possible independent variations may be much smaller . The present independent variables Al, ..., are specially appropriate in this problem, because each of them has the same value in all the phases . Now each phase has its own characteristic equation, giving a relation between bp, ST, and See also:

• BILL

bill, . Sµn, or such of the latter as are independent; if r phases coexist, there are r such relations; hence the number of possible independent variations, including those of v and T, is reduced tom—r+2, where m is the number of independently variable chemical constituents which the system contains . This number of degrees of constitutive freedom cannot be negative; therefore the number of possible phases that can coexist alongside each other cannot exceed m+ 2 . If m+ 2 phases actually coexist, there is no variable quantity in the system, thus the temperature and pressure and constitutions of the phases are all determined; such is the triple point at which See also:

• ICE (a word common to Teutonic languages; cf. Ger. Eis)

ice, water and vapour exist in presence of each other . If there are m+i coexistent phases, the system can vary in one respect only; for example, at any temperature of water-substance different from the triple point two phases only, say liquid and vapour, or liquid and solid, coexist, and the pressure is definite, as also are the densities and potentials of the components .

Finally, when but one phase, say water, is present, both pressure and temperature can vary independently . The first example illustrates the case of systems, physical or chemical, in which there is only one possible state of equilibrium, forming a point of transition between different constitutions; in the second type each temperature has its own completely determined state of equilibrium; in other cases the constitution in the equilibrium state is indeterminate as regards the corresponding number of degrees of freedom . By aid of this phase rule of Gibbs the number of different chemical substances actually interacting in a given complex system can be determined from observation of the degree of spontaneous variation which it exhibits; the rule thus lies at the foundation of the modern subject of chemical equilibrium and continuous chemical change in mixtures or See also:

• ALLOYS (through the Fr. aloyer, from Lat. alligare, to combine)

alloys, and in this connexion it has been widely applied and developed in the experimental investigations of Roozeboom and See also:

• VAN

van 't Hoff and other physical chemists, mainly of the Dutch school . Extent to which the Theory can be practically developed: It is only in systems in which the number of independent variables is small that the forms of the various potentials,—or the form of the fundamental characteristic equation expressing the energy of the system in terms of its entropy and constitution, or the pressure in terms of the temperature and the potentials, which includes them all,—can be readily approximated to by experimental determinations . Even in the case of the simple system water-vapour, which is fundamental for the theory of the See also:

• STEAM
• STEAM (0. Eng. steam, vapour, smoke, cf. Du. steam; the origin is unknown)

steam-engine, this has not yet been completely accomplished . The general theory is thus largely confined, as above, to defining the restrictions on the degree of variability of a complex chemical system which the principle of Carnot imposes . The tracing out of these general relations of continuity of state is much facilitated by geometrical diagrams, such as See also:

• JAMES
• JAMES (Gr. 'IlrKw,l3or, the Heb. Ya`akob or Jacob)
• JAMES (JAMES FRANCIS EDWARD STUART) (1688-1766)
• JAMES, 2ND EARL OF DOUGLAS AND MAR(c. 1358–1388)
• JAMES, DAVID (1839-1893)
• JAMES, EPISTLE OF
• JAMES, GEORGE PAYNE RAINSFOP
• JAMES, HENRY (1843— )
• JAMES, JOHN ANGELL (1785-1859)
• JAMES, THOMAS (c. 1573–1629)
• JAMES, WILLIAM (1842–1910)
• JAMES, WILLIAM (d. 1827)

James See also:

• THOMSON, JAMES (1822-1892)
• THOMSON, JAMES (1834-1882)
• THOMSON, JAMES (r700-1748)
• THOMSON, JOHN (1778-184o)
• THOMSON, JOSEPH (1858-1895)
• THOMSON, SIR CHARLES WYVILLE (1830-1882)
• THOMSON, THOMAS (1773-1852)
• THOMSON, WILLIAM (1819-189o)

Thomson first introduced in order to exhibit and explain Andrews' results as to the range of coexistent phases in carbonic acid . Gibbs's earliest thermodynamic See also:

• SURFACE

surface had for its co-ordinates volume, entropy and energy; it was constructed to See also:

• SCALE
• SCALE (1) A

scale by Maxwell for water-substance, and is fully explained in later See also:

• EDITIONS

editions of the Theory of Heat (1875); it forms a See also:

• RELIEF

relief See also:

• MAP
• MAP (or MAPES), WALTER (d. c. 1208/9)

map which, by simple inspection, reveals the course of the transformations of water, with the corresponding mechanical and thermal changes, in its three coexistent states of solid, liquid and gas . In the general case, when the substance has more than one independently variable constituent, there are more than three variables to berepresented; but Gibbs has shown the utility of surfaces representing, for instance, the entropy in terms of the constitutive variables when temperature and pressure are maintained constant . Such graphical methods are now of fundamental importance in connexion with the phase rule, for the experimental exploration of the trend of the changes of constitution of complex mixtures with interacting components, which arise as the physical conditions are altered, as, for example in modern metallurgy, in the theory of alloys . The study of the phenomena of condensation in a mixture of two gases or vapours, initiated by Andrews and developed in this manner by van der Waals and his pupils, forms a case in point (see CONDENSATION OF GASES) . Dilute Components: Perfect Gases and Dilute Solutions.—There are, however, two simple limiting cases, in which the theory can be completed by a determination of the functions involved in it, which throw much See also:

• LIGHT

light on the phenomena of actual systems not far removed trom these ideal limits .

They are the cases of mixtures of perfect gases, and of very dilute solutions . If, following Gibbs, we apply his equation (2) expressing the pressure in terms of the temperature and the potentials, to a very dilute solution of substances m2, m,, . . . M,, in a solvent substance m,, and vary the co-See also:

• ORDINATE

ordinate mr alone, p and T remaining unvaried, we have in the equilibrium state din dµ, dg„ mrdnr+m'dmr+ ... + m„dmr =o, in which every m except m1 is very small, while dµ,/dmr is presumably finite . As the second See also:

• TERM

term is thus finite, this requires that the total potential of each component mr, which is mrdµr/dmr, shall be finite, say kr, in the limit when mr is null . Thus for very small concentrations the potential µr of a dilute component must be of the form krlogmils, being proportional to the See also:

• LOGARITHM (from Gr. Abyos, word, ratio, and apx0µos, number)

logarithm of the density of that component; it thus tends logarithmically to an See also:

• INFINITE (from Lat. in, not, finis, end or limit; cf. findere, to deave)

infinite value at evanescent concentrations, showing that removal of the last traces of any impurity would demand infinite proportionate See also:

• EXPENDITURE

expenditure of available energy, and is therefore practically impossible with finite intensities of force . It should be noted, however, that this argument applies only to fluid phrases, for in the case of deposition of a solid in, is not uniformly distributed throughout the phase; thus it remains possible for the growth of a crystal at its surface in aqueous solution to extrude all the water except such as is in some form of chemical combination . The precise value of this logarithmic expression for the potential can be readily determined for the case of a perfect gas from its characteristic properties, and can be thence extended to other dilute forms of matter . We have pv=R/m.T for unit mass of the gas, where m is the molecular See also:

• WEIGHT

weight, being 2 for See also:

• HYDROGEN [symbol H, atomic weight x-oo8 (0=16)]

hydrogen, and R is a constant equal to 82 X toe in C.G.S. dynamical See also:

• UNITS, DIMENSIONS OF
• UNITS, PHYSICAL

units, or 2 calories approximately in thermal energy units, which is the same for all gases because they have all the same number of molecules per unit volume . The increment of heat received by the unit mass of the gas is aH=psv+KaT, K being thus the specific heat at constant volume, which can be a function only of the temperature . Thus $ = jdH/T = R/m. See also:

• LOG (a word of uncertain etymological origin,possibly onomatopoeic; the New English Dictionary rejects the derivation from Norwegian lag, a fallen tree)

log v+ f sT-1dT; and the available energy A per unit mass is E—Tclt+Tito where H;=E+fed'l', the integral being for a standard state, and e being intrinsic energy of chemical constitution; so that A = E+doT+ f rcdT —T f sT -1dT — R/m.T log v .

If there are v molecules in the unit mass, and N per unit volume, we have my = Nmv, each being 2 where v' is the number of molecules per unit mass in hydrogen; thus the free energy per See also:

• MOLECULE (from mod. Lat. molecula, the diminutive of moles, a mass)

molecule is a'+R'TlogbN, where b=m/2v', R'=R/2v', and a' is a function of T alone . It is customary to avoid introducing the unknown molecular constant v' by working with the available energy per " gramme-molecule," that is, for a number of grammes expressed by the molecular weight of the substance; this is a constant multiple of the available energy per molecule, and is a+RT logy, p being the density equal to bN where b=m/2v' . This formula may now be extended by simple summation to a mixture of gases, on the ground of Dalton's experimental principle that each of the components behaves in presence of the others as it would do in a vacuum . The components are, in fact, actually separable wholly or partially in reversible ways which may be combined into cycles, for example, either (i.) by diffusion through a porous See also:

• PARTITION

partition, taking account of the work of the pressures, or (ii.) by utilizing the modified constitution towards the top of a long See also:

• COLUMN (Lat. columna)

column of the mixture arising from the action of gravity, or (iii.) by reversible absorption of a single component . If we employ in place of available energy the form of characteristic equation which gives the pressure in terms of the temperature and potentials, the pressure of the mixture is expressed as the sum of those belonging to its components: this equation was made by Gibbs the basis of his analytical theory of gas mixtures, which he tested by its application to the only data then available, those of the equilibrium of dissociation of See also:

• NITROGEN [symbol N., atomic weight 14.01, 0=16]

nitrogen peroxide (2NO2 N204) vapour . Van 't Hoff's Osmotic Principle: Theoretical Explanation.—We proceed to examine how far the same formulae as hold for gases apply to the available energy of matter in solution which is so dilute that each molecule of the dissolved substance, though possibly the centre of a complex of molecules of the solvent, is for nearly all the time beyond the See also:

• SPHERE (Gr. vclsa?pa, a ball or globe)

sphere of direct See also:

• INFLUENCE (Late Lat. influentia, from influere, to flow in)

influence of the other molecules of the dissolved substance . The available energy is a function only of the co-ordinates of the matter in hulk and the temperature; its change on further dilution, with which alone we are concerned in the transformations of dilute solutions, can depend only on the further separation of these molecular complexes in space that is thereby produced, as no one of them is in itself altered . The change is therefore a function only of the number N of the dissolved molecules per unit volume, and of the temperature, and is, per molecule, expressible in a form entirely independent of their constitution and of that of the See also:

• MEDIUM

medium in which they are dissolved . This suggests that the expression for the change on dilution is the same as the known one for a gas, in which the same molecules would exist free and in the main outside each other's See also:

• SPHERES, MUSIC OF THE

spheres of influence; which confirms and is verified by the experimental principle of van 't Hoff, that osmotic pressure obeys the laws of gaseous pressure with identically the same physical constants as those of gases . It can be held, in fact, that this See also:

• SUGGESTION

suggestion does not fall short of a demonstration, on the basis of Carnot's principle, and independent of special molecular theory, that in all cases where the molecules of a