T H E ?rlriSS E X I T OF T H E CHEMICAL POTESTIAL B Y WILDER D. B P S C R O F T
It may be carrying coals t o Kewcastle t o explain selected passages from Gibbs to the chemists; but I have found no chemists-either hand-picked or run-of-mine-who would explain them to me and consequently I feel temporarily like a discoverer, quite regardless of whether I am one or not. For the sake of convenience the references are to the first volume of “The Collected Papers of J. Killard Gibbs,” edited by H. A. Bumstead and R. G. Van Name. “If we consider the amount and kind of matter in a homogeneous mass as fixed, its energy is a function of its entropy q, and its volume v and the differentials of these quantities are subject to the relation de = tdq - pdv, (11) t denoting the (absolute) temperature of the mass, and p its pressure. For tdv is the heat received, and pdv the work done by the mass during its change of state. But if we consider the matter in the mass as variable, and write ml, m z j. . , m, for the quantities of the various substances SI,S2,. . . S, of which the mass is composed, o will evidently be a function of q, v, m,, m2,. , . m,,, and we shall have for the complete value of the differential of e de = tdq - pdv p l , fi?,
. .
,
+ p1 dml + p2 dml . . . + p, dm,,
pn denoting the differential coefficients of
.
E
(12)
taken with respect
. m,. t o m l , m2,. “The substances SI, S2, . . . S,, of which we consider the mass composed, must of course be such that the values of the differentials dml, dmz, , . , dm, shall be independent, and shall express every possible variation in the composition of the homogeneous mass considered, including those produced by the absorption of substances different from any initially present. I t may therefore be necessary to have terms in the equation relating to component substances which do not initially occur in the homogeneous mass considered, provided, of course, that these substances, or their components, are to be found in some part of the whole given mass. “If the conditions mentioned are satisfied, the choice of the substances which we are to regard as the components of the mass considered may be determined entirely by convenience, and independently of any theory in regard to the internal constitution of the mass. The number of components will sometimes be greater, and sometimes lese, than the number of chemical elements present. For example, in considering the equilibrium in a vessel containing water and free hydrogen and oxygen, we should be obliged t o recognize three components in the gaseous part. But in considering the equilibrium of dilute sulphuric acid with the vapor which it yields, we should have only two components in the liquid mass, sulphuric acid (anhydrous, or of any particular degree of corcentration) and (additions) of water. If, however, we are considering sulphuric acid in a state of maximum concentra-
io
WILDER D. BANCROFT
tion in connection with substances which might possibly afford water to the acid, it must be noticed that the condition of the independence of the differentials will require that we consider the acid in the state of maximum concentretion as one of the components. The quantity of this component will then be capable of variation both in the positive and in the negative sense, Tvhile the quantity of the other component can increase but cannot decrease beyond the value zero, “For brevity’s sake, we may call a substance S,an actual component of any homogeneous mass, to denote that the quantity ma of that substance in the given mass may be either increased or diminished (although we may have so chosen the other component substances that ma = 0);and we may call a substance Sb a possible component to denote that it may be combined with, but cannot be subtracted from the homogeneous mass in question, In this case, we must so choose the component substances that mb = 0. “The units by which we measure the substances of which we regard the given mass as composed may each be chosen independently. To fix our ideas for the purpose of a general discussion, we may suppose all Substances measured by weight or mass. Yet in special cases, it may be more convenient to adopt chemical equivalents as the units of the component substances,” p. 63. “If we call a quantity p x , as defined by such an equation as (IZ), the potential for the substances S, in the homogeneous mass considered, these conditions may be expressed as follows-The potential for each component substance must be constant throughout the whole mass,” p. 65. ‘[Whenever, therefore, each of the different homogeneous parts of the given mass may be regarded as composed of some or all of the same set of substances, no one of which can be formed out of the others, the condition which (with equality of temperature and pressure) is necessary and sufficient for equilibrium between the different parts of the given mass may be expressed as follows:“ T h e potential for each of the component substances m u s t have a constant value in all parts of the given mass of which that substance i s a n actual component, and have a value of not less than this in all parts of which it i s a possible component,” p. 67, “In the definition of the potentials p1,p 2 jetc., the energy of a homogeneous mass was considered as a function of its entropy, its volume, and the quantities of the various substances composing it. Then the potential for one of these substances was defined as the differential coefficient of the energy taken with respect to the variable expressing the quantity of that substance. NOWas the manner in which we consider the given mass as composed of various substances is in some degree arbitrary, so that the energy may be considered as a function of various different sets of variables expressing quantities of component substances, it might seem that the above definition does not fix the value of the potential of any substance in the given mass, until we have fixed the manner in which the mass is to be considered as composed. For example, if we have a solution obtained by dissolving in water a certain salt containing
THE MASS UKIT OF THE CHEMICAL POTENTIAL
71
water of crystallization, we may consider the liquid as composed of ms weight units of the hydrate and mw of water or as composed of m, of the anhydrous salt and m, of water. It will be observed that the values of ms and mBare not the same, nor those of mw and m,, and hence it might seem that the potential for water in the liquid considered as composed of the hydrate and water, viz.,
(&)vj v, m s ’ would be different from the potential for water in the same liquid considered as composed of anhydrous salt and water, viz., The value of the two expressions is, however, the same, for, although mwis not equal to m,, we may of course suppose dmwto be equal to dm,, and then the numerators in the two fractions will also be equal, as they each denote the increase of energy of the liquid, when the quantity dmwor dm, of water is added without altering the entropy and volume of the liquid. Precisely the same considerations will apply also to any other case. In fact’,we may give a definition of a potential which shall not presuppose any choice of a particular eel. of substances as the components of the homogeneous mass considered. “Dt$nition.-If t,o any homogeneous mass we suppose an infinitesinal quantity of any substance to be added, the mass remaining homogeneous and its entropy and volume remaining unchanged, the increase of the energy of the mass divided by the quantity of t,he substance added is the potential for that substance in the mass considered. (For the purposes of this definition, any chemical element or combination of elements in given proport’ions may be considered a substance, whether capable or not of existing by itself as a homogeneous body,” p. 92. “Between the potentials for different substances in the same homogeneous mass the same equations will subsist as between the units of these substances. That is, if the substances, Sa, S b , et,c. &, SI,etc., are components of any given mass, and are such that as, PSb etc. = K S ~ XSI etc., (120) S,, S h , etc., K, A, et,c., denoting numbers, then if p%,p b , etc., pk, pi, etc., denote the potentials for those substances in the homogeneous mass. etc. (Yka @C(b etc. = Kpk xpl (121) To show this, we will suppose the mass considered to be very large. Then t.he first member of (IZI) denotes the increase of t,he energy of the mass produced by the addition of the matter represented by the first member of ( IZO) and the second member of ( I z I ) denotes the increase of energy of the same mass produced by the addition of the matter represented by t,he second member of ( 1 2 0 ) the entropy and volume of the mass remaining unchanged in each case. Therefore, as the two members of ( 1 2 0 ) represe- the same matter in kind and quantity, the two members of (121) must be lual.
+
+
+
+
+
+
+
72
WILDER D. BANCROFT
“But it must be understood that equation (120) is intended to denote equivalence of the substances represented in the mass considered, and not merely chemical identity; in other words, it is supposed that there are no passive resistances to change in the mass considered which prevent the substances represented by one member of ( 1 2 0 ) from passing into those represented by the other. For example, in respect to a mixture of vapor of water and free hydrogen ant1 oxygen (at ordinary temperature), we may not write
+
9 SA^ = I SH 8So hut water is to he treated as an independent substance, and no necessary relation will subsist between the potential for water and the potentials for hydrogen and oxygen,” p. 94. If we work at temperatures at which the reaction of hydrogen and oxygen to form water is a reversible one or if we produce a stabe of reversible equilibrium by the addition of a suitable catalytic agent (pp. 139, 141, 184))then we can write 9 s .=~ ~ I S H 4- 8So or 9g’aq = IP’H
+8 ~ ~ 0 .
Gibhs says, p. 363, that “when the proximate components of a gas-mixture are so related that some of them can be formed out of others, although not necessarily in the gas-mixture itself at the temperatures considered, there are certain phases of the gas-mixture which deserve especial attention. There are the phases of dissipated energy, Le., those phases in which the energy of the mass has the least value consistent with its entropy and volume. An atmosphere of such a phase could not furnish a source of mechanical power to any machine or chemical engine working within it, as other phases of the same matter might do. Kor can such phases be affected hy any catalytic agent. A perfect catalgtic agent would reduce any other phase of the gasmixture to a phr,se of dissipated energy. The condition which mill make the energy a minimum is that the potentials for the proximate components shall satisfy an equation similar to t,hat,which expresses the relation between the units of weight of these components. For example, if the components were hydrogen, oxygen, and water, since one gram of hydrogen with eight grems of oxygen are chemically equivalent to nine grams of water, the potentials for these substances in a phase of dissipated energy must satisfy the relation” 9P‘Aq = IP’H f 8P’O
If we express these units in gram-molecular weights, as chemists usually do, we shall write for a phase of dissipated energy, which is another way of saying a state of reversible equilibrium, ZFH*O
= PHi
+ Poi
m-here g’ is the chemical potential referred to grams and p the chemical potential referred to molecular weights. In order to avoid confusion, Lash Millor suggests using the words specific potential for the chemical potential referred to grams, the molecular potential (or mol-potential) for the chemical potential referred to molecular weights, and the equivalent potential for the
THE MASS U S I T O F THE CHEMICAL POTESTIAL
73
chemical potential referred t,o equivalent weights. The coulomb potential might be used for the chemical potential referred to the electrochemical equivalent. This is a bit puzzling to the chemist, especially if he wishes to deduce the mass law from the relation between the potentials a t equilibrium. The difficulty is due to the fact that Gibbs defines the chemical potential in a purely formal way. “The potential for any substance in any homogeneous mass is equal to the amount of mechanical work required to bring a unit of the substance by a reversible process from the state in which its energy and entropy are both zero into combination with the homogeneous mass, which a t the close of the process must have its original volume, and which is supposed so large as not to be sensibly altered in any part. XI1 other bodies used in the process must by its close be restored t,o their original state, except those used to supply the work, which must be used only as t,he source of the work. For, in a reversible process, when the entropies of other bodies are not altered, the entropy of the substance and mass taken together will not be altered. But the original entropy of the substance is zero; therefore the entropy of the mass is not altered by the addition of t’he substances. Again, the work expended will be equal to the increment of the energy of the mass and substance taken together, and therefore equal, as the original energy of the substance is zero, to the increment of energy of the mass due to the addition of the substance, which by the definition on p. 93 is equal to the potential in question,” p. 9 j. Under this definition the potential of any given substance is a function of the mass unit, adopted. The molecular chemical potential or the chemical potential of a gram-molecular weight of oxygen taken as the mass unit, is thirty-two times of the specific chemical potential or the potential of the same amount of oxygen with one gram taken as the mass unit. In other xords =, 1 8 p ’ ~ ~Substituting . these values in the PH, = Z P ’ H , p0,=32p’o and p ~ ? p equation 2pH20=z ~ H , + ~ o ,we , get 36j~”~=4pH+3zpoor 9 ~ ‘ . ~1 ~p ’=~ ~ + 8 p ’ o ~ . Lash Miller points out. that when hydrogen, oxygen and water are in a state of reversible equilibrium, the following three equations all say exact,ly the same thing:9X spec’fic potential water = 8X specific potential oxygen
+ specific potential hydrogen
X molecular potential water = molecular potential water 2 X molecular potential hydrogen Equivalent potential water = equivalent potential osygen 2
+ + equivalent potential hydrogen.
All these equations are identical under the formal definition of the potential. The question then arises whether this definition is a desirable one in the general form which Gibbs has given it. I have not, yet found that Gibbs has discussed this question definitely anywhere. Really t,he mass unit to be taken for the chemical potential should be the one which is the chemical unit for the process under consideration, because this is the one which leads
74
WILDER D. BASCROFT
directly to the mathematical formulas describing the equilibria, When we are considering pressure-volume-temperature relations for gases, the obvious units are the molecular weights. Ignoring variations from the gas law, two grams of hydrogen are equivalent to thirty-two grams of oxygen, eighteen grams of water, or thirty- six and a half grams of hydrochloric acid. So far as I can see, it is quite impossible to pass directly from the relation 9 ~ = ’ IP’H,, 8p’oX to the equation KC2H,o = C 2 ~. CO,. , We must pass through the relation 2pHz0 = 2pH9 p o 2 . When considering mass law relations, we must, therefore, refer the chemical potentials to the molecular weights as units. Gibbs actually does this, p. 362, and assumes that we shall follow him. “From the physical properties which we attribute to ideal gases, it is easy to deduce their fundamental equations. The fundamental equation in E, 9, u and m for an ideal gas is €-Em m clog = y+H+alog-; cm m V t,hat in J., t , v, and m is J. = Em mt (c-H-c log t a log m/v. (IS) that in p , t and p is
+
+
~
+
H-c-a ~
+
pzE
a f c
p=ae a t a e at (19) where e denotes the base of the Saperian system of logarithms. As for the other constants, c denotes the specific heat of the gas at constant volume, a denotes the constant’ value of pv + mt, E and H depend upon the zeros of energy and entropy. The two last equations may be abbreviated by the use of different constants. The properties of fundamental equations mentioned above may easily be verified in each case by differentiation. “The law of Dalton respecting a mixture of different gases affords a point of departure for the discussion of such mixtures and the establishment of their fundamental equations. It is found convenient, to give the law the following form :“ T h e pressure i’n a wiixture of different gases is equal to the s u m of the pressures of the different gases as existing each by itself at the same temperature and with the same value of its potentials.” If we take the logarithm of equation (19) and consider the temperature as constant, we get, p = at log p constant which is identical with the one that the modern physical chemist would write, p = R T log p C. If we express the chemical potentials in terms of molecular weights, as mass units, the relation for the reversible equilibrium bet’ween hydrogen, oxygen, and water becomes Z ~ H 4?- P O , = 2pH20 or, in terms of pressures, 2 RT log pi 2 C1 RTlog p2 +210g C z = 2 R T log p3 2 Cal which reduces t o pi2 . pz = Kp?, the regular mass law equation for this case.
+
+
+
+
+
~
~
T H E MASS CNIT O F T H E CHEMICAL POTESTIAL
75
If we are dealing with electromotive forces, Gibbs has shown that the electromotive force of a reversible cell is a measure of the difference of the chemical potentials. If we take the gram as the mass unit, the electromotive force is proportional to the difference of the specific chemical potentials. If we take the electrochemical equivalent as the mass unit, the electrgmotive force is equal to the difference of the coulomb chemical potentials, p. 33 I . “We know by experience that in certain fluids (electrol>+c conductors) there is a connection between the fluxes of the component substances and that of electricity. The quantitative relation between these fluxes may be expressed by an equation of the form Dm, Dmb Dm Dmh De = etc. - - etc., (682,) aa a b a, a h where De, Dm,, etc. denote the infinitesimal quantities of electricity and of the components of the fluid which pass simultaneously through any same surface, which may be either at rest or in motion, and aa,a b , etc., a,, a h , etC. denote positive constants. We may evidently regard Dm,, Dmb, etc., Dm,. Dmh, etc. as independent of one another. For, if they were not so, one or more cou’d be expressed in terms of the others and we could reduce the equation to a shorter form in which all the terms of this kind would be independent. “Since the motion of the fluid as a whole will not involve any electrical current, the densities of the components specified by the suffixes must satisfy the relation.
+
~
+
+
These densities, therefore, are not independently variable, like the densities of the components which we have employed in the other cases. “We may account for the relation (682) by supposing that electricity (positive or negative) is inseparably attached to the different kinds of molecules, so long as they remain in the interior of the fluid, in such a way that t,he quantities‘ CY.,a b , etc. of the substances specified are each charged with a unit of poeitive electricity, and the quant’it’ies ag,a h , etc. are each charged with a unit of negative electricity. The relation (683) is accounted for by the fact that the constants a,, ag,etc. are so small that the electrical charge of any sensible portion of the fluid varying sensibly from t’he law expressed in (683) would be enormously great, so that the formation of such a mass would be resisted by a very great force. “It will be observed t,hat the choice of the substances which we regard as the components of the fluid is to some extent arbitrary and that the same physical relations may be expressed by different equations of the form (682), in which the fluxes are expressed w th reference to different sets of components. If the components chosen are such as represent what we believe to be the actual molecular constitut on of the fluid, those of which the fluxes appear in the equation (682) are called the ions and the constants of the equation are called their electro-chemical equivalents. For our present purpose, which
76
WILDER D. BASCROFT
has nothing to do with any theories of molecular constitution, n e may choose such a set of components as may be convenient, and call those ions, of which the fluxes appear in the equation of the form (682) without farther limitation.” When we let V‘, V” denote the electrical potentials in pieces of the same metal connected with the two electrodes of a reversible cell, “when the great effect of gravity may be neglected and when there are but two electrodes, as in a galvanic or electrolytic cell, n-e have for any cation y’’- F” = cy, (687) and for any anion - T” = % (PP” - Fe’) (688) where V“-V‘ denotes the electromotive force of the combination. That is:“ W h e n all the conditions of eyuilibriuifi ore fulfilled in a galvanic or electrolytic cell, the electromotite force is equal to the diference in the d r i e s o j the potential f o r a n y i o n or apparent ion at the surfaces of the electrodes mztltiplied b y the electrochemical eguizalent of that i o n , the greater potential of a n a n i o n being at the same electrode a s the greater electrical potential, a n d the rezberse being true of a cation,” p. 3 3 2 . Oibbs thus states explicitly that, we can iefer the chemical potentials to grams or to electrochemical equivalents and that. the second is simpler because then equations (687) and (688) become V”-T” = and Y“-T‘ = (P)~”- (p)&’. Actually, n e do not, use either of these expressions, because we are concerned with the osmotic pressure of the ions and consequently we refer the chemical potentials to gram ions and bring the valence of the ion in elsewhere in the formula. This gives nF(V” - T’) = (p)’ - (pea)" = (P)~”- ( ~ 1 ~ ’ . For t’he single potential difference at the surface of an electrode reversible with resFect to the cation, this becomes nFE = pn, - p s = R T log P, p. which is the Nernst equation, p,,, being the potential of the metal and p s the potential of t,he cation in solution. The measurement of an electromotive force, as usually c a h e d out, does not involve the measurement or knowledge of the current in any portion of the apparatus, except that there must be no current flowing through the telephone, galvanometer, or electrometer. Since the value of the coulomb chemical potential will depend on the va‘ue assumed for the coulomb, it may seem to some that the electromotive force, which i apparently independent of the value of the coulomb cannot necessarily be a measure of the coulomb chemical potential. This is an error, because the value of the ampere (coulomb per second), the volt, and the ohm are so connected by Ohm’s law that me have only two independent variables. If we keep the ohm constant and vary the value of the coulomb, we shall necessarily change the value of the volt’. While it might be possible on paper t.0 keep the unit of potential difference constant. and to vary simultaneously the ohm and the coulomb. this is not practical, however, because we have a relation between the joule (voltcoulomb) and the calorie, so that we cannot very well vary the coulomb
THE MASS UNIT O F THE CHEMICAL POTENTIAL
77
without varying the volt, Actually, this discussion is academic because nobody has any intention of doubling or halving the value of the coulomb. It may be asked what the use of this whole paper is when Gibbs hasalready stated clearly that the chemical potential may be referred to any mass unit, meaning thereby that in equat’ons one may use specific potentials, molecular potentials] etc., provided one introduces in each case the appropriat’e factors. One reason for writing this paFer is that Ostwald has apparently never been clear on the question of the mass unit and it is probable that others have been equally confused. I know that I have been. In an early paper Ostwaldl gives the combining weight as the capacity factor of chemical energy and says that “the capacity factor of the chemical energy, like that of heat is proportional to the mass and the weight. From the name for it, t,he combining weight, one must not, conclude that the chemical capacity is fundamentally a weight. I t is not that any more t’han it is a mass. It is only proportional to these two and the proportionality factor changes with the nature of the substance. A ‘dimension’ in the usual units of length, time, and mass can no more be given for the capacity factor of chemical energy than for any of the other energy factors of the t’able;all assumpt’ions to the contrary rest on arbitrary omissions and therefore on error.” A little later there is a paragraph? which may or may not have a bearing. “It must finally be mentioned that several capacity fact,ors are equal for chemically comparable amounts of different substances. This mae first suggested by LeChatelier and was emphasized strongly by Meyerhoffer3. Thus the value K of the gas equation PV = RT is t,he same for molecular amounts, the heat capacity for atomic weights, and the electrochemical capacity for equivalent weights. These three esamples show, howerer, that the weights in question are not proportional from case to case, for the gas content, the heat capacity. and the electrochemical capacity are proportional respectively to the molecular weight, the atomic weight, and the electrochemical equivalent‘, and these are not in general proportional for different substances. Other capacities, such as the mass, do not show the relation and consequent.ly the rule is by no means general.” On page j o o of the same volume, Ostwald says: “While the intensity factor of heat, the temperature, is well known to us and the other factor, the entropy, is difficult to grasp, the converse is true in the case of the chemical energy. It appears to us so obvious that the chemical energy is proportional to the amount of the substances in question, other things being equal, that at first we do not trouble to formulate such a hypothesis. Kevertheless, it is necessary to be clear that this proportionality of chemical energy to mass and weight is an empirical fact, not in any way necessary a priori. Mass is a magnitude which belongs only in the field of kinetic energy and is connected wit’hchemical energy only through this proportionality law. This latter holds Z. physik. C h e m . , 10, 370 (1892). 2
Lehrliuch allgem. Chemie, 2 I, j o (1893) 2. physik. C h e m . , 7, 544 (1891).
78
WILDER D. BAXCROFT
also for weight. In addition, we know from experience that the proportionality factor depends t o a great extent on t,he special (chemical) nature of the substances. “The factor of the chemical energy which is proportional to the mass of the substances in question, does not have the character of an intensity because it determines neither the occurrence nor the form of the chemical changes. We shall, therefore, have to set the chemical capacity proportional to the mass. It is not desirable to make the proportionality factor between the mass and the chemical capacity equal to unity and thus refer the chemical energy to the unit of mass of the different substances as happened in the earlier development of thermochemistry. The law of combining weights suggests to us to multiply the values referred to the unit of mass by combining weights and thereby to evaluate the chemical energy for such quantities of different substances wh’ch ttand in t,he ratio of the combining weights. In this way the neceeeary calculations are reduced to a minimum and the relations between the chemical energy of different, substances are easier to see.” On p. 302 Ostwald says: “In regard to the choice of the chemical capacity factor we have to make a definite, more or less arbitrary, agreement, just as we had to do in regard to the chemical combining weights. In that case it was possible so t.o determine these latter values that a number of other relations assume a simple and clear form, whereby a definite system of values, the atomic weights now in use, have proved the most serviceable in every way. Since the numerical values of the chemical capacity factors have no other condition to satisfy than that their sum for the substa,nces and the amounts of the substances shall be equal on both sides of a chemical equation for the reaction, a condition which the atomic and combining iveights satisfy, we m a y take these Lsalues as the units of the chemical capacity for the di,fferent substances.” All this is a bit vague, but it seems to mean that the combining weights are the mass uni:s for the chemical capacity factor. This does not seem to me to be right at all. Gibbs has shown that in a completely reversible voltaic cell the electromotive force is a measure of the chemical potential (or rather of the difference of chemical potential) when we take as the mass unit the electrochemical equivalent, which is quite a different, thing from the equivalent weight, the lat,ter being approximately 9 6 j o o times the former. On the other hand, we must take the molecular neights as the mass units if we are to deduce the mass law directly from the relation between the chemical potentials a t equilibrium. At first eight this seems to make the choice of the mass units an entirely arbitrary one; but I think that this is not the case, and Ostwald has himself shown the wal‘ out, apparently without realizing it. The difficulty is in the Gibbs definition of the potential, p. 93. “If to any homogeneous mass we suppose an infinitesimal quantity of any substance to be added, the mass remaining homogeneous and its entropy and volume remaining unchanged, the increase of the energy of the mass divided by the quantity of the substance added is the potential for that substance in the mass considered.”
THE MASS UNIT O F THE CHEMICAL POTENTIAL
79
This definition is absolutely right; but, as Ostwald' has pointed out, we cannot determine the absolute value of the chemical potential of any substance because we do not know the change of t'he energy with the change of mass. All we can do is to measure differences of chemical potential. As I see it, this means that we are always measuring the chemical potential of a reaction and that consequently we must use the mass units of the reaction. If we are making electrometric measurements the unit is the mass associated with the unit quantity of electricity, the coulomb in electrochemical work. If we are studying pressure relations in a gas or a solution, the unit is the molecular weight of the substance or the ion as the case may be. I have thought that the electromotive force was a measure of the chemical potential of the system and that there should be a chemical potential for any organic compound if we could only measure it. It is now clear that this was a mistaken point of view. Since we measure only the difference of chemical potent'ial for a given reaction, there are really as many chemical potentials in an organic compound as there are bonds which can be broken. I t is these values which must be measured some day. Another reason for writing this article is that we do not yet make the use of the chemical potential in chemistry that I am sure we ought to make. It seems probable that this is due in part to our misconceptions of the chemical potential. Gibbs deduced the phase rule from the relation among the potentials, and Lash Miller has made use of it to a limited extent in discussing changes of solubility, and it is recognized that electromotive force determina. tions measure differences of chemical potential; but that is about as far as we go. Guldberg and Waage deduced the mass law from assumptions in regard t'o reaction velocity and Gibbs deduced the mass law from the relation between the potentials at equilibrium; but no one, so far as I know, has ever deduced the reaction velocity formulas from the chemical potentials, though Xernst has suggested that the reaction velocity may be proportional to the difference of chemical potential divided by the unknown chemical resistance. Of course one can substitute from one set of equations to the other. If we consider the reacting substances as present in equivalent quantities, we can write for monomolecular, bimolecular, and trjmolecular reactions the equations dp,'dt = R, dpjdt = R(A-x) and dpjdt = R ( A - X ) ~respectively; or for t,he general case dk/dt = k ( A - x ) ~ - ' where n is the order of t,he reaction; but we cannot deduce these equations and they are not what we want anyhow. By subtraction, one can also wr.te the general equation p1 + P ? . . . pn dx , which is of no value until we can deduce it and - = e R dt show that this is the necessary formula and that has not yet been done. Consequently, a further study of the chemical potential is essential. I feel sive also that some day the history of chemistry will be written as showing the devc!cpment (conscious and unconscious) of the theory of the Lehrhrtch allgem. Chernie, 2 11, 123 (1902).
80
WILDER D. BANCROFT
chemical potentials; but we are certainly a long way from that still. I hope that this article may stimulate somebody better qualified than I am, and that it will thereby bring that day nearer. The general conclusions of this paper are:We never measure any absolute chemical potential but only the difI. ference of potential due to some reaction. The chemical potentials may be referred to any mass units; but 2. general relations are brought out more clearly if the mass units appropriate to the reaction used. 3 . When we are studying react'ions involving the pressures of gases or the osmotic pressures of solutions, the appropriate mass units are the molecular weights of the reacting substances or ions. 4. If one is studying electrochemical reactions, the appropriate mass units are the electrochemical equivalents, the masees which are transferred with one coulomb. These are equal to the equivalent weights divided by the Faraday constant. 5 . Kobody has worked out what t,he appropriate mass units are in the case of electrical endosmose or of frict.iona1 electricity. 6 . In the caFe of organic compounds one can have as many chemical potentials as there are bonds which can be broken or activated. j . The mass law equations have been deduced from the reaction velocity equations and from the relations between the chemical potentials; but apparently nobody has deduced the reaction velocity equations from the relations between the potentials although the difference of chemical potential must be one of the factors determining the reaction velocity, and in spite of t,he fact that Ohm's law holds absolutely for reactions taking place under certain clearly-defined conditions. 8. The history of chemistry will undoubtedly be nritten some day as a study of the conscious and unconscious development of the theory of the chemical potential. Cornell University.
