ADSORPTION EQUATIONS
A REVIEW OF T H E LITERATURE BY E S O C H SWAN AXD ALEXAKDER ROBERT VRQUHART
Contents
INTRODUCTION. Komenclature : Evaluation of the Quantities; Classification. I. THEGENERAL ADSORPTION EQUATION. Empirical : Freundlich ; Muller. Theoretical : Gibbs, and related equations of Harkins, Thompson, Warburg, Lewis, Porter, and Polanyi; Langmuir; Frenkel; Volmer; Henry; Eucken; Lorentz and Lande; Polanyi; Iliin.
11. THEISOTHERM. Empirical:
Theoretical:
The “exponential” equation, and related equations of Schmidt and Hinteler, and McGavack and Patrick; Gurwitsch; Freundlich’s equations; Rakovski ; Kayser; Schmidt; Homfray; Trouton. Deductions of the “exponential” equation by Robertson, Kolthoff, and Henry; Ostwald and Izaguirre; Gurwitsch; Stadnikoff; Pavlov; Reichinsteinj Duhem; Schmidt’s first equation ; Arrhenius; Schmidt’s second equation; Williams ; Henry; Gorbatschew.
111. THEISOBAR. Empirical: Kayser; Ostwald; Freundlich. Theoretical: Williams; Iliin. IV. THEISOSTERE. Empirical : Freundlich ; Homfray Theoretical: Williams; Henry. V. RATEOF ADSORPTIOX. Empirical : Mills and Thomson; Bergter; Lagergren; Pickles; Burt and Bangham; Zacharias; Freundlich; Dietl; Davis and Eyre; Fisher. Theoretical: McBain; Marc; Gustaver; Henry; Iliin. VI. HEATOF ADSORPTION. Empirical: Freundlich; Homfray; Lamb and Coolidge; Katz and Holleman. Theoretical: Williams; Eucken; Lorentz and Lande; Polanyi: Harkins; Iliin; Tarasoff. CONCLUSION. Note on t h e Formulation of Adsorption Laws.
252
E N O C H S W A N AXDALEXAXDERROBERTURQCHART
Introduction The great importance of the part played by adsorption phenomena in natural processes is now well recognised, and in consequence of this recognition a large amount of work has been done with a view to discovering the lams which underlie such phenomena, and more particularly to make these laws more concise and readily understandable by expressing them in mathematical form. As a result of this work many adsorption equations are extant, and though none is completely satisfactory, most are valuable as approximations to the truth in a more or less restricted region. In the present paper an attempt has been made to collect in one place the various equations that have been proposed, and to consider in how far they are applicable to actual adsorption systems. In its strictest sense the word adsorption designates the concentration of one phase at its interface with another, but the filling of pores in the adsorbent, which may accompany true adsorption, is occasionally included in a looser use of the term. In this paper these capillary effects will be considered adsorption just as far as the authors concerned have so considered them, but the adsorption of a gas by a liquid and of a liquid by a liquid mill not be discussed.
Nomenclature. As it is desirable to retain one set of symbols in order to avoid the frequent repetition of definitions, the following nil1 as far as possible be used throughout, irrespective of the actual synibols used by the various authors in the original papers. Let m = ma% of adsorbent. v = volume of medium surrounding adsorbent. X = mass of adsorbate adsorbed by m grams adsorbent. x = mass of adsorbate adsorbed by I gram adsorbent. s = mass of adsorbate adsorbed by I gram adsorbent at saturation. p = equilibrium pressure of adsorbed gas or vapour. P = saturation pressure of adsorbed gas or vapour. co = initial concentration of solution. c = equilibrium concentration of solution. 8 = temperature Centigrade. T = temperature Absolute. t = time. H = total heat of adsorption. R = gas constant. n, k, kl, etc., constants. Evaluatzon of the Quantzties. The experimental determination of most of the above quantities presents no great difficulty, b , the evaluation of the amount adsorbed, 2, has not always been accurately >erformed. When adsorption is from the gaseous phase a knowledge of the to volume of the system and the initial and equilibrium
ADSORPTION EQUATIONS
253
pressures is sufficient to determine r. It seems hardly necessary to add that account should he taken of the volume of the adsorbent, yet this has sometimes been neglected. This method of determining r neglects the volume of the adsorbed gas, hut this error is ueually quite unimportant. In the majority of investigations of adsorption from solution z has been taken as equal to (co - c)w or (c, - c ) ~ where , c denotes the concentration in grams Fer unit u-eight or volume, and w and 0 are respectively the original weight and volume of solution. JTilliams,107however, has pointed out that this calculation of z assumes that the mass or volume of the solvent remains constant during adsorption, and that no solvent is adsorbed. Keither of these assumptions is justifiable, though the error introduced by the former is sufficiently small to De neglected when the solution is very dilute. Wjlliams has further shown that if RI is th? mass of solution, and concentrations are expressed in grams solute per gram solution, then
where x0 is the amount adsorbed calculated on the assumption that no solvent is adsorbed. If z and ?/ represent the true adsorption value? of solute and solvent resrectively , then s=x0+y- C I - c
In general z and y cannot be determined, hut Williams has given two empirical equations from which approximate values may be deduced. ThiP analysis, which leads to a simple explanation of negative adsorpf' 1,lo9 has more recently been substantiated by Gustafson.J? It is of interest to note that Mecklenburg68has recently proposed a graphical method for the determination of the equilihrium concentration when the amount adsorbed is known. ClassiJication. For convenience the problems t,o be solved may be regarded as the formulation of the following six functions: The General Adsorption Equation, x = p(c, T). The Isotherm, s = p(c). 3. The Isobar, s = p(T). 4. The Isostere, c = p(T). j. Rate of Adsorption, s = p(t). 6. Heat of Adsorption, H = ~(x), I.
2.
I. The General Adsorption Equation Enipiiical. FreundlichZ2has developed a method for the representation of adsorption from solution at any concentration and temperature, depending on thk use of the eauations
2 54
ENOCHSWANANDALEXANDERROBERTURQUHART
and in combination with his X equation. (4. v. section 11.) Muller'? found that the absorption of water vapour by textile fibres could be expressed by the equation
+
x = (ki kz p / P ) d n This equation is valueless except perhaps over a very limited pressur? and temperature range, both the assumptions involved in it-that of a linear relationship between 5 and p / P and that of zero adsorption at the boiling point-being quite unsound. Theoretical. The earliest theoretically derived equation is that deduced by G i b b ~ . ~ ~ In 1874 Gibbs showed thermodpamically that if r is the surface excess, u the surface tension, and p the chemical potential of the adsorbed substance in solution, then
Writing p = RTlog(d
+ k) where A is the activity of the solute then d p = R T d log
and
r=-
A
do
R T d log A
.
c do For very dilute solutions A = c, and hence I' = - - R T dc which is the usual form of the Gibbs equation. This equation is usually interpreted qualitatively as meaning that if a dissolved substance lowers the surface tension at an interface then it will be positively adsorbed at that interface. Bancroft* has shown that this interpretation is not a logical conclusion for the interface solid-liquid. Wo. O ~ t w a l d ' ~ extends the usual interpretation of the equation as follows: "Bdsorption will take place whenever there exists a t a surface a difference in energy potential which can be decreased by a change in the concentration of the dispersed material bordering the surface." Amongst other authors who have published proofsof the Gibbs equationare Milner,'l Harlow and W l l o ~ s , *Williams,111 ' I~edale,'~ and TT. Ostwald.7s A deduction of the equation has also been given by Harkins,86who has shown that c' du r = - - c- =d u RT dc RT dc' where c' is the concentration of the solute in the vapour. Thompsonlw in 1888 deduced the relationship
+
_- I
d-
c r = ce R T d c which yields the usual Gibbs form on expanding and neglecting higher terms.
255
ADSORPTION EQUATIONS
That the surface layer cannot be homogeneous with the remainder of the liquid was shown by Warburg,lo6who for sodium chloride solutions obtained a relation for the surface deficit similar to that of Gibbs. Lewis,m using a somewhat simpler method than Gibbs, has found the e quation of electrocapillary adsorption to be
r salt
+r
cation
+r
anion =
[c
da
- RT dc
+ (a + b) g]
where a and b are the electrochemical equivalents of anion and cation, and T is the potential difference a t the interface. A slight error in Lewis’ original proof has been pointed out by P a r t i n g t ~ n . ’ ~ Porters7is the only author who has derived the equation for concentrated solutions. He finds that =
-
(I
- ac)*c d-a KT
dc
where c is the ratio of solute molecules to solvent molecules (n/N), and a is Callendar’s hydration factor, obtained from the equation
It is of interest to note t.hat P ~ l a n y has i ~ shown ~ ~ ~ ~that for a swelling gel
where H is the swelling pressure. In a later paperaesome results of Loeb have been used to provide experimental confirmation of this equation. FreundlichZa considers the Gibbs equation to be of limited application, and has stated that the more general equation would be
r
=
- f(c, T) du dc
in which all that is known about f(c,T) is that it is positive. The usual form of the equation is obtained by assuming the van’t Hoff Laws, and hence is limited with these laws to dilute solutions. Even for dilute solutions, however, Freundlich is doubtful of the soundness of the deduction, since the surface tension depends on factors, such as solvation, which are not considered in the van’t Hoff laws. L a n g m ~ i ron , ~ the ~ other hand, believes that the deduction is sound for dilute solutions, since the gas laws in some of the derivations,for example Milner’s-are applied only to the interior of the solution. It should be noted, however, that the equation is definitely limited to true solutions, and cannot be applied to colloidal solutions, a point which has been emphasized by Bancroft.’ All the deductions of the equation assume the thermodynamic reversibility of the adsorption process, the validity of which assumption may in some instances be in slight doubt.
2 56
ENOCH SWANANDALEXANDERROBERTURQUHART
Many attempts have been made to obtain confirmation of the equation, but the great experimental difficulties have prevented any definite conclusion being reached, Donnan has recently said “the question whether the simplified form of the Gibbs equation yields a sufficiently accurate value for the excess surface concentration can scarcely be decided without further experimental evidence.’’ According to L a n g m ~ i r ’ stheory ~ ~ of adsorption the adsorbate is held by the unsatisfied secondary valences of the adsorbent to form a continuation of the space lattice of the latter, so that the adsorbent forms a true chemical compound with the adsorbate. It is assumed that all molecules striking a surface condense, and that evaporation takes place subsequently, so that adsorption is the equilibrium condition representing the time-lag of condensation over evaporation. Since the force of attraction is much less at the second layer than at the first, molecules will evaporate much more quickly from the second layer, and hence it 1s to be expected that a monomolecular layer will be exceeded only at high pressures or concentrations. Langmuir’s theory is derived specifically for true adsorption at a plane surface, and neglects capillary effects. A consideration of the dynamic equilibrium between rate of evaporation and rate of condensation leads to a number of equations according to the conditions obtaining at the s u r f a ~ e . ~ ~ 1 ~ ’ (I)
The plane surface has only one kind of space lattice.
Then where 0 is the fraction of the surface covered v
is the rate at which the gas would condense if the surface were completely covered
p is the number of moles of the gas striking each square centimetre
per second. From the kinetic theory p =
P
d
m
h- is Avogadro’s Constant Sois the number of elementary spaces per square centimetre 7 is the number of moles of gas adsorbed per squate centimetre Under these conditions adsorption should be large and nearly independent of pressure at low temperatures, while at higher temperatures adsorption should be small and proportional to pressure. The plane surface contains more than one kind of elementary space. (2) Then where PI, 8 2 , . . . etc., represent the fractions of the total number of elementary spaces occupied by each kind.
ADSORPTIOX EQUATIONS
(3)
Amorphous surfaces-all
257
the elementary spaces may be unlike.
0
where a is a function of p. In Langmuir’s opinion this equation should be true for porous bodies. (4) Each elementary space may hold more than one adsorbed molecule. r\’
kil
+ 2kikzF’ + 3kik2k3pa + . . .
K q = + k+ + klk2pz + k1k2k3p3 + . I
.
.
This equation may also be expected to hold when adjacent molecules influence each other’s rate of evaporation. ( j) Atomic adsorption-the elementary spaces are occupied by individual atoms of the adsorbate.
If 0 + I , i.e. adsorption is small, q oc p %
(6) Adsorbed films more than one molecule thick N k -7 = No I/P a bp cpZ where a = kl - zkz b = kz(4kz - 3k3 - ki) c = zkZ(6kzka - 2k3k4 klkz - klk3 - 4kz)
+ + + + +
This equation shows that at very low pressure q is proportional to p , but a t pressures close to saturation q begins to increase rapidly, and is infinite when saturation is reached. Langmuir’s equations have not yet received adequate experimental confirmation over large pressure ranges, which may perhaps be attributed to the difficulty of expressing the equations in terms of measurable quantities, and also to the fact that the conditions under which they may be expected to hold are ideal rather than readily attainable. In adducing much evidence in favour of the existence of monomolecular films of oil on water from the exper mental work of, amongst others, Milner, Syskovski, and Traube, Langmuirs6has made use of a new relation between r and c, viz., where X is the decrease in potential energy which occurs when one mole of solute passes from the interior of the solution into the surface layer. A theory similar to that of Langmuir, has recently been put forward by Frenkel,21while equations of the same type have been obtained thermodynamically by V ~ l m e r . ~ ~ ~
258
E N O C H S W A N A N D A L E X A N D E R R O B E R T URQCHART
Henrysg has developed equations based on the conceptions of surface energy introduced by Hardy and Langmuir. The fundamental assumptions are, first, that the range of action of adsorption forces is comparable with atomic diameters; second, that the impact of a molecule is completely inelastic. By applying to the kinetic equilibrium Jeans’ equations for the rate of impact and Langmuir’s equation for the rate of evaporation, general equations for the rate of adsorption, the isotherm and the isostere are developed for the adsorption of each of n gaseous components. Examples of these equations are given later. According to Eucken,]’ there is around an adsorbent an “atmosphere” of adsorbed gas in a state of compression, the density of any layer of this atmosphere varying with its distance from the surface. I t is assumed that the adsorbed molecules have no interaction, and that the relation between force (F) and distance (h) is
where p is a constant. In a manner similar to the derivation of the barometric height formula the equation representing the adsorption of a perfect gas by a plane surface of area 0 is obtained in the form
where 6 is the mean molecular diameter, and C, is the number of moles of gas in unit volume at a distance from the surface corresponding to zero attractive force. By evolving in series and integrating an approximate solution is obtained
Other equations are given for the adsorption of a perfect gas by a segmented surface, the adsorption of a real gas by a plane surface, and the adsorption of a vapour by a segmented surface. In a more recent paper Euckenl* has rederived those equations making use of Boltzmann’s relation between the potential energy of a single particle and its distance from the surface. Throughout this work it is assumed that p = 3, which is hardly in agreement with modern opinion.I6 Xevertheless, the theory is in good agreement with experimental data; the same value for k2 is obtained whether it is calculated from (I) the temperature coefficient of gaseous viscosity, ( 2 ) van der Waals’ theory or (3) the heat of adsorption. Polanyis5has adversely criticised Eucken’s theory, and has concluded that it is not applicable t o any of the markedly curved isotherms. Lorente and Lande59 believe that Debye’s work on the dipolarity of molecules is opposed to the conception of a purely molecular force of adsorption, and they conclude in contradistinction to Eucken, that adsorption
ADSORPTION EQUATIONS
259
potential should in general vary with temperature. For the temperature range in which z varies as p they deduce the equation
which is very similar to that obtained b; Eucken. It is also shown that the apparent independence of potential on temperature found by Eucken, cannot be adduced as evidence against the dipolar molecular theory. Polany+ by considering with relation to the Third Law the heat and free energy of wetting has deduced that at low temperatures an adsorption system behaves as an ideal concentrated solution, whilst with increasing temperature the behaviour gradually approaches that of a dilute solution. By assuming that the adsorbed layer is highly compressed by the molecular forces and that the “exponential” equation holds, he has concluded that the molecular force varies m the distance. (cf. EdseP). In a later paper PolanyijB2still assuming a highly compressed layer more than one molecule thick, has shown how to calculate the adsorption at any temperature and pressure if one isotherm is known. As a result of the compression a vapour will exist in the adsorption layer in the liquid state; if it is assumed that the internal pressure of this liquid is identical with that of the liquid in bulk then from thermodynamics it follows that ex = R T log P ’p where e, is the adsorption potential at a distance I., corresponding to an amount x adsorbed. If now an isotherm is known for any temperature below the critical, the relation cp, = x/6, where 6, is the density of the compressed layer, assumed equal to that of the liquid, and p, is the volume of the adsorption space to a distance z from the surface, allows of the plotting of a curve showing the relation between E and cp. I t being assumed that this relation does not vary with the temperature, a curve can be drawn showing E, against 6,, provided the equation of state of the adsorbed substance is known. This calculation of 6, varies according as ( I ) the gas is condensible, ( 2 ) the gas is not so condensible (3) the gas is in the transition region between ( I ) and ( 2 ) . Lastly a curve can now be drawn showing 6, against p,, and mechanical integration of this curve gives the amount adsorbed, since
x’ =
$6””6.
dp
- d, . pman
where x’ is the adsorption excess, d, is the density of the free gas space, and 9max is the volume of the total adsorption space. According to this theory the adsorption excess isotherm is divisible into a first part where the ideal gas laws are applicable and z varies as p , a second part where the a ’v2 of van der Waals’ equation predominates and the compressibility decreases more slowly than I /p, and a third part where van der Waals’ b predominates and the compressibility decreases more quickly than I/P. The theory also indicates that the saturation value should vary with the temperature, and in a later paper Polanyisj claims superiority of his theory over that of Eucken on this account.
ENOCHSWANANDALEXANDERROBERTCRQUHART
260
Calculations according to the theory are in excellent agreement wit,h experimental data, and this agreement is regarded as proof of the validity of the assumption that the adsorption potential is independent of the temperature.ss Assuming the electrical origin of attraction, Polanyi therefore concludes that the cause of cohesion is a deformation of the molecules to bring opposite # charges as near as possible. The theory has been extended to solutions84but theri is not sufficient suitable experimental evidence to allow of the verification of this extension; qualitatively, however, it is in agreement with Freundlich’s rule that substances which are strongly adsorbed from solut,ion permit only of weak adsorption from solutions in which they act as solvents. Berenyi6,7 has developed two new methods of calculating the E - cp diagram, one by the use of the pressure-density tables of Xmagat, the other depending on the Nernst rule of corresponding states. The latter method can be applied only at temperatures between 6 and 1.4 times the boiling point, to substance? whose boiling points lie between 170’ and 400’ A. As sbated before, Lorentz and disagree with the assumption of temperature independence of potential, but, they have shown that Polanyi’s theory is in agreement with the application of the theorem of corresponding state? t o adsorption in the p - 0 region where Henry’s Law holds but where the pressure is not greater than the critical pressure and the temperature not’ less than the critical temperature. Polanyi’s theory is well supported by the experimental data tested, yet it is based on three unproved assumptions; ( I ) I t is assumed that the potential e and with it van der Waals’ a is independent, of the temperature. ( 2 ) It is assumed that the potential is independent of whether the remaining space is empty or not. (3) I t is assumed that, the equation of state is applicable to an adsorbed gas. Berl and SchwebelQ have combined Polanyi’s equations with the “exponential” equation to find the quantity of steam required to expel a gas from an adsorbent. The agreement with experiment is excellent. Considering the forces of adsorption to he of electrical origin Iliin4’ has obtained, by a method similar t o that used by Eucken. a complicated expression for t,he amount adsorbed z in terms of the surface, temperature, etc., and a number of electrical quantities. This equation requires that the graph of log (x
.E
against
-
I
where e is the dielectric constant of the adsorbed gas, plotted
‘5-1 I
~
should be a straight line. I n this manner the equation is con-
€
firmed by the experimental results of Homfray and Titoff. 11. The Isotherm h typical isotherm for adsorption from the gaseous phase is shown in
Fig. I . From this curve it is obvious that dx dp approaches zero as p increases, d2x/dp2being negative. The adsorption of vapours may be repre-
ADSORPTION EQUATIONS
261
sented by one or other of the two curves of Fig. 2 . For both these curves dxidp increases as the partial pressure approaches its saturat,ion value, but for curve 2 d2x/dp2is positive, whilst for curve I d2x/dp2passes from negative to positive values. Little information is available for the construction of isotherms illustrating adsorption from the liquid phase over any extended range of concentration, but what there is indicates that the curve is of the form shown in Fig. 3. It should be noted, however, that this curve shows the variation of x o not x, with the concentration.
co Ft9.3.
T \-c
From a consideration of the curves shown above it seems clear that one equation cannot be expected to represent these various types of isotherms. Empirzcal. The equation x = kc’ ”, which is usually but erroneously called the “exponential,” is perhaps the best known of all the equations which have been proposed to represent an adsorption isotherm. According to Firth,Ig De Saussureg4in 1814 was the first to apply this equation to the adsorption of gases; its application was extended to solutions by Boedecker’O in 1859, and in more recent years it has been used by hundreds of workers. Experiment has shown that the exponent I/n is always less than I : hence dx ’dc decreases as c increases, and d2x:dc2 is negative. Thus the equation may represent the isotherm for a gas shown in Fig. I , but cannot represent any part of curve I of Fig. 2 , while it is applicable only to the first portions of the other two isotherms. I t would seem a t first glance, therefore, that this equation does not agree very well with experimental data, but it should be noted that it can be applied to most adsorption phenomena in the low pressure or low concentration region, and as much of the experimental work which has
262
ESOCH SW‘AN A S D ALEXASDER ROBERT CRQUHART
been carried out did not extend beyond this region, the equation was found to be of very general applicability. Above the region in which the equation may safely be used I ’n decreases with increasing pressure or concentration. Consideration of a large amount of data has shown that the smaller the adsorbability of the gas the greater is r{’n, a conclusion which agrees with Seeliger’slooobservation that, r,‘n a 1/TCwhere T, is the critical temperature of the adsorbate. I t is also generally found that I/n increases with increasing temperature; in fact’, Berl and Schwebelg have shown from Polanyi’s theory that I/nccT. TTXams,112 in a critical discussion of much data, has shown that at low concentrations I,/n approaches I , i. e. the system obeys Henry’s Law. All of these observations may be coordinated in the statement that’the smaller the amount adsorbed the more nearly does 1,’n approach I . I t is also found that Iyn increases with molecular complexity, runs parallel to the strength of solute acids, and is independent’ of the degree of dispersion of the adsorbent. k, on the other hand, increases with the degree of dispersion of the adsorbent, and varies inversely as 1’ n, where I is the solubility of the solute. Schmidt and Hintelerjg9in applying the equation to organic vapours used it in the form z = s(p,’P)‘ n, while a similar modification has been used by v McGavack and Patrick,67 viz= k(p,’P)’#”,where 2’ is the volume of u1 ” condensed vapour and u is the surface tension. According to Gurwitscha! the equation x = k, klcT’’’ agrees better with the results of experiment than the usual form, but the superiority of this equat,ion has been disputed by K ~ l o s o v s k i . ~Ostwald ~. and I~aguirre’~ have pointed out that the curves are identical, the only difference being in the disposition of the axes of reference.
+
It was found by KroeckeP that dxjdm= ___ - where a is the original v
amount of solute and X is a constant.. From this result FreundlichZ2,”deduced his “X equation” v a x = -log -__ m a-x in which X is independent of m,but is a function of a l v or C,. He also found that
from which by expanding in series the “exponential” equation can be derived. In conjunction with Gibbs’ equation (q.v.) it yields an expression for the variation of surface tension with concentration which agrees well with experiment. The X equations have been criticised by McBain,“ who concludes the system of X formulae introduced recently by Freundlich to represent the amount of adsorptior, does not lead to any definite number characteristic of each set of substances, and further, i t s results are irreconcilable with each
263
ADSORPTION EQUATIONS
other and with experiment. It should be noted, however, that in his argument McBain erroneously used zero instead of z as the limit when a/c + I of the expression dlogc -~(a - c) log a/c (a - c) - cloga/c d logX/m Nevertheless, the X equation is not unexceptionable, as is shown below. We have v a a - X X = -log&, co = - , c = m a-X v V X = mx = v(c, - c) x log c o / c = X l o gc 2 = -~ c, - c c c, I - c/co and d log X - d log x cic, I d log c ogc logco/c I - c/c, Now if X is independent of c when co is constant, as is claimed, then d log -
X
Hence
+-
Z
C
-
and therefore d-=-log x d log c
c/co
I
log
C,/C
I
-
I
-
- log c/co
-
(I
-
+
CiCd
%(I
I
-
c/c,
- c/co)2 + % ( I
ww-
l/Iz(I
-
C/C,
c/co - c:c,
I
I
-
I
-
+
- c/Ico)* c/co)-
. . .+
. . .+ 1 ......
I
I
-
I
-
I
- c/co
A
- c/c,
( I - c’co) = where all succeeding terms are negative. Now if adsorption is positive c k p , Wjllia~ms’equation gives a curve of the type shown in Fig. I . The general equation to the isotherm obtained from Henry’sJg theory (q. v.) is not soluble, but for the adsorption of a single gas the equation bs-
+
comes x
=
kip(
I
- &)k3, which for small pressures is identical with Williams’
equation. The relationship between a number of the isothermal equations has recently been demonstrated by Gorbatschew,so who, by superimposing various simplifying assumptions on a few general relationships, has derived the equations of Langmuir, Reichinstein, Schmidt, Williams, and Henry, together with Freundlich’s X equations, and the “exponential” equat,ion. The lack of a complcte mathematical interpretat,ion of the isotherm has recently been expressed by McBain,6ewho believes that the “exponential” equation is still the best representation of the existing data.
ADSORPTION EQUATIONS
111. The Isobar Empirical. K a y ~ e found r ~ ~ that if L’ is the volume of adsorbed gas at temperature 0 , then approximately, v = kl k2 0 Ostwald7s has used two equations, viz.
+
Freundlichs- 28 found that for not too high temperatures and pressures the log x, log p, curves at different temperatures spread out in a fan shaped diagram. In these circumstances log xe = log xo - (< -
5 log p)B
Ao,41 successful1y used the Ramsay-Young vapour pressure law To Ti
To’ - R(T, - TO’) Ti‘ where TI and TI’are the absolute temperatures corresponding to two selected pressures on any one isostere, and To and To’are the temperatures at which any saturated vapour exerts the same pressures. It ia shown that this relationship may be derived from Bertrand’s empirical vapour pressure formula p
=
kl(
q)ki ,
whichitself expresses the isostere results. It is of interest
t o note that the Ramsay-Young equation may be derived from the Clapeyron equation by assuming that the molecular latent heat I C Independent of temFerature. I t may also be derived from I a n der Waals’ equation.
Theoretzcal By equating t u o eypressions for the change of energ) consequent to the adsorption of I mole of gas obtained from the kinetic theory and the o~tlinarr theory of attiaction, Killiams”8, 11‘ deduced the equation of the isostere to lie log?= B + A T
where B and -1are independent, of temperature. A1lthoughthis equation is an approximation it is in good agreement with experimental data. In the cleciuction it is assumed, first, that the kinetic energy of a moIecule is unchanged by adsorption, second, that n(6 - a) is independent of temperature, where (I is the area of the adsorbent. b is the thickness of the adsorption layer, and u is t.he diameter of an adsorbed molecule. From a long theoretical discussion a method is derived of placing the internal cohesion of gases. and of evaluating the range of molecular action and the thickness of the adsorbed layer. The values thus obtained are in good agreement with those obtained 11)- other met’hods. The isostere equation obtained from H e n r ~ - ’theory s ~ ~ (q. v.) is identical with that of Williams. Henry has also indicated a method of deriving this equation from Perrin’s radiation hypothesis.
V. Rate of Adsorption Whilst values from a few seconds to as many years have lieen recorded as representing the time required for the attainment of equilibriuni in an aclsorption system, there is a large amount of evidence to show that equilibrium is reached in a finite time. The curve showing the relation between 5 and t should therefore be of the form shown in Fig. I , dx dt decreasing with increasing t , and becoming zero at a finite value of t. Empirical. The equations of Mills a n d . T h o m ~ o n ’and ~ Mills and Takaminesg are probably the first to be proposed for the representation of the rate of adsorp-
269
ADSORPTION EQUATIONS
tion. These investigators found thpt the rate of adsorption of dyes from solution by textile fibres could be expressed in one of the forms:
y = .4a'
* B, y = A d J or y = A at * B B'
where y is the per cent dye remaining in the bath after time t , .A initial amount of dye (taken as 1 0 0 ) ~and (Y and B are constants. For the adsorption of gases Bergter* has used the equation
x=-
kit a+t
* B is the
+-bkt+ tt +
+
as well as esponential equations of the type x = ki kZeh: k4ek:. Similar equations have also been used by other authors, including R a k o ~ s k i , ~ ~ and Bateman and T0wn.j The best-known rate equation is probably that first used by Lagergred' dx - = k(s, - x), where sm is Lhefinal value of x. This equation does not condt form Xvith the general requirements of the rate equation, and GustaveP has pointed out that it is inapplicable to high concentrations, since it assumes the constancy during adsorption of the concentration. Pickles7s has used both for adsorption from a gas and from a liquid the equation a - s1 s 2 - s, -k=' log--a - x p - .4343 tz - t l where a is the original amount of adsorbate. From their data, which are probably the most accurate figures known for the rate of adsorption of a gas, Burt and Banghami2 have successfully used k?. Beyond the range in which this equation the equation log x = kl log t is applicable Bangham and Severf have shown that the equation
+
log
s,
- s
=
kith'
may be used. Za~harias,"~ using an equation due to Korner found that the rate of adsorption from solution was given by
--k(c,
- s) (c - s) dt. FreundlichzB,27 and his co-workers have represented the rate function by the "equations of negative auto-catalysis," either
-d s_ - zkt(1 + bx)
dS
+
( I - s)' or - - Zkt(1 bx) ( I - X) dt dt They have further found that the variation of k with temperature is given by .lrrhenius' equation k log k = - -2 kl, T
+
270
ENOCH SWAN AND ALEXANDER ROBERT URQUHART
and its variation with concentration by the equation k = kixk' Dietl14 has used these equations in preference to those of Zacharias and Lagergren and in preference to the equation dx = k(a - x) dt X and he has shown that one of them can be derived from the Nernst-Brunner diffusion equation dx - _ - 0 D(a - x) dt 6 ' if 6, the diffdsion layer, changes with time in the senss found by Moderstein and Fink for the kinetics of the contact process for the manufacture of sulphuric acid. Recently Davis and Eyre's have claimed that for colloids the rate curve is discontinuous, each portion being represented by an equation of the type x kit k2t2 k3ts The rate of evaporation from the very ill-defined system soil-water has been investigated by several workers, and various empirical formulae have been proposed. A thorough analysis of this work has been made by Fisher,*O who finds that the rate curve is discontinuous. I t will be note? that a number of these equations do not conform with the general requirements of the rate equation, and that for some of them x cc as t -+ m , which renders them applicable to a limited region only.
-
+
+
---f
Theoretical. By considering an adsorption system as a solid solution, and assuming the applicability of Fick's Law, McBainG derived an equation to represent the rate of adsorption. From a consideration of the properties of this equation, however, McBain concluded that any theory of arrested diffusion was untenable. Marc," assuming that the rate of adsorption in solution depends on the attractive force, the osmotic pressure of the solute, and the kinetic energy of the adsorbed molecules, writes dx/dt = k3x-k' - kS kz (a - x) where the first term represents the effect due to the attractive force, the second term the kinetic energy of the adsorbed molecules and the third term the osmotic pressure of the solute. For values of z not too near ZCC, an approximate solution is given by
+
k i + ~
+
(2x2 klxk') = k3 t This equation has also been used by Arendt.' By assuming that the amount of adsorption is proportional to ( I ) the free surface, and ( 2 ) the total amount,of adsorbate in the gaseous or liquid phase, GustaverSSobtained the equation k = I (log-. a - x t(a - x ) m ,x - x a
")
ADSORPTION EQUATIONS
271
for adsorption from a liquid, a similar equation obtaining for the adsorption of a gas. These equations can also be deduced from Langmuir's theory. Henry,*8 from the theory already discussed, has deduced the general equation for the rate of adsorption of one component (7)from a mixture of n gaseous components to be
dc Iliin42has obtained an equation - = kle-k'' which agrees well with experidt mental results. This equation is of the same form as the empirical equations of Rakovski and Bergter.
VI. Heat of Adsorption In this section, following Williams,llo h is used to denote the heat of adsorption, defined as the ratio of the heat evolved to the gas adsorbed when only a very small quantity of gas is adsorbed; the total heat of adsorption will be, in general, H =
L2 h,
. dx.
This integral can usually be obtained
only by graphical or mechanical integration. Empirical. Whilst there are many early observations of heat effects accompanying adsorption, FreundlichZ5was the first to formulate equations for the heat of adsorption, distinguishing two kinds: Integral heat of adsorption, corresponding to heat of solution, evolved (I) when a gas is brought into contact with just sufficient adsorbent to take it up. Differential heat of adsorption, evolved when one equilibrium con(2) dition is transferred to another. According to Freundlich this transformation can take place either isosterically (z constant) or isopneumatically ( p constant). Combining the Clapeyron equation with his empirical equation to the isostere (q. v.) he obtained the equation nRT2 - h, = ( 5 - c: log PI 2 2 4 1 0 loglue for the differential heat of adsorption under isosteric conditions. Williams11o has criticised the above classification as being confusing and erroneous, and has pointed out that the Clapeyron equation really refers to (h)p,T. h r h e n i q 2 on the other hand, believes that the Clapeyron equation can be applied in the region where h does not vary greatly with p, since he supposes the adsorbate to behave as a liquid under its own saturated vapour. Combining the Clapeyron equation with Bertrand's vapour pressure equa-
IOOXT
tion, Hornfraydo,d l obtained the equation (h)p,T= -where r is the T - x amount of adsorbate expressed in per cent of adsorbent. Lamb and Coolidgej3 from a study of the heat of adsorption of eleven organic oapours find that the heat of adsorption is given by h = klx'" where h
272
ENOCH SWAN AND ALEXANDER ROBERT URQEHART
is the heat of adsorption per normal cubic centimetre of vapour and z is the number of cubic centimetres adsorbed. Since k? is approximately I , the heat of adsorption decreases but slightly with increase in z. It is also found that the difference between total heat of adsorption and heat of evaporation is nearly constant for a11 liquids, and that these differences are closeIy proportional to the heats of compression under high pressures. Katz and Holleman4?have recently shown that for water and carbon from x = 5 to 8 1 7 ~(saturation = 93%) the variation in h is of the same order as the variation in free energy obtained from the equation
A
1252
__ loglo p/P.
18
Theoretical. To Williams1D*. is due the exact definition of h under different conditions, as follows: (I) At equilibrium (h)T; where adsorption proceeds with the vapour phase constantly in equilibrium with the adsorbed phase. (2) At constant pressure, (h)p,T;where the gas is adsorbed a t constant but not necessarily equilibrium pressure. ( 3 ) A t constant volume (h)v,T; where the total volume of the system is constant and the gas is adsorbed with a fall in pressure. In a later paper"' expressions are developed thermodynamically for (h)T, (h)p.T, and (h)",T. From these equations it follows as a first approximation that (h)P,T= (h)",T 4-RT. The application of the equationfor (h)p,Tto Titoff's measurements leads to the conclusion that the surface area alters during adsorption. Where a vapour is adsorbed as a liquid it is shown that (h) P , is ~ given approximately by the relation llol
Expressions are also developed for the heat of immersion of a powder in a liquid when (I) The powder is free from the adsorbate before immersion The powder is partially saturated before immersion (2) Corresponding to these two expressions two others are deveIoped for similar conditions, but allowing for a change in the surface area of the adsorbent. By finding the amount of work entailed in the formation of a single layer of gas on an adsorbent and integrating by reduction E u ~ k e n ' ~has ~ obtained a simple equation which when coupled with the Second Law yields the simple result that the heat of adsorption per mole of gas is equal to the adsorption potential e, defined as the amount of work involved in bringing a mole of gas from an infinite distance to the surface. Thus H = E = Rkp, where kg is obtained from Eucken's adsorption equation (9. v.) for the total amount adsorbed. In a further discussion in which it is assumed that the adsorbing force is molecular attraction it is deduced that kz should vary as the square root of
ADSORPTION EQUATIOXS
273
The boiling point of the gas van der Waals' a. (3) The heat of evaporation of the adsorbed gas. These expectations were realised. By considering the orientation of the molecules resultant on dipolarity Lorentz and Landeljghave also found the adsorption potential to be equal to the heat of adsorption. They have also shown from the theorem of corresponding states that the heat of adsorption per mole H = ROT,, where 8 is (I)
(2)
k
a universal constant = - , k being the constant of their adsorption equation TC (9. v.). k may also be obtained from a knowledge of the dipolar moment of the molecule. This application of the theorem of corresponding states only applies in the p, 8 region where Henry's Law holds, and where p>p., and
e < e,. Polanyi,82 making the assuniptions previously considered, found that - R T log p/P, but differing from Eucken, set the heat of adsorption in a non condensible region H = e, ux, where ux is the molar heat of compression, equal to as, where a is van der Waals' constant and 6 is the density of the gas. For a gas in the condensible region the molar heat of evaporation (A) must also be added, so that H = e, a, X. On the assumption that the adsorbed vapour is compressed by the molecular forces HarkinsSGdeduced th rmodynamically that for a saturated vapour -H = E, - E, X, where - H is the heat of adsorption of sufficient vapour to form a liquid layer covering the solid surface, X is the heat of evaporation of unit volume, E. is the total surface energy of the solid, and E, is the total energy of the interface formed by the adsorption. This equation, which really yields the heat of immersion, is deduced on the assumption that the surface area of the liquid is negligible in comparison with the area of the interface. From the electrical theory of adsorption previously mentioned Iliin43has obtained for the heat of adsorption the equation ex =
+
+ +
+
where E, is the tension of the electrical field in vacuo, and e is the dielectric constant of the adsorbed gas. This equation is in good agreement with the experimental results of Homfray and Titoff. The same equation has been derived by Tarasoff,lo2who has also shown that Hv2 is a constant for a given adsorbent, where Y is the ionisation potential of the adsorbed gas.
Note on the Formulation of Adsorption Laws For a theoretical investigation of the laws of adsorption two general methods are available; first, a development from the fundamental laws of thermodynamics, and second, a discussion of the kinetics of the adsorption process. Thermodynamic treatment has the great advantage that it does not
274
ENOCHSWANASDALEXANDERROBERTURQUHART
require a knowledge of the intimate mechanism of the process, and in particular it involves no assumptions with regard to molecular attraction. On the other hand no relation for t,he rate of adsorption can be obtained from thermodynamics alone. Kinetic treatment permits of the deduction of a rate of adsorption equation, but when an attempt is made to allow for the interaction of molecules the mathemat'ics becomes so involved as to be impossible of solution. The condibions obtaining in a solution are so complicated that it will probably be of little avail attempting to postulate any theory of adsorption from solution until our knowledge of the laws underlying adsorption from the gaseous phase is much more complete. It would seem that an adequate theory of gaseous adsorption should distinguish between a gas above and below its critical temperahre. For the former there is no possibility of condensation to liquid, so t'hat the adsorpt,ion layer would consist of gae, probably compressed, and would vary in tbickness according to the pressure. For such conditions t'hermodynamic treatment has been succCssfully used, as witness the remarkable agreement between Briggs' experimental data and Williams' equation for the kotherm. For a gas below the critical t,emperature a layer of gas would be expected a t low pressures, but as the pressure is increased condensation might occur, with the formation of a liquid film. If the adsorbent were porous a simultaneous condensation would take place in the pores,60obeying, probably, quite other laws. Anderson, Zsigmondy, Fisher, Trouton, and Gustaver believe that this condensation takes place according to Kelvin's equation z uDv r= D, Plog P/p where r is the radius of the liquid meniscus, D, is the vapour density at pressure P, and D, is the liquid density. Wilson115 considers that this equation is only applicable to pores of radius great,er than 5 x 10-8 centimetres. I t may well be, however, that the breakdown for very small pores is due not so much t,o an imperfection in the equation as to the insertion of the wrong value of u in it, since it is not permissible to assume that the surface tension of such very thin films is identical with that of the liquid in bulk. Most of the evidence with regard to an adsorbed liquid film (as distinct from a column of liquid in a pore) tends to show that it is very thin. Thus Katz4Si 46 has shown from the equation F = R T log p/P, where F is the molecular attraction, that the force of attraction diminishes rapidly with increase of thickness, whilst Hatscheks8 by means of the Einstein-Hatschek viscosity equation has shown that the thickness of the adsorbed film on the spheres of a suspension in sodium chloride solutions is of the order 8 X IO-* centimetres, which is about twice the diameter of a water molecule. On the other hand, reference must be made to some recent work of Hardy,s4 which shows that a surface may exert a specific effect through a laysr of liquid probably some hundreds of molecules thick. For a reconciliation of these divergent views we must look to the future.
ADSORPTION EQUATIONS
275
BIBLIOGRAPHY Arendt: Kolloidchem. Beihefte, 7, 212 (1915). Arrhenius: Medd. K. Vetenskapsakad. Nobel-Inst., 2, No. 7 (1910). 3. Bancroft: J. Franklin Inst., 185,218 (1918). Bangham and Sever: Phil. Mag., (6) 49,938 (1925). Bateman and Town: J. Ind. Eng. Chem., 15,371 (1923). 6. Berenyi: Z. physik. Chem., 94,628 (1920). Berenyi: Z. angew. Chem., 35,237 (1922). &. Bergter: Ann. Physik, (4) 37,472 (1912). 9. Berl and Schwebel: Z. angew. Chem., 36,541, 552 (1923). IO. Boedecker: quoted by van Bemmelen, “Die Absorption.” p. I I O (1910) Briggs: Proc. Roy. SOC.Edin., 41, (2), 119 (1920-1). 11. 12. Burt and Bangham: Proc. Roy. Soc., 105A,481 (1924). 13. Davis and Eyre: Proc. Roy. SOC.,104A, 5 1 2 (1923). 14. Dietl: Kolloidchem. Beihefte, 6, 127 (1914). 15. Duhem: J. Phys. Chem., 4,65 (1900). 16. Edser: “Fourth Report on Colloid Chemistry,” British Association for the Ad van.cement of Science, p. 40 (1922). 17. Eucken: Ber. deutsch. phyeik. Ges., 12,345 (1914). 18. Eucken: Z. Elektrochem., 28, 6 (1922). 19. Firth: Z. phvsik. Chem., 86,294 (1914). 20. Fisher: Pro;. Roy. SOC.,103A, 139, 664 (1923); 105A,571 (1924). Frenkel: Z. Physik, 26, 117 (1924). 21. 22. Freundlich: Z. physik. Chem., 57,385 (1906). 23. Freundlich and Losev: Z. physik. Chem., 59,284 (1907). 24. Freundlich: KolloidTZ.,3 2 1 2 (1908). 2 5 . Freundlich: Z. physik. Chem., 61! 249 (1908). 26. Freundlich and Schucht: Z. physik. Chem., 85,641 (1913). 27. Freundlich and Hase: Z. physik Chem., 89,417 (1915). 28. Freundlich: “Kapillarchemie,” (1922). 29. Gibbs: Trans. Conn. Acad., 3, 439 (1874). 30. Gorbatschew: Z. physik. Chem., 117, 129 (1925). 31. Gurwitsch: Z. physik. Chem., 87, 323 (1914). 32. Gustafson: Z. physik. Chem 91,385 (1916). 33. Gustaver: Kolloidchem. Beigefte, 15, 185 (1922). 34. Hardy: “Mond Memorial Lecture,” Manchester, 192j, and private ccImmuni cati on. (Work not yet published). 35. Harkins and King: J. Am. Chem. SOC.,41,971 (1919). 36. Harkins and Ewing: J. .-lm. Chem. SOC.,43, 1787 (1921). 37. Harlow and Willows: Trans. Faraday Soc., 11, 52 (1915). 38. Hatschek: Kolloid-Z., 11, 280 (1913). 39. Henry: Phil. Mag., (6) 44,689 (1922). 40. Homfray: Proc. Rov. SOC.,84,99 (1911). 41. Homfray: Z. physig. Chem., 74, 129 (1910). 42. Iliin: Z. physik. Chem., 107, 145 (1923). 43. Iliin: Phil. Mag., (6) 48, 193 (1924). 44. Iredale: Phil. Mag., (6) 48, 1088 (1923). 45. Katz: Chem. Ztg., 36,607 (1913). 46. Kata: Proc. K. Akad. Wet. Amsterdam, 15, 445 (1913). 47. Kats and Holleman: Proc. K . Akad. Wet. Amsterdam, 26, 548 (1923). 48. Kayser: \Vied. Ann., 12,526 (1881). 49. Kolosovski: J. Russ. Phys. Chem. SOC.,47,717 (191j). 50. Kolthoff: Kolloid-Z., 30,35 (1922). j I . Iiroecker: Dissertation, Berlin (1892) ; quoted by Freundlich. 5 2 . Lagergren: Bih. K. Svenska Vetenskapsakad. Handl., 24, (z), 4 (1898). 53. Lamb and Coolidge: J. Am. Chem. SOC.,42, 1146 (1920). 54. Langmuir: Phys. Rev., ( 2 ) 6,79 (1915). 55. Langmuir: J. Am. Chem. SOC.,38, 1 2 2 1 (1916). 56. Langmuir: J. Am. Chem. SOC.,39, 1848 (1917). 57. Langmuir: J. Am. Chem. SOC.,40 1361 (1918). 58. Lewis: Z. physik. Chem., 73, 129 t191o). 59. Lorentz and Lande: Z. anorg. Chem., 125,47 (1922). 60. Lowry and Hulett: J. Am. Chern. SOC.,42, 1393 (1920). 61. Marc: 2. physik. Chem., 76,j 8 (1911). Marc: 2 . physik. Chem., 81,641 (19x3). Marc: Z. Elektrochem., 20, 515 ( 1 9 1 4 ) . McBain: J. Chem. Soc., 91, 1685 (1907). McBain: Z. physik. Chem., 68,471 (1909). I. 2.
::
276
ENOCH SWAN AND ALEXANDER ROBERT URQUHART
66. McBain: J. SOC.Dvers and Colourists, 39,233 (1923). 67. McGavack and Patrick: J. Am. Chem. Soc., 42,946 (1920). 68. Mecklenhurg: Naturwiss. Wochensch. 15,4 ’9 ‘1916’. 69. Mills and Takamine: 3. Chem. Soc., 43,142 (1883). 70. Mills and Thomson: J. SOC.,Chem. 35,26 (1879). 7 1 . Milner: Phil. Mag., (6) 13,96 (1907). 72. Muller: J. Soc. Chem., Ind. 1, 356, (1888). 73. Ostwald, Wi., “Outlines of General Chemistry,” 3rd English edition, p. 499 (1912). 74. Ostwald, SVo., “Grundriss der Kolloidchemie,” 1st edition, p. 434 (1909). 75. Ostwald and Izaguirre: Iiolloid-Z., 30,279 (1922). 76. Partington, ”Text-book of Thermodynamics,” p. 473 (1913). 77. Pavlov: Kolloid-Z., 35, 3, 87, 89, 156, 159, 221, 225 (1924). 78. Pickles: Chem. News, 121,2 5 (1920). 79. Polanyi: Biochem. 2. 66,258 (1914). 80. Polanyi: Z. physik. &em., 88,622 (1914). 81. Polanyi: Ber. deutsch. physik. Ges., 16, 1012 (1914). 82. Polanyi: Ber. deutsch. physik. Ges., 18,5 5 (1916). Z. Elektrochem., 26,370 (1920). 2. Physik, 2, 111 (1920). Z. Elektrochem., 28, I I O (1922). Z. physik. Chem., 114,387 (1925). 87. Porter: Trans. Faraday Soc., 11, 51 (1915). 88. Rakovski: J. Russ. Phvs. Chem. Soc., 43, 186 (1911). 8Q. Rakovski: J. Russ. Phks. Chem. Soc., 43, 1762 (1911). 90. Rakovski: J. Russ. Phys. Chem. Soc., 44,836 (1912). 91. Rakovski: J. Russ. Phys. Chem. Soc., 45, 13, I837 (1913). 92. Reichinstein: Z. physik. Chem., 107, 119 (1923). 93. Robertson: Kolloid., Z., 3, 49 (1908). 94. De Saussure: Gilb. Ann., 47, 113 (1814). 95. Schmidt: Z. physik. Chem., 74,689 (1910). 96. Schmidt: Z. physik. Chem., 77,641 (1911). 97. Schmidt: Z. physik. Chem., 78,667 (1912). 98. Schmidt: Z. phvsik. Chem., 83,674 (1913). ,99. Schmidt and Hinteler: 2. physik. Chem., 91, 103 (1916). 00. Seeliger: P h p k . Z., 22, 563 (1921). 101. St.adnikoff: olloid-Z., 31, 19 (1922). 102. Tarasoff: Physik. Z., 25,369 (1924). 103. Thomson, “Applications of Dvnamics to Physics and Chemistry,” p. 191 (1888). 104. Trouton: Proc. Roy. Soc., 77A,292 (1906). 10s. Volmer: Z. physik. Chem., 115. 253 (1925). 106. Warburg: Wied. Ann., 41, I (1890). 107. Williams: Medd. K. S’etenskapsaksd. Sobel-Inst., 2,2 7 (1913). 108. Williams: Trans. Faraday Soc., 10, 155 (1914). 109. Williams: Trans. Faraday Soc., 10, 167 (1914). Williams: Proc. Roy. Soc. Edin!, 37, 161 (19171. I IO. 111. Williams: Proc. Roy. Soc. Edin., 38,24 (1918). Williams: Proc. Roy. Soc. Edin., 39,48 (1919). 112. 113. Williams: Proc. Roy. Soc., 96A,287 (1920). I 14. Williams: Proc. Roy. Soc., 96A,298 (1920). 115. Wilson: Ph s R e v , ( 2 ) 16,8 (1920). I 16. Zacharias: i.’physik. Chem., 39, 468 (1902). 1 1 7 . Zacharias: quoted by Dietl.
SeFt. 4, 19%6.
