258 PHILIPS TECHNICAL REVIEW VOL. 16, No. 9

ENTROPY IN SCIENCE AND TECHNOLOGY

I. THE CONCEPT OF ENTROPY

by.J. D. FAST.

The a.lIIhor,who is not unknown to our readers by virtue of his many articles on metallurgy,has made a thorough study of the concept of entropy and written a widely·read book on thissubject. It is his intention, in a series of articles in this Review, to consider the significanee ofthe concept of entropy in various fields of science and technology, Owing to the abstract way inwhich entropy is dealt with in classical thermodynamics, it is a less familiar concept thanthat of energy, though nó less important. The author devotesparticular attention to the statisticalbackground of entropy, in an attempt to make it as readily understandable as the concept ofenergy. It is surprising how many widely divergent problems may be resolved with the aidof the concept of entropy. A general survey like this may co,{tribute in some measure towardsa general synthesis, all the more urgently needed as specialization in the exact sciences b~comesmore acute.

The present article, the first of the series, is devoted to the essential meaning of entropy.The subsequent articles will be mainly concerned with examples of the application of theentropy concept.

Introduetion

All phenomena in nature are subject to the lawsof thermodynamics, These well-established. lawsenable us not only to calculate the maximum POSR-

ible efficiency of engines and to predict the directionand the maximum yield of chemical reactions, butthey are also of fundamental importance in almostevery field of science and technology.

The first and second laws may' be formulated inmany different ways. At first sight the various for-mulations seem to bear little or no relation to eachother, but essentially they are equivalent. Whenapplied to an isolated system, i.e, a system withoutinteraction with the outside world, they may, forexample be worded as:First law: The total energy of an isolated systemis constant.Second law: The entropy of an isolated system tendstowards a maximum.

The first law of thermodynamics

The first law of thermodynamics, which is some-times called the law of conservation of energy,finds its origin in the empirical knowledge that heatand mechanical work are both forms of energy andthat the one may be converted into the other.If a system is not isolated (e.g. a quantity of gas

in a cylinder under a movable piston), then anamount of heat dQ may be added to it, or an amountof work dW may be done on it. According to thefirst law the whole of this added energy must appearin the system as an increase in its internal energy U,i.e.,

dU = dQ+ dW. . .... (1,1)

536.75

From the point of view of classical thermodyna-mies - i.e. independent of the state of agregationof matter or the precise physical form of the energy- the concept of internal energy gains a significanceonly by virtue of this mathematical definition. Ifthe atomic state of agregation of matter is consi-dered, the internal energy of a system is the s~m ofthe kinetic and potential energies of all the elemen-tary particles of which the system eonsists. Theinternal energy depends solely"on the thermodynamicstate of the system, i.e. on its pressure, temperature,volume, chemical compositïon, structure, etc. Thehistory of the system does not influence its value.

For this reason U is called a thermodynamicfunction. Wand Q are not thermodynamic functions,since according to equation (I, 1) the same changein internal energy dU can be brought about eitherby supplying only heat, or by only doing work onthe system. It is therefore possible to speak of theinternal energy of a system, but not of the quantityof work or the quantity of heat of that system.In other words: dWand dQ are only infinitesimalquantities of work and heat, and' not differentialsof thermodynamic functions.

T~e ~econd law of thermodynamies

Although W is not a thermodynamic functionand dW is not a differential, the latter can generallybe expressed as the product of an intensive propertyof the system and the differential of an extensiveproperty of the system. The meaning of these termsis given by the fact that a system in equilibriumcan always be divided into two equal parts suchthat those thermodynamic properties which are

This formula, apart from providing the definitionof the thermodynamic function S (and, strictlyspeaking, also of the absolute temperature T), alsorepresents the mathematical formulation of thesecond law of thermodynamics.For readers without a previous knowledge of

thermodynamics the foregoing will still leave theconcept of entropy completely obscure. One alsofeels the lack of any connection between the givenformulation and the above-mentioned tendencyof the entropy towards a maximum. To bring somelight into this darkness it may be useful to leavethe path of pure thermodynamics and to considerthe atomic aspect of the matter. Only after ex-plaining the atomic aspect of the concept of entropy,shall we return to the thermodynamical dcfinition(1,2), and demonstrate how this should be modifiedin order to express the tendency of entropy to- Fig. 1. A regular distribution of white and red billiard balls.

wards a maximum.

MRACH 1955 ENTROPY, I

extensive (e.g. volume) are halved, while those whichare intensive (e.g. pressure) remain unchanged.The work done on a gas by compressing it can, forexample, be expressed as:

dW = -pdV,

in which p, the pressure of the gas is the intensiveproperty of the system, and V, its volume the exten-sive property. In so far as p refers to the internalpressure, this formula applies only to a reversiblechange of volume, i.e. such a change that the exter-nal pressure always differs only infinitesimally fromthe internal pressure. Analogously, dQ may beexpressed as:

dQ = TdS,

in which T, the temperature of the system, is theintensive property and S, the entropy, the extensiveproperty. This formula, too, applies only if thc heatis supplied in a reversible manner, i.e. suppliedfrom a source whose temperature is only infinitesi-mally higher than that of the system. We may thuswrite:

dQrevT

. . (1,2)dS =

I rreoersible processes

The second law of thermodynamics finds itsorigin in the experience that all spontaneonslyoccurring processes take place in one direction onlyand are, therefore, irreversible,If the previously considered isolated system

consisted, say, of a vessel containing neon and he-lium under such conditioris that they may be con-sidered perfect gases, then the first law would per-mit any imaginable distribution of the gas molecules

259

In the available space: pressure and temperaturedifferences may exist between different parts of themixture without affecting in any way the internalenergy of the system. Experience tells us, however,that irrespective of the initial state, the ultimatestate of the system left to itself (the "equilibriumstate") is always one in which the gases have mixedhomogeneously and in which pressure and temper-ature are uniform. After reaching this ultimate state,the system will nevel spontaneously return to oneof its previous states.

How can we explain this tendency towards homo-geneous mixing? May we say that the helium andthe neon atoms have a certain "preference" forhomogeneous distribut.iou and, if so, on what isthis preference founded?In order to deal with these questions, let us imaginethat a small number of white and red billiard balls(e.g. 50 + 50) are substituted for the helium andneon atoms. As the initial state we select a given,regular distribution of the balls in the vessel (fig.l),and the thermal motion of the atoms is simulated by

thoroughly shaking the vessel for some time. Weknow from experience that after shaking, theorderly initial state will never be found again; arandom distribution of the white and red balls willalways be found. Yet we must take it as axiomaticthat each separate random distribution is just asprobable or improbable as the initial distribution.In actual fact, however, and here we hit the coreof the problem, there are so many more possiblerandom distributions than regular ones that vir-tually only random distributions will be found

260 PHILlPS TECHNICAL REVIEW VOL. 16, No. 9

after shaking. This point m!ly be illustrated withthe aid o.f a simple calculation.

A statistical calculation

If all 100 balls could be distinguished from oneanother (e.g. because they were numbered), therewould be 100 X 99 X 98 X ... X 2 X 1 = lOO!different ways of arranging them in tbe 100 availablespaces in the vessel. .

Since, however, the 50 red balls are, in fact,indistinguishable (not numbered), any interchan-ging of 2 red balls leaves the distributton unaltered,so that the number of possible arrangements thatcan be distinguished by eye (m) is far smaller.This number, nevertheless, is still enorm.ous.Taking into account the fact that the 50 whiteballs are also mutually indistinguishable, it is givenby 1) lOO! .

m = ---- = 0.08 X 2100 = l.01 X 1029•50! 50!

The chosen number of balls .is too small forapplying the roughest approximation of Stirling'sformula 2)

InN!R::i NlnN-N . . .. (I,3)

since then the result for m would be

m = 2100 = l.26 X 1030,

a value which is too large by more than a factor of 12.Itwill be readily appreciated that the number of 2100includes not only all possible distributions of 50white and 50 red balls, but also all distributions ofa total of 100 balls irrespective of the ratio of redto white, i.e. all distributions of 49 white and 51red balls, of 48 white and 52 red balls, etc. Withoutthe condition of the 50/50 ratio, we have the follow-ing situation. The first place to be occupied by aball provides a choice of'two possibilities (white orred); for occupying two places, each of the twocolours may be combined with each of the twocolours, so that here one has 22 possibilities (w-w,w-r, row and r-r), and so on.According to the result m = '1029, an avelage of

1029 shakings are necessary to obtain one givendistrihution of the balls. If each shaking action ismade to last one minute, this means that on theaverage one would have to shake for 1023 years inorder to obtain one given distribution. This con-

. stitutes a sufficient explanation of t he empirical

1) Exact values of N! for integers up to N = 100 are givenin Barlow's Tables, E. and F. N. Spon, London. Fairlyexact values are given by Stirling's formula in the ap-

. NN ,--proximation N! ="[i{ 1 2nN.e

2) Throughout this article we shall use In in place of themore cumbersome loge, to denote natural logarithms.

knowledge that the chance of obtaining by shaking aperfectly regular distribution, e.g. an arrangementof the white and the red balls in separate layersor one in which each white ball is exclusively sur-rounded by red ones and vice versa, is practicallynil in view of the comparatively small number ofregular arrangements compared to the number g. of the irregular distributions (g ~ m).

The foregoing considerations were concerned withthe small number of 100 balls. Returning to ourgas mixture and assuming that it consists of 0.5gram-atom of Ne and 0.5 gram-atom of He, wearrive at a total number No = 0.6 X 1024 atoms.A calculation analogous to the one carried out for100 "atoms", now gives for m the value

N. I ,m = o· = 2N,-40 R::i 10(2xIO")-12.

~(1/2 No) q2The number 40 in the exponent ~f 2 is negligible

with respect to No = 6X 1023; moreover, the valueof Avogadro's number is known to so few decimalsthat from a physical point of view there is nopoint in distinguishing between No and No - 4·0.We may thus write:

and this is, according to the foregoing, nothingbut the total number of distributions of No atomsof two types. In other words: at large values of Nand with the ratio 1: 1 there occurs so sharp amaximum in the curve of the number of distribu-tions as a function of the mixing ratio that there ispractically no difference whether we take intoaccount the number belonging to this maximumor the total number of distributions. It is of someimportance that the approximation (I, 3) also givesthe result m = 2N•• Hence this formula is a per-fectly satisfactory approximation when applied tothe numbers of atoms normally dealt with inpractice.

Such a staggering number as 2N, is, of course,quite beyond human comprehension; the numberof orderly distributions of the atoms is negligiblysmall when compared to this total number m.

Macro and micro-states

As already stated, we are bound to assume thatwith our shaking experiment all m = 1029 distribu-tions ofthe balls (all "micro-states") possess an equalprobahility tv, 1

W=-.m

. (I,4)

,The various micro states may be assembled intogroups ("macro-states") each of which is character-

MARCH 1955 ENTROPY,I

ized by a certain extent of disorder. A regular.arrangement as shown in fig. 1, is only possible inone way, although one might regard it as beingpossible in two ways (horizontal layers w-r-w-r orrow-row). The same applies to a three-dimensionalcheckerboard pattern in which each red ball hasonly white and each white ball has only red ballsas 'its nearest neighbours. An "imperfection" maybe "introduced into any orderly distribution byinterchanging a red and a white ball. Since theposition of each of the 50 white balls can be inter-changed with that of each of the 50 red balls, thisparticular macro-state with one imperfection com-prises 2500, micro-states. If the number of imper-fections is increased to two or more, then weobtain macrostates comprising considerably moremicro-states. The probability tv of each macro-state is determined by the number of micro-statesor a priori equally-probable arrangements ginherent to this state:

gw = - (1,5)

m

The significanee of formula (I, 5) may be illustrated bya very simple example. When throwing dice, the probabilityof the macro-state"even ", comprising the three micro-states2, 4 and 6, for a single die, is given by

3 1w (even) = '6 = '2'

obviously with the provision that all 6 faces of the die have anequal pro~ability of coming on top, i.e. that the die is properlymade.

Returning to the atomic case, the various atomconfigurations or micro-states may be combined ingroups, again designated macro-states, the pro-bability of each being determined by formula (I, 5).The choice of the groups is dependent upon theproperties of the system under consideration.If the foregoing is applied to our mixture of

ideal gases, it will be clear that at a given momentit is in a 'given micro-state. Due 'to the motion ofthe gas molecules this micro-state is continuouslychanging. It is justifiable to assume that in thecourse of time the system passes through all spatialdistributions (micro-states) that are possible withinthe scope of the available volume. There is, how-ever, one particular MACRO-state 3), in which thegases, for as far as can be ascertained by macro-

3) As implied above, a macro-state is taken, in this article,to mean any group of micro-states. Sometimes, 'however,it is used to describe a thermodynamical state which ischaracterized by a small number ofmacroscopie quantities,such as temperature, pressure, volume etc. Such MACRO-states, which as a rule comprise many macro-states, willhenceforth be designated by MACRO in capitals. TheMACRO-states can be physically distinguished; the macro-states generally cannot.

261

scopic measuring equipment, are homogeneouslymixed and of uniform density. This MACRO-statecomprises such an enormously greater number ofmicro-states thàn all other macrostates put togetherthat after any interval, even if short, it is alwaysfound to be present to the exclusion of all the othermacro-states. This is called the state of equilibrium, .because the system always returns to it of its own'accord, irrespective of its initial distribution. Thisstate of equilibrium is also the state of maximumentropy.

Flucuuuion. phenomena

The answer to the questions put under the headingIrreuersible processes, concerning the tendency to-wards forming a homogeneous mixture, shouldapparently be that the reason for this spontaneouslyoccurring state is just the fact that there exists nopreference for any partienlar micro~state. In thestate of equilibrium the system passes continuouslyfrom one micro-state into another but as a rule theyare macroscopically indistinguishable. Only undervery special conditions can fluctuations aroundthe state of maximum entropy be observable.

The blue colour of the clear sky, for instance,reveals the occurrence of local fluctuations in thedensity of the air, whilst also the well-knownBrownian movement in a colloidal suspension is dueto the irregular thermal agitation of the mole-cules of the medium.A similar fluctuajion phenomenon occurs in a

conductor due to the thermal agitation of theelectrons. Thus extremelysmall alternating voltagesarise spontaneously between the ends of a resistor.The arithmetic mean of this voltage averaged overa considerable period of time is of course zero, butthis is not the case with its r.m.s. value. This pheno-menon is termed thermal noise, because aftersufficient amplification these alternating voltages'can he heard as noise through a loudspeaker.These spontaneous voltage fluctuations may beresolved into components with various frequencies.In 1928 Nyquist demonstrated that, with theexception of the very high frequencies, all fre-quencies are uniformly represented in the fluctuationspectrum. He further showed that the effect ofthe fluctuations in an electrical network can becomputed by assuming in series with each resistoran imaginary electromotive force E such that E2 =4kTrLl'V. In this, kis Boltzmann's constant, T theabsolute temperature, r the resistance and Ll'V thefrequency range [handwidth} occurring under thegiven conditions. For a network consisting of aresistance T and a parallel capacitance C, this relat-

262 PHILIPS TECHNICAL REVIEW VOL. 16, No. 9

ion gives a mean square voltage V2 = kT/C. Thisis in accordance with the theorem of equipartitionof energy: 1/2CV2 = 1/2 kT.

On the basis of the above-mentioned equivalentcircuit it can readily be demonstrated that themaximum power of the noise arising across a resistoris kT L.!v. This applies not only to a normal resistorbut also to an aerial or a cable in which no endreflections occur. A noise voltage is, thereforesuperimposed upon every signal voltage. The flue-tuating character of this noise renders it impossibleto observe the finer details of the signal voltage;it puts a fundamental restrietion on the amount of"information" that may be transmitted by electricsignals in given circumstances. We shall return tothis in the last article of this series.

, Quantum states

In the foregoing we have mainly concerned our-selves with the number of micro-states in relationto the mixing of two kinds of particles. Of' evengreater importance are the micro-states corres-ponding to the thermal energies of the particles.

Consider a system of identical atoms, in the formof a crystal, or say, as a gas confined to a certain volu-me. In the crystal each particle is allotted a volumeof the order of 10-23 cm'' in which it can execute itsthermal oscillations; in the gas, however, each par-ticle can move throughout the entire gas volume of,e.g., 102 or 103 cm''.

Modern physics teaches us that a particle con-,fined within a restricted space, can only exist incertain, discrete quantum states. Correspondingto each quantum state are a specific energy levelof the particle, and a specific wave function, thelatter being related to the probability of finding therelevant particle in the different regions of theavailable space. One of the fundamental problemsof statistical thermodynamics is that of determiningthe distribution of a system of N identical particlesamong the various quantum states of the systemat a given value U of the total energy. The determ-ination is based on the hypothesis that all micro-states, by analogy With the example of the billiardballs, have an equal a priori probability. Thishypothesis has been confirmed by the successes ofstatistical thermodynamics.

In order to demonstrate the various distributionpossibilities among the available quantum stateswe shall consider a greatly simplified model of asolid, known as an Einstein solid, in which theatoms execute their thermal vibrations virtuallyindependently of one another. Since a certain inter-action is necessary to attain the thermal equilibrium,

there is assumed to exist a negligibly small inter-atomic coupling, enabling the atoms to exchangetheir energies.

For the present we shall overlook the fact thatan atom in' a solid has three vibrational degrees offreedom. In our model this number is reduced toone, i.e. we are concerned with an idealized solid-in which the atoms behave as linear harmonicoscillators vibrating about fixed centres. Thesecentres are arranged in ~pace according to the pointsof a crystallattice. According to quantum mechanicsthe energy levels of these localized oscillators arespaced equally from· one another, i.e. in additionto their lowest energy EO they may take, up amountsof energy El= h», E2 . 2 lw, etc., where v representsthe frequency of their fundamental vibration andh is Planck's constant. Each energy ievel corre-sponds to one given quantum state of a particle.

We shall first consider a very small number ofoscillators viz. 25, represented by one of the fourhorizontal layers of balls of one colour in fig. LAta temperat:ure of absolute zero, all oscillators arein the state of energy EO' i.e. at the lowest energylevel. When the temperature is raised by supplyingenergy to the system, particles are raised from theground state to higher quantum states, i.e. to higherenergy levels. The essential point in our consider-ation is again the number of different ways in whichthe energy can be distributed. If we supply a totalof 25 energy quanta h» to the 25 oscillators, eitherby applying heat dQ or by exerting work dW uponthe system, we wish to know the number of possibleways in which the energy dU = 25 Iw can be distri-buted among the oscillators. As with the shakingexperiment we shall ignore for the moment the factthat the system under consideration is too smallfor a profitable application of statistical-thermo-dynamics. For the time being our aim is only todemonstrate the method of counting. If each of theoscillators receives one quantum, then 'the inter-changing of two atoms does not create a new state.In other words, the uniform distribution of theenergy can only be realized in one way; it representsone micro-state and will, consequently, occur veryrarely. .A less regular distribution shows an entirely

different picture. An example of a distribution ofthis type is schematically represented in fig. 2. Thesix oscillators in the positions C3, C4., C5, Dl, D2,D3 have each absorbed one quantum, the fouroscillators in D4, D5, El, E2 each two quanta, thetwo oscillators in E3, .E4 each three quanta andthe oscillator in E5 five quanta. The remainingtwelve oscillators have not taken up any energy.

:MARCH 1955 ENTROPY, I 263

Because each oscillator has its own position in the"lattice", any interchanging of two ~scillatorshaving different quantum numbers will create a newmicro-state. (This is not the case in a gas, in whicheach particle has access to the whole gas volume.)

ABC 0 E

2

0 0 0 ( 2

0 0 0 ( , 2

0 0 f f 3

0 0 f 2 3

0 0 f 2 5

3

5

Fig. 2. Schematic representation of a certain distribution of25 energy quanta among 25 oscillators. Each small squarecorresponds to one oscillator. The' number in it shows thenumber of quanta per oscillator. The numbers shown can bedistributed among the squares in approximately 1012 differentarrangements; each arrangement corresponds to one micro-state. The total number of possible distributions of 25 quantaamong 25 oscillators is substantially greater, viz. approxi-mately 6 X 1013•

The number of distributions in which any sixoscillators have absorbed one quantum each, anyfour oscillators two each, any two oscillators threeeach, and anyone oscillator five quanta is foundfrom 25!

g = = 9.40 X 1011 ~ 1012•12! 6! 4! 2! 11

Instead of speaking of a distribution' of 25 quantaamong 25 oscillators as above, we may just ~s wellstate that the 25 oscillators have been distributedamong the available energy levels in such a waythat 12 are at the lowest level, 6 at level I, etc. Wethus obtain the diagram in jig. 3. In view of the

~~-------------------~---------------ê3 ............ -------e2 ••••e1 ••••••eo ••••••••••••

(J1I16

Fig. 3. Macro-state comprising the 1012 micro-states, one ofwhich is shown in fig. 2.

fact that the spatial positions AI, A2, ·etc. of theoscillators cannot be derived from this diagram,this schematic representation corresponds to amacro-state comprising the 1012 micro-states dis-cussed above, one of which was shown in fig. 2.

The most probable macro-state and the total numberoj' micro-states

The general expression for the number of micro-states forming a given macro-state of our

idealized solid is given, according to the foregoing,by

N!. . (1,6)

where N is the number of oscillators, and thenumbers no, nI' n2", • • represent the populationsof the corresponding energy levels, i.e. the numberof oscillators with energies eo, el' e2 . . . (cf.' fig. 3,in which no = ,12, nl = 6, etc.). Only those setsof populations are allowed which satisfy theauxiliary conditions: .

2:ni ' N,2:niei = 'U,

. (1,7)

.- (1,8)

in which U = qh» is the total energy supplied ov~rand above the zero-point energy (q = number ofquanta absorbed).

The most probable macro-state, according to (I,5), is the one containing the largest number ofmicro-states. Without entering into the calculation,we mention the fact that this maximum in g occurs(observing the auxiliary conditions (I, 7 and I, 8»,if the populations Tio, nI' n2 • • • form a descendinggeometrical series. For the numerical example underconsideration, fig. 3 shows one of the most probablemacro-states. (The most probable distribution forN = 25, q = 25, has the set of populations: no =11, nl = 7, n2 = 4, na= 2, n4 = 1.) Different macro-states are characterized by different ni-series. Thetotal number of micro-states m is determined bythe sum of all expressions of the form (I, 6) whichsatisfy the auxiliary conditions (I, 7) and (I, 8).In our example, m mayalso be evaluated in a farsimpler way by directly counting the number of

emicro-states which form the MACRO-state definedby the number of oscillators N and the total energyU (= qhv). We may imagine that the q quanta andN - I of the N oscillators are arranged in arbitrary~rder along a straight line. At the right-hand endwe place the last, the Nth oscillator. If the quantabetween two oscillators are considered as belongingto the oscillator to the right, then each sequencerepresents a complete distrihution, since the lastposition is always occupied by an oscillator.

The total number of micro-states is thus givenby ihe total number of different arrangementspossible on a straight line. Noting that the oscillator'last placed in position has no ambiguity of locationand that the interchange of two oscillators or of twoquanta .leaves the sequence unaltered, the totalnumber of possible arrangements is given by:

(q+N-l)!m = ..... (1,9)

q!(N-l)!

264. PHILIPS TECHNICAL REVIEW VOL. 16, No. 9

If this formula is applied to our numericalexample, we find:

(25 + 24) im = = 6.3 X 1013•

25! 24! / .

With the two problems considered above, viz.the mixing of two kinds of particles and the dis-tribution of quanta among localized oscillators itis permissible, if the number of particles or the num-ber of quanta and oscillators is very large, to sub-stitute the total number of micro-states for the num-ber of micro-states of the most probable arrange-ment. This is justifiable for the same reason thatit was permissible in the former example to replace2N.-40 by 2N, notwithstanding the fact that thelatter is 240 times greater. This reasoning is furtherreinforced by the fact - to be explained below -that we are not so mu~h interested in g and m them-selvesas in thenaturallogarithms ofthese quantities,and while gmaxlm becomes smaller and smaller withan increasing number of particles, the value of(Ingmax)/(lnm) approaches closer andcloserto unity.We shall demonstrate this by calculating these

ratios 'for three different sets of populations of thethree lowest levels, viz. for the following numbersof oscillators N and quanta q:

1 2 I 3

N 111 1110 11100

q 12 120 1200

The most probable distributions for these 'threecases are given by the populations:

I 1 2 3

no

I100 1000 10000

n1 10 100 1000n2 1 10 100

With the aid of (I, 6) and (I, 9), and using moreaccurate factorials 4) than could be obtained via(I, 3), we can now obtain:

NI m gmnx/m I (lngmnx)/(In m)--111 5.2X 1015 1.3 X 1016 4X 10-1 0.975

1110 l.0 X 10168 2.5x 10169 4X 10-2 0.99211100 3.6 X 101699 8.7 X 101705 4X 10-7 0.996

Hence' we see that as the number of particles.increases, the value of (In gmax)/ln m) approachesunity. Real crystals usually contain at least 1020,and in most cases from 1021 to 1024 atoms, For such

4) Here we use the more accurate approximation n! =nn1'2nn.en

numbers, of oscillators m is enormously greater thangmax, whilst at the same time the difference betweenIn gmax and In m is negligible.

The statistical (atomic) definition of entropy

The state of greater entropy towards which asystem strives according to the .second law, is, ashas been demonstrated in the foregoing, the MACROstate with the largest number of equally probablearrangements, i.e. the most probable state. Thisbrings us to the statistical definition of the' conceptof entropy:,

s= kIng, (1,10)

ill which g is the number of micro-states fromformula (I, 5) and k is Boltzmann's constant,derived from

k = RINo• • • • • '. • • (I,ll)

(R = gas constant; No = Avogadro's number).As explained in detail above, one may in many

cases substitute the total number of micro statesm for the number of micro-states g of the mostprobable arrangement, One then obtains the alter-native statistical definition of S:

S = k In m. . . . . . . (1,12)

The statistical evaluation of the. entropy thusamounts to counting nu~bers of micro-states, Asan example of such a procedure we shall calculatethe increase in entropy which accompanies iso-thermal expansion of a perfect gas. Ifwe attemptedthis starting from the quantum states of the' mole-cules, the derivation would become rather complic-ated, dué to the fact that in quantum mechanics eachmicro-state bears a relation both to the energy and

oto the distribution of the molecules in space (seeabove), and the intervals between energy levelswill decrease due to the enlargement of the volume.In the case in question it is permissible and moreconvenient to use the cl~ssical method of consider-ing separately the number of possible moleculardistributions in the available space and their dis-tribution among the various velocities. So far asthe velocity distribution is concerned, the entropydoes not change on isothermal expansion. The in-crease in entropy to be calculated is entirely thatdue to the larger space that becomes available tothe molecules. Let us imagine the volume contain-ing the gas to be divided into a very large number ofunit cubes, so small that the large majority of thecubes are empty, whilst a small fraction of themcontain one gas molecule each, Due to the thermalmotion, the arrangement of occupied and unoccupiedcubes is continuously changing. If there are z cubes

MARCH 1955 ENTROPY, I

a~d n molecules, then there are n occupied and'(z-n) unoccupied cubes.' Because the mutual inter-changing of two empty cubes as well as that oftwo occupied cubes leaves the distribution un-altered, the total number of micro-states is givenby

or,

z!m= ,

n! (z-n)!

using formula (I, 3):

. . . . (1,13)

In m = zln z -'- n In n - (z-n) In (z-n).

Using the approximation In (1 ___:n/z) = -n/zfor nl z ~ 1, we arrive at:

zlnm=nln-+n.

n... (1,14)

If the volume is increased by a factor r = V2/ VI'then rz has to be substituted for z in the formula.The increase of the entropy is thus, according to(1,12):

rz z..1S = Ic In m2 - le In mI = kn In - - kn. In -

n nV2= kn In r = kn In -VI

For 1 gram-molecule of gas, u[>on reference to(I, 11), we arrive at:

. . . . (1,15)

The two definitions of entropy

With the formulae (I, 2) and (I, 10) or (I; 12) wehave given two definitions of entropy that seemat first sight unrelated and even appear to lead to. contradictory conclusions. If, for instance, wedouble the volume of a gram-molecule of a perfectgas by allowing the vessel in which it is containedto communicate with an equally large evacuatedvessel, then the entropy will increase accordingto (I, 15), although there is no flow of heat into orout of the system. At first sight one might be tempt-ed to think that application of formula (1,2) wouldlead to an entropy change of zero.

On further' consideration, however, one seesthat formula (I, 2) cannot be simply applied to thistypically irreversible process. This formula appliesonly to reversible process~s; and in order to calculatethe change in entropy, a reversible process mustbe found that leads from an identical initial stateto the same final state. Such a process is as follows.The gas is contained in a cylinder having a pistonthat can move without any friction. In constanttemperature surroundings, the expansion is madeto take place in su~h a way that the back pressure

265

on the piston is at all times an infinitesimallysmall fraction less than the gas pressure. In thesecircumstances the process can be regarded as beingreversible, because the system is in equilibrium inany stage of the process, so that an infinitesimallysmall change in the back pressure is sufficient toreverse the process. During the reversible expansionthe perfect gas performs work on its surroundings,givenby

JdW= - fpdV.

(Work done on a system and he~t applied to a sys-tem are designated as positive). Since the internalenergy of the gas is not changed by the isothermalexpansion, according to (I, 1) a quantity of heat willbe absorbed from the surroundings, given by

f dQ = + f pdV.

According to (1,2) the change in entropy is given byv.' v. v.

LIS = f dQ = j' pdV = i= dVT TV'

V, v,, V2LIS = Rln-,

VI

which agrees with the result (I, 15).The same change of entropy is bound to occur

with the irreversible expansion from VI to V2, inview of the fact that S is a thermodynamic function.During this irreversihle change, however, no heatis exchanged with the surroundings, so that

i.e.,

dS dQ> T'

and this has, general application to all irreversibleprocesses. The second law of thermodynamics can,therefore, be expressed in a form more general than(I, 2) as:

. . (1,2')

III which the symbol = applies to, a reversiblechange of state, and the symbol > applies to anirreversible change. For an isolated system dQ = 0is always valid, and hence, according to (I, 2),dS ~ O. Formula (I, 2'), is therefore the mathe-matical formulation of the written form of thesecond law as in the introduction, viz. the entropyof an isolated system strives towards a maximum.It was the analogy between the classical thermo-dynamical picture of an isolated system strivingtowards maximum entropy, and the atomic pictureof the system striving towards the state with themaximum' number of equally probable arrange-ments g, that led, in the 19th century to the ass-umption (Boltzmann) that a relationship should

266 ' PHILlPS TECHNICAL REVIEW VOL. 16, No. 9'

exist between Sand g. This relationship could notbe otherwise than of a logarithmic nature, sincethe entropy is an additive variable, whereas thenumber of equally probable arrangements is amultiplicative variable, as we have already seen.That Sis 'an additive variable is a direct conclusionfrom the fact that the total amount of heatnecessa,ry in order to raise the temperature of thesystem A + B, in a reversible manner from T toT + dT, is the sum of the quantities of heat requiredto raise the temperatures of A and B separately tothe same extent (cf. formula (I, 2». The value ofthe proportionality constant k in (I, 10) and (I, 12)could then be directly derived from the applicationof this formula to a perfect gas, as described above.

A different formula for the entropy

According to formula (I, 6) the entropy of asystem ofN oscillators with energy level populationsni can also be written as:

S= kIng = k [Nln N - I ni In ni] •

Since N = .E ni we may write:

S = -k Ini In (ni/N)

and for the entropy per oscillator:

s = -kIpi lnpi, (1,16)

in which the fractions Pi = nil N represent thefractions of the total number of oscillators at dif-ferent energy levels i.

In some books on thermodynamics the lastformula is chosen as the statistical definition ofentropy. Outside the field of statistical thermody-namics an entropy formula similar to (I, 16) is usedin information theory. The values Pi. then relate tothe probability of occurrence of certain possibleevents. We shall return to this subject in the lastarticle (IV) of this series.

Justification of the first and second laws

Up to now we have paid hardly any attention tothe historical path leading to the mathematicalstatements

dU= dQ + dW,

TdS > dQ .

(1,1)

(1,2')and

of the first and second laws. From a logical pointof view it is perhaps most satisfactory to regardthese relationships as postulates, and find theirjustification in the fact that all conclusions derivedfrom them are confirmed by experiment. Indeed,if a single experiment were to be devised whose re-sults contradicted one of the two laws, the whole

admirable structure of thermodynamics wouldcollapse. The concept of absolute tempera'ture Tintroduced in (I, 2') finds its justification in ananalogous way:, it proved to be identical to thetemperature scale derived experimentally by meas-urement with a gas thermometer. .The statistical definitions (I, 10) and (I, 12) (if

entropy are justified by the fact that in all casesstudied up to now they are found to lead to thesame results as those derived from the thermodyn-amical definition (1,2). This has been demonstrat-ed with one example, viz. that of a perfect gasexpanded isothermally from VI to V2•

Free energy

In order to enlarge on the significànce of theconcept of entropy we have in the mixing experi-ments so far considered only the perfect gas state,i.e. a state, of matter in which there is negligibleinteraction between the atoms. In this' conditionthe state of equilibrium always corresponds to adisordered distribution of the various types of atoms.This is not necessarily the case if forces of attractionare present between the atoms, or if the distribu-tion of the atoms is influenced by external forcesor fields. If, in our case of the 100 billiard balls,the 50 white balls suffered strong mutual attraction(e.g. by magnets incorporated inside each of theballs), then after a shaking experiment one would,in most cases, find a distribution such that thesystem is separated into two phases, one containingalmost exclusively white balls and the other almostexclusively red balls. In this case the disordereddistribution of the balls corresponds to a stateof greater energy. The striving towards maximumentropy can, therefore, in certain cases be apparentlycounteracted by another tendency, viz: the strivingtowards a minimum energy. To formulate a quanti-tative relationship, the formulae (I, 1) and (I, 2')are combined to give

dU-TdS ~ dW, . . . . (1,17)

01', for a constant value of T:

d(U - TS)r ~ dW. . (1,17a)

If in the course of the change of state not onlythe .temperature remains constant, but also thevolume, (and any other ,parameter whose changewould lead to the performance of external work),then dW= 0, so that (I, 17a) for an irreversibleprocess can be written as

d(U - TSh,v < O .••••• (1,18)

. Hence, where we are concerned not with an

MARCH 1955 ENTROPY, I

isolated system, but with a system in which thetemperature T and the volume V are kept constant,then the tendency towards maximum entropy isreplaced by the tendency of the function (U - TS)towards a minimum. This thermodynamic functionis called the Helmholtz free energy or the free energyat constant volume, and is indicated by the symbol F.If instead of t.emperature and volume, tempcrature

and pressure are kept constant, then even if the pro-cess is irreversible, work is done by the system, viz.the expansion work - dW = pd V (p representingnot the internal, but the external pressure). Refer-ring back to (I, 17a), it will be seen at once that hereanother thermodynamic function, viz. the function(U - TS + p V) tends towards a minimum. Thisfunction is called the Gibbs free energy, or alternati-vely the free energy at constant pressure, the freeenthalpy, or the thermodynamic potential, and isusually indicated by the symbol G.We thus obtainfor irreversible processes, depending on the auxiliaryconditions, the two relations

(dF)r, v < 0 and (dG)r,p < o. . (1,19)

Finally formula (I, 17) provides the followingrelations applying to an irreversible change ofstate:

(dU)s,v < 0 and (dS)u,v> 0 . . (1,20)

subject, again, to the condition that the volume andany other parameters whose change would lead tothe performance of external work are kept constant.The relation (I, 18), viz. d(U __,_TSh,v < 0 may

be more or less arbitrarily divided into two parts,(dUh,v < 0 and (dSh,v > 0, which may be con-sidered as the mathematical statement of the twoopposing tendencies mentioned abovein this section.If only the latter (dS > 0) were operative, wewould expect to find only those processes andreactions in nature in which the number of equallyprobable arrangements (the "disorder") increases.If, on the other hand, only the former (dU < 0)were operative, we would expect the occurrence ofonly those processes in which the opposite happens,i.e. whereby heat is liberated and, generally speaking,the degree of order increases. To determine thedirection of a process it is therefore necessary totake into account the free energy which involvesboth thermodynamic functions U and S. From (I,18) it follows that at low temperatures, the tendencytowal:ds minimum energy and the correspondingorder predominates, whereas at high temperatures(violent shaking of the billiard balls) the tendencytowards maximum entropy and the correspondingdisorder, prevails. '

267

These conclusions are borne out in practice. Atlow temperatures the atoms and molecules, underthe influence of .their mutual attraction, form theordered, periodic structures we knowas crystals.At high temperatures, however, all matter isultimately transformed into the chaotic state of agas. The temperature range in which this entropyeffect begins to predominate depends on the mag-nitude of the attractive forces between the gasparticles. Even in the gase~ a certain degree oforder is often present in the form of orderedgroups of atoms (molecules). At high enough tempe-ratures, however, this order also disappears, and atthe surface of the sll:n (temperature approximately6000 °C) matter only exists as a gaseous mixtureof the atoms of the various elements.

Even this is not complete chaos, however; as thetemperature assumes higher and higher valueseven the order of the electron shells is finally com-pletely destroyed due to thermal ionization. Attemperatures of a few million degrees the ionizationis complete. This state in which the atoms areentirely split up into naked nuclei and electronsoccurs. in the interior of the sun and other stars.The only order remaining is that of the nuclei. Acomplete disintegration of these into protons andneutrons would require even higher temperaturesthan seem to occur in the hottest stars.

Zero-point entropy

After this brief digression into the field of ex-tremely high temperatures we shall now considerthat of extremely low temperatures. According toNernst's heat theorem, the entropy of all systemsin a stable or nietastable equilibrium tends to zeroon approaching the absolute zero of temperature.This means, according to (I, 10) or (I, 12) that atabsolute zero a system in equilibrium can exist inonly one micro-state. This situation can be readilyinterpreted in terms of the quantum-mechanicalpicture: all particles' are in their lowest quantumstate at absolute zero. In the diagram of fig. 3 thehigher levels gradually empty as the temperaturedecreases, until ultimately àll 25 particles areat level 80, Nernst's theorem would not apply,however, if the lowest energy level correspondedto two or more quantum states, in other words, ifthis level was "degenerate". In that case, the particlesin the state of equilibrium would be uniformlydistributed among these quantum states even atabsolute zero. Nernst's theorem thus necessarilyincludes the postulate of the non-degenerate stateof the lowest energy level. The validity of thistheorem can he tested experimentslly in many cases.

268 PHILlPS TECHNICAL REVIEW VOL. 16, No. 9

One of the classical examples is the transformationof white into grey tin. Grey tin is stable below 13 oe(286 OK), white tin is stable above this temperatme.Owing to the fact that white tin can be supercooledright down to the lowest temperatures attainablein the laboratory, it has heen possible to measurethe specific heat cp of both modifications at lowtemperatures. The validity of Nernst's theorem canthus he tested as follows. From (I, 2) the entropyof white tin at 286 oK can be found in two ways:directly from the cp-m~asurements on white tin, andindirectly from the value of cp for grey tin and thechange in entropy occurring with the transformationof grey into white tin. Both ways should lead to thesame result, i.e. the following equation should hevalid:

286 286

f cP(WT)dT = J~cp(g)dT QT + 286'

(1,21)

o o

in which cp(w) and cp(g) represent the specific heatper gram-atom of white and grey tin respectively,and Q stands for the heat of transformation, i.e.the quantity of heat absorbed during the isothermal

, and reversible transition of 1 gram-atom of tin fromthe grey to the white modification. It has been foundthat (1,21) is satisfied within the limits of experimen-tal accuracy. Unfortunately the heat oftransforma-tion Q is not known with sufficient accuracy toattach very much value to this agreement. More-over, even if complete agreement were establishedthis would only prove that the difference in entropybetween the two modifications at 0 "K is equal tozero. The heat theorem is therefore often wordèdin a somewhat more cautious form, e.g.: at zeroabsolute temperature all entropy differences betweenthe states of a system in internal equilibriumvanish. This formulation has the same significancein practice as that which states that the separateentropy values approach zero, for one can nowjustifiably define the zero point entropy of allsubstances in stable or metastable equilibrium, tohave the value zero. If, for example, in a chemicalreaction of the type A + B~ AB, the entropychange is zero at 0 OK, then it is logical to assignto A and B as well as AB a-zero-point entropy ofzero. For the energy, this is not possible, as theextrapolations to T = 0 clearly demonstrate thatthere is no question of the heat of reaction disap-pearing at zero temperature.Stronger evidence for the validity of Nernst's

theorem is derived from measurements on gases.With the aid of the statistical thermodynamicalexpression S = k In m (or S = k In g) the entropy

of many gases can be evaluated, using informationon their molecular rotational and vibrational statesderived from their spectra.

On the other hand, assuming the validity ofNernst's theorem, the gas entropy mayalso becalculated with the aid of the classical formuladS = dQrev/T, employing existing data on thespecific heats c , of these substances in the solid,liquid and gaseous states and that on heats oftransformation, heats of fusion, and heats ofevaporation. The entropy of a substance in thegaseous state at temperature T, assuming notransformations occur in the solid state, can hewritten as:

ill which Ti and Te respectively are the meltingpoint and the boiling point, and Qf and Qe are theheat of fusion and the heat of evaporation.

This "caIOl~imetric .entropy" is thus obtainedentirely without reference to the existence of atoms,being based merely on the results of calorimetriemeasurements; the "statistical entropy" on theother hand is evaluated by methods entirely in-dependent of the de facto existence of the liquid andthe solid states. It is most satisfying that the twoways as a rule lead to the same result, whilst the fewexceptions that have been found can be satisfactorilyexplained. The nature of these exceptions may betwofold. Some only appear to be exceptions, causedby the fact that the measurements of the specificheat were not extended to a sufficiently low tem-perature; other exceptions are caused by the non-attainment of equilibrium at decreasing tempera-ture. Neither of the two types of exceptions iscontradictory to Nernst's theorem, which onlyclaims validity for the absolute zero temperature,and even then only for systems in a state of jnternalequilibrium.These apparent exceptions to the agreement

between calorimetrie and statistical entropy occur.if the "lowest energy level" of the particles, uponfurther scrutiny, is found to consist of a group ofenergy levels at intervals Lie which are small com-pared to kT even at the lowest temperatures ofmeasurement 5). When this is the case, the particles

5) The distribution among the availabl~ energy levels isentirelydeterminedby the ratioL1elk'I', where k is Boltz-mann'sconstant.

MARCH 1955 ENTROPY, I

are still uniformly distributed over the aforemen-tioned group of energy levels, even at this lowestmeasuring temperature. The states corresponding tothese energy levels will, therefore, not be manifestin the specific heat. The gradual emptying of thehigher levels of the group will only start when thetemperature reaches a value for which kT is of thesame order of magnitude as LI 8; not before tempe-ratures are reached for which kT is appreciablysmaller than LI 8 will all particles have settled atthe lowest level 80, This regrouping will manifestitself by a peak in the specific heat temperaturecurve (see fig. 4). If such a peak has not beenestablished because the measurements have notbeen extended to a sufficiently low temperature,the value of the calorimetrie entropy derived fromthem will be too low.The fact that too small a value for the calori-

metric entropy is sometimes found, may thus be

c

Ir:«I \I ,I ' ___..,.-

/ --/ _-~ _-81117 -T

Fig. 4-,The calorimetrie entropy can be calculated if the specificheat c is known as a function of the temperature T. Extra-polation from the lowest attainable temperatures to absolutezero is of course necessary, but it must be noted that this in-volves the risk of overlooking a peak in the specific heat curvesituated below the temperature range in whichmeasurementscan be made.

due to the extrapolation of the specific heat from.too high a temperature. The cause may, however,also be a different one, namely the non-attainment

269

of equilibrium, mentioned earlier. An example ofsuch a "frozen-in" distribution of molecules amongvarious energy levels, is solid carbon monoxide(CO) at low temperatures. The value found for thecalorimetrie entropy is smaller by an amountRln 2 = kin 2N• cal/mol. degree than the statisticalentropy derived from the CO-spectrum. Thisdiscrepancy corresponds to a number of micro-states m = 2N•• This immediately suggests thatthe molecules in' the crystal have two possibleorientations. The CO-molecules are assumed tohave such a small electric moment and to be sonearly symmetrical that the crystallattice does notshow a pronounced preference for the one or theother orientation, CO or OC. As a consequence ofthis, the two opposed directions of orientation wouldremain irregularly distributed among the positionsof the lattice down to the lowest measurable tem-perature. It is highly improbable that in solid COthe rotational shift through 1800 required to producea state of equilibrium, is at all possible. In otherwords, it is not to be expected that an extensionof the measurements to lower temperatures willeliminate the discrepancy. If this assumption istrue, then the discrepancy between the statisticaland the calorimetrie entropy can only mean thatthe energy difference between the two CO-positionsat the freezing point is still too small with respectto kT! to determine a given orientation. Otherexamples of systems possessing a zero-point entropydue to the fact that the internal equilibrium is notattained at low temperatures, can be found amongthe many disordered solid solutions of metals.Further examples are the glass-like substances,which may be considered as supercooled liquids.

Nernst's theorem cannot be derived from the twomain laws and is therefore often referred to as the"third law of thermodynamics".

In the three subsequent articles we shall demon-strate with the aid of examples that the conceptof entropy plays an important part in widelydivergent fields of science and technology.

270 P HILIPS TECHNICAL REVIEW VOL. 16, No. 9

ABSTRACTS OF RECENT SCIENTIFIC PUBLICATIONS OFN.V. PHILIPS' GLOEILAMPEN FABRIEKEN

Reprints of these papers not marked with an asterisk .. can be obtained free of chargeupon application to the Administration of the Philips Research Laboratory, Eindhoven,Netherlands.

2122: w. J. Oosterkamp: Die Dosierung weicherRöntgenstrahlung, insbesondere bei Kon-takttherapie (Strahlentherapie 91, 591-494).(Dosimetry of soft X-rays, with particularreference to contact therapy; in German.)

A description is given of ionization chambers forthe measurement of the heterogeneous, extremesoft radiation. emitted by X-ray tubes with aberyllium or mica-beryllium "window. At'tenuationcurves in aluminium and depth dose curves in awater phantom for 10,15,20,30 and 50 kV, measur-ed on a new constant-poten,tial contact therapyapparatus are given.

2123: J. Fransen and W. J. Oosterkamp: A univer-sal X-ray dosemeter (Trans. Instr. Meas.Conf. Stockholm 1952, p. 93-94).

Brief description of ionization chambers anda dynamic electrometer for measuring dose rateand integrated dose for a wide rangc of radiationqualities and intensities.

2124: W. J. H. Beekman: An X-ray spectrometerfor the 200 kV region using a scintillationcounter (Trans. Instr. Meas. Conf. Stockholm1952, pp. 84-87).

An X-ray spectrometer using a scintillation coun-ter is described, constructed' for the purpose ofestimating the peak voltage of X-ray generators inthose cases where conventional methods fail. Theprinciple is. that of measuring the shortest wave-length of the X-rays generated and determining thepeak voltage by means of the Duane-Hunt relation.This instrument has been used in the case of anunorthodox form of X-ray generator, consisting ofa grid-controlled X-ray tube, the anode of which isconnected to a D.C. source via a high inductancecoil. By applying negative pulses to the grid, thestray capacitance of the tube and coil can be alter-nately charged by the current through the. coil anddischarged through the tube.

2125: H. G. van Bueren: The formation of latticeeffects during slip (Acta Metallurgica I,464-465, 1953).

Tentative explanation of the formation of ele-

mentary structure (slip lines with a length of theorder of 10-3 cm and 10-50 inter-atomic distancesapart), as found by Wils dorp and others in plastic-ally deformed aluminium. The explanation is basedon the effects of the formation of vacancies andinterstitials during plastic flow.

2126: A. van Weel: Susceptance valves and re-actance valves as phase modulators) J. Brit.Inst. Radio Engrs. 13, 315-320, 1953).

Triode valves may be used in three different waysto give variable-impedance circuits. The most usedform, usually known as a "reactance-valve circuit"behave much more as a variable "susceptance-valve circuit". Of the other two circuits, one is alsoa "susceptance-valve circuit", while only the thirdis a "reactance-valve circuit" proper. All three kindscan be used successfully in phase modulator stages,with the feature that, for introducing phase varia-tions of up to 45 degrees, only one valve is necessary.Moreover, this may be realized with mutual-con-ductance variations of not more than 0.5 mA/V,from which follows both a high sensitivity and agood linearity.

2127: H. C. Hamaker: Beispiele zur Anwendungstatistischer Untersuchungsmethoden in derIndustrie (Mitt. Math. Statist. 5, 211-229,1953).

The application of "variance analysis" to indus-trial problems is explained with the aid of a suitableexample, viz. the measuring of the thickness of theoxide coating deposited on nickel bars by electro-.phoresis. The results of the analysis are presentedin a simple manner which can be understood bypersons not specially trained in statistics. The samemethod is then applied to some other examples,viz. the measuring of the heat of combustion of 9different coal samples by 9 different laboratories,the measuring of the diameters of 6 different bicyclebearing-balls with 7 different micrometers, andsmelling-tests for the selection of an odour-judgingpanel in the perfume industry. The article alsodeals in some detail with the concept of interactionand how its existence can be established by statisticalanalysis.

MARCH 1955 ABSTRACTS OF' RECENT SCIENTIFIC PUBLICATIONS 271

2128: H. P. J. Wijn: Ferromagnetic domain wallsin Ferroxdure (Physica 19, 555-564, 1953).

In the preparation of BaO.6Fe203 (Ferroxdure)it is possible to distinguish between the contrib-utions to the initial permeability of Bloch-wall dis-placements and of Weiss-domain rotations. Frommeasurements of these contrihutions as functions ofthe frequency, there is evidence of a resonanceeffect of the Bloch-walls at about 300 Mc/s, and thecontribution of the rotations remains independentof the frequency to above 3000 Mc/s. The possibilityof Bloch-wall resonance has several times been pro-posed in the literature. The resonance frequencyto be expected from theoretical considerationsagrees well with the observed results.

2129: W. Hoogenstraaten: The chemistry of trapsin zinc sulphide phosphors (J.Electrochem.Soc.l00, 356-365,1953).

Trap characteristics of zinc sulfide phosphors arestudied as a function of chemical composition. Thesimplest phosphor systems are found to have simple,single-peaked glow curves. They contain only activa-tors and coactivators in pure sulfide base materials.Coactivators are defined as impurities necessary tostabilize the activators in the zinc sulfide lattice.They are found to exert a major influence upontrap characteristics. The trap depths are found to be0.37 electron volt for Cl", Br", and AI3+, 0.51 eV,for Sc3+, 0.62 eV for Ga3t-, and 0.74 eV for In3+ ascoactivators in ZnS-Cu. Additional glow peaks andtraps are produced by oxygen and by the killerscobalt and nickel. The formation of mixed crystalswith cadmium sulfide or zinc selenide generallyresults in a shift of the glow curves toward lowertemperatures.

2130: K. ter Haar and J. Bazen: The titration of"Complexone Ill" with thorium nitrate atpH = 2.8 - 4.3 (Anal. chim. Acta 9, 235-240,1953).

The reaction between thorium and "ComplexonelIl" (disodium salt of ethylenediaminetetra-aceticacid) at pH = 2.8 - 4.3 has been developed to aquantitative method. As indicator alizarin-S is used.In the first place the reaction is suitable to back-titrate an excess of "Complexone Ill" in the pHrange mentioned and it is the basis for an AI, Ni andBi determination still to be published; .moreover, itprobably presents the possibility of a simpledetermination of thorium.

2131: A. J. W. M. van Overbeek and F. H. Stieltjes:Bandwidth limitation of junction transistors(Proc. Inst. Radio Engrs. 40, 1424, 1952).

Proceeding from results obtained by Steele, afundamental limiting value to the Q-factor forwide-band amplification is derived for a junctiontransistor. The output capacitance is neglected inthis analysis.

2132: J. Feddema and W. J. Oosterkamp: Volumedoses in diagnostic radiology (from: Moderntrends in diagnostic radiology, 2nd series,edited by J. M. M. McLaren, Butterworth,·London, p. 35-42, 1953).

The importance of dose measurement in diagnos-tic radiology is pointed out. Methods are describedfor arriving at the volume dose. A table is includedof data on the average volume doses relating to anumber of different diagnostic conditions. A surveyis given ofthe number of exposures to which patientshave been subjected during the course of their life.Case histories show no sign of radiation damage tothe patients.

2133: J. L. Meijering: On a statement by C. S.Smith concerning an upper limit to the shar-ing of corners in aggregates (Acta Metall. 1,607, 1953).

Contrary to the proposition put forward by C. S.Smith, it is asserted that the number of corners in acrystal aggregate can be greater than 6 times thenumber of crystals, even if all the crystals areconvex polyhedra with flat boundaries.

2134: H. G. Van Bueren: Relation hetween plasticstrain and increase of electrical resistivity ofmetals (Acta Metall. 1, 607-609, 1953).

The increase of electrical resistivity of metals dueto plastic strain at low temperatures is attributedto the formation of vacancies, interstitial atoms anddislocations. On the basis of the considerationstreated in Abstract No. 2125, it is demonstratedthat the influence of dislocations gives rise to anincrease proportional to e'J, (13 = elongation) whilstvacancies and interstitial atoms cause an increaseproportional to e'J,. Manintveld's experiments in-dicate a 3/2-power relationship, which shows thatdislocations have little influence, a conclusion whichappears ~lso theoretically justified. This helps toexplain the influence of annealing on the resistivity.

2135: W. Hoogenstraaten and H. A. Klasens: Someproperties of zinc sulfide activated withcopper and cobalt (J. Electrochem. Soc.100,366-375,1953).

Some properties of ZnS-Cu-Co phosphors under3650 A excitation are described, viz., the thermalglow, decay and build-up of fluorescence, tempera-

272 PHILlPS TECHNICAL REVIEW VOL. 16, No. 9

ture dependence, and light sum. Most 'of the ex-perimental results can he explained by a model inwhich cobalt levels act both as electron traps witha trap depth of 0.5 electron volt, and as acceptorsfor holes, ejected thermally from copper centerswith an activation energy of 1.1 eV. The possibilityof excitationby 3650 A radiation ofelectrons fromtraps to the conduction band is introduced to,,explain the observed intensity dependence on thelight sum. .

2136: J. D. Fast: Erzeugung von reinem undabsichtlich verunreinigtem Eisen und Unter-suchungen an diesen Metallen (Stahl undEisen 73,1484-1496,1953).

To investigate the causes of various phenomenaoccurring in steels, a high-vacuum 300 kc/s induc-tion furnace was developed which is suitable for-melting very pure iron in quantities up to 2 kg.Some general directions regarding the melting-procedure and the choice of crucible material aregiven. (See also Philips tech. Rev. 15, 114-121,1953/1954). Starting with iron in its purest form,investigations were carried out into the individualand collective influence of carbon (up to 0.04%),oxygen (up to 0.03%), nitrogen (up to 0.02%) andmanganese (up tb 0.50%), on (a) quench ageing,(b) strain ageing, (c) blue-brittleness, (d) grain-boundary brittleness. Regarded from an atomicviewpoint, a common cause is found for a numberof phenomena that seem at first sight to be hardly,if at all, interrelated. These are, the greater solubili-ty of carbon and nitrogen in y-il'on, compared tothat in a-iron; the greater solubility of nitrogenin both phases compared with that of carbon; thepresence of the dissolved carbon and nitrogen atomsin the octahedral interstices of both phases; theSnoek-damping; the formation of martensite; theoccurrence of an upper and a lower yield point inthe stress-strain curve of mild steel; the strain age-ing; the, fact that carbon and nitrogen are lesssoluble in silicon iron than in pure iron; and thepreference of the dissolved carbon and nitrogenatoms for the grain boundaries of iron.

2137: J. 1. de Jong and J. de Jonge: The chemicalcomposition of some condensates of urea andformaldehyde (Rec. Trav. chim. Pays-Bas72, 1027-1036, 1953).

Some condensates of urea and formaldehyde wereprepared from solutions of pH 2-7, at temperat-ures of 20-76° C. These products have been ana-

Iysed with respect to their content of methylenegroups, methylol groups and urea groups. Theaverage molecular weights could be estimated. Theanalytical data are in harmony with the occurrenceof methylene bridges between the urea fragments.The condensates will he formed by a stepwise con-densation reaction. "Methylene urea" may 'he amixture of condensates with an average molecularweight of 300 __:_500.

2138: J. S. C. Wessels and E. Havinga: Studies onthe Hill reaction, II (Rec. Trav. chim.Pays-Bas 72, 1076-1082. 19,53).

The influence of the presence of oxygen on theHill reaction has been investigated by redoxpotential measurements. A very simplè reactionscheme, implying primary formation of a specificreductant reacting subsequently with the oxidantadded, seems to fit the kinetic and other data of thereaction. Some of the results of investigations on theinfluence of inhibitors and biochemically importantsubstances are reported and discussed; possiblecauses for the discrepancies in the literature con-cerned with the action of inhibitors are indicated.

2139: K. F. Niessen: Ratio of exchange integralsin connection with angles between partialmagnetizations in ferrimagnetic spinels(Physica 19, 1035-1045,1953)

Allowing the spin direction ofmagnetic ions in onesublattice of (octohedral) B-sites to differ from thatin the other sublattice of B-sites (as assumed byYafet and Kittel for a spinel with only one kind ofmagnetic ions) a mixed crystal spinel containingtwo kinds of magnetic ions is considered, taking intoaccount the different physical nature of the ions.Here a situation may be realized where the partialmagnetizations are neither parallel nor antiparallelbut where in one sublattice (say BI) two specialspin directions occur for the two kinds of magneticions and in the other (BIl) another couple of spindirections lying 'with the former set symmetricalwith respect to the single spin direction of ions onthe (tetrahedral) A-sites. The A-A interaction isneglected and consequently a subdivision of theA-lattice is not taken into account. Compositionsof the mixed crystal are possible where such a non-rectilinear case is just possible, i.e. where the mutualdeviations of the spin directions on the B-sites' arevery small, their angles with the spin directions onA-sites being nearly n, From three such compositionsthe ratios of exchange integrals can be determined.