HIGH-PRESSURE REGIONOF CERTAINSORPTION ISOTHERMS
1611
Analysis of the High-pressure Region of Certain Sorption Isotherms
by Akira Takizawa Faculty of Teztile Technology, Tokyo Imtitute of Technology, Tokyo, Japan
(Received M a y 9, 1966)
In order to describe the high vapor pressure region of the sorption isotherm of polar polymer-polar small molecule systems, theoretical investigations of the statistical thermodynamics of solution accompanied by adsorption were made. The result was applied to the experimental isotherms of protein-water and cellulose nitrate-acetone systems. The theory of solution accompanied by adsorption seems to be adequate for the polymersolvent systems, because it not only gives a constant interaction parameter, XI, for the polymer-solvent systems over the whole relative pressure range, but it also yields an interaction parameter which agrees with that obtained by another method.
I. Introduction In order to develop a new concept that will be able to describe the high-pressure region of the sorption isotherm of a polar polymer-polar small molecule system, several possibilities were first examined for protein-water systems. Brunauer, Emmett, and Teller’s theory’ can describe the experimental isotherm of an insulin-water system at 25’ with the aid of a restricted number (five to eight) of adsorbed layers. It is, however, only one among several possible descriptions and lacks concrete physical meaning. The generalized adsorption equation that is obtained from a fundamental statistical treatment is expressed as3r4
A / A m = z+’/+
+
+
+
+
+
where = 1 CIZ C1C2z2 C1C2C& . . . and Ct is the ratio of partition function for the ith layer to that for the liquid state. By setting up the experimental equation that might describe the equilibrium sorption isotherm of an insulin-water system over the whole pressure range and comparing it with the above theoretical equation Cc values are determined. The result shows that some of the Cr values are negative. Because partition functions cannot be negative, the result means that the “adsorption” theory cannot explain the isotherm over the whole range of relative vapor pressure. Rowen and Simha’s treatment5 based on the FloryHuggins’ “solution” theory was examined for insulin-
water (25’) and egg albumin-water (26.5’) systems.6 The interaction parameter, xl,increased with the relative pressure or volume fraction of water and the initial stage of the sorption showed a large affinity (low XI) between protein and water. This tendency means that a high interaction energy by adsorption is evolved at the initial stage of mixing and this exothermic tendency gradually diminishes as the adsorption mechanism decreases. Because of the strong interactions between polar sites of the polymer and small polar molecules, small molecules adsorbed at the initial stage of sorption must be far from a randomly mixed state even at high vapor pressures. Therefore, it is most desirable to develop the statistical thermodynamics of solution accompanied by adsorption for the description of these systems. The results of the theoretical investigation will be examined by the experimental isotherms of proteinwater and cellulose nitrate-acetone systems.
11. Theory In the following, we shall assume that small molecules are sorbed to polymer molecules in two different ways: one is the Langmuir-type adsorption with strong inter(1) s. Brunauer, “Adsorption of Gases and Vapors,” Princeton University Press, Princeton, N. J., 1943,p 154. (2) W. Heller and A. Takizawa, unpublished work. (3) T. L. Hill, J. Chem. Phys., 14,263, 272 (1946). (4) M.Dole, ibid., 16, 25 (1948). (5) J. W. Rowen and R. Simha, J. Phys. Colloid Chem., 53, 921 (1949). (6) W.Heller and A. Fogiel, J. Phys. Chem., 70,2039 (1966).
Volume 71I Number 6 May 1967
AKIRATAKIZAWA
1612
action and the other is the Flory-Huggins type of random mixing of small molecules with polymer segments. As has been customarily done with the statistical mechanical treatment of a d ~ o r p t i o n , ~the - ~ ,sorption ~ isotherm in the present paper is derived by setting up the partition function of the two-component system, assuming that adsorption is accompanied by solution. Furthermore, since the present case is essentially the two-component theory, the result was examined by applying the Gibbs-Duhem equation to the derived chemical potentials of vapor and polymer. We suppose that there are PB polar sites per P polymer molecule, that X of these sites are occupied by adsorbed molecules, and that (A - X) small molecules are mixed randomly with polymer segments PA[. P is the number of polymer molecules and M is the number of segments per molecule. ( 1 ) Thtlorg I . The Case Where B qn/(qlqm)
The chemical potential, PA, of the small molecules in the system can be calculated by the same procedure as eq 5 . Equating PA and pa of eq 6 ln (qle - Cwii/2k-T) In vl’ v2’ -
-
+ In
212
- (M - 1 ) s +
+
+
bz’ - (X/(A
x1MvI2 - In K’
(Czo/2kT)[v2’ - ( X / ( A
(10)
+ PM))I/M -
+ PM))I2 = (a/kT) + In p (14)
where
For the pure liquid, vl’, v2’, and A become 1, 0, and respectively. Hence and X in v1 and v2 is to be obtained by eq 4. It can be seen that eq 5 and 10 satisfy the GibbsDuhem equation AdpA
+ Pdpp = 0
(11)
by using eq 8 for the first term of eq 10 and by expressing ,UA and ,UP as functions of v2. (2) Theory I I . The Case Where B or X Is of Comparable Order with M or PM ( A - X ) , Respectively. In this case we cannot neglect the effect of combined X molecules on the dimension of the polymer molecule. In the following, we assume that X molecules adsorbed to strong polar sites of polymer molecules participate as parts of polymer segments. Thus the volume fractions are defined as VI’, volume fraction of small molecule = ( A - X ) / ( A P M ) , and vz’, volume fraction of polymer segment (including adsorbed small mole cules) = ( X P M ) / ( A P M ) . The partition function for the X molecules is the same as eq 1. The partition function of the polymer solution in this case is
+
+
+
QL
=
+
(Ea/kT)-(CwiilZkT)
(qBe
(qpeP-1
Il [ A i-0
1X h e - Cw:wli/ZkT> A -
>
Cwzzl2kT P M
X
x
x
+ PM - ( ( X / P ) + M ) i ] ( X ” P ) +XM PI
A
e(Cw/2kT)IAPM/(A+PM11
(12)“
The X value which corresponds to the maximum term in the complete partition function of the system eq 3, where Q. and QL are given by eq 1 and 12, respectively, is In [ ( P B - X ) / X ]
+ In vl’ + In K
=0
(13)
In
2
= In vl’
a,
+ v2’ X In VI‘
+
A+PM
VZ’
- (VZO’/M)+ ~ 1 ~ 2 0 (15) ‘ ~
+
where v20’ = v2’ - ( X / ( A P M ) ) is the true volume fraction of polymer (excluding the adsorbed small molecules) of the entire system. From eq 13 and 15
In the region of relatively low 2, v2’ and no’of the right-hand side of eq 16 approach unity. Then, neglecting the effect of the vzO’/M term and putting K/elfx’ K‘ lim X = P d O
PBK’x 1 K’x
+
Equation 17 has the same form as the Langmuir equation. For the whole pressure range, we have to calculate an x us. A relationship by eliminating X using eq 15 and 16. (9) T. L. Hill, “An Introduction to Statistical Thermodynamics,” Addison-Wesley Publishing Co., Inc., Reading, Mass., 1960, pp 372, 405. (IO) T. L. Hill, see ref 9, p 478. (11) In the derivation of eq 12, average interaction potential energies are assumed as
wla’ =
WlnPM PM
+ WllX
+X
where subscript 2’ means polymer segment including adsorbed small molecules.
Volume 71, Number 8 M a y 1967
AKIRATAKIZAWA
1614
The chemical potential of the polymer in this case is calculated as pp/kT =
-a
= Bln
In QsQL/dP
PB - X PB
Mv2’- ~
2
- ~ 2 0 ’ )-~ In K’
(18)
111. Application of the Present Theory to Experimental Isotherms (1) Insulin-Water System. The experimental isotherm at 2502 ( A moles sorbed/100 g of insulin vs. relative vapor pressure x relation) is shown in Figure 1 by open circles. (Experimental details were given in a former papere6) For the determination of the Langmuir-type isotherm of eq S, we assume X (adsorbed) is nearly equal to A in the low-pressure region because of the large adsorption energy. Applying the least-square method to the six points up to x = 0.05136 of the x/X vs. z relation
+ 0.48352
X values calculated by this equation correspond to the Langmuir-type adsorption. Subtraction of the X value from A to each x gives the water molecules that are randomly mixed with polymer segments according to eq 7. Transforming the unit of ( A - X)from moles per 100 g of insulin to centimeters per g of insulin and using the specific volume of insulin, 0.73513 cc/g,13 one can calculate the volume fraction of freely mixed water molecules, VI. Then, by using eq 7‘ and neglecting the term 1/M, values of XI at various x value are obtained. The result of calculation is shown in Figure 1 by closed circles. XI values obtained by this theory show constancy over the wide range of x. The average value of XI is 1.063. Using this value, the calculated adsorption isotherm is also included in Figure 1. The theoretical curve explains the experimental data quite satisfactorily. (2) Eg17AZbumin-Water System. The experimental isotherm a t 26.5’ is shown in Figure 2 by open circles. By the same procedure as in the case of insulin, XI values at various x values are calculated and are shown The Journal of Physical Chemistry
:
1.4
where K‘ is equal to that of eq 10 and X in VI’, VZ’, and vzo’ is to be obtained by eq 13. In this case, too, it may be seen that the PA (lefthand side of eq 14) and the p~ of eq 18 satisfy the GibbsDuhem equation (eq 11) by using eq 16 for the first term of eq 18 and by expressing both PA and pp as functions of v2‘.12
= 0.1579
1.2
2s -
’
xlM(1
x/X
a w 2 1.0 c‘& GI-
+a
+ In v2’ - (M - 1) + - In (02’/v2~’) + ~
0
... -
..
1
0 -
I
I
02
I
I
0.4
X
I
I
I
I
0.8
0.6
-
a :EXPERIMENTAL VALUES
-
x
:ADSORPTION
A : SOLUTION ACCOMPANIED WITH ADSORPTION 0 0
//
0
a2
0.4
0.6
a0
RELATIVE VAPOR PRESSURE
1.0
x
Figure 1. Interaction parameter (O), xl,and, theoretical isotherm (-), calculated by theory I for insulin-water system (25’); 0,experimental isotherm.
in Figure 2 by closed circles. In this case, too, the result shows almost constant x1 values (average 1.087) over the wide range of x. (3) Cellulose Nitrate-Acetone System. Using Campbell and Johnson’s i ~ o t h e r m of ’ ~ nitrocellulose of nitrogen content 11.2% plus acetone system (reproduced in Figure 3), Rowen and Simha’s treatment5 and the analysis by the present theory I were made. Though the present theory I gives more uniform XI values (-0.093 a t z = 0.2 to -0.016 at x = 0.9) than those (-1.41 at x = 0.2 to -0.18 at x = 0.9) of Rowen and Simha’s treatment, the difference between XI values of the osmotic pressure determination and the present (12) The terms relating to VZO‘ are expressed by using the relation dvzo’ = (dvzo’/dvz’)dvz’. (13) J. D. Edsall, “The Proteins,” Academic Press Inc., New York, N. Y., 1953, p 562. (14) H. Campbell and P. Johnson, J. Polymer Sci., 4 , 247 (1947).
HIGH-PRESSURE REGIONOF CERTAINSORPTION ISOTHERMS
1615
B-X
X L
-
1.4
g ~~
,
,
02
,
,
,
,
0.6
0.4
X
,
,
,
+ xl)v1’ = 1
(13’)
Using eq 13’ and 15, we can determine values of both x1 and X a t each x by a trial-and-error method as follows. Putting an arbitrary value of XI into eq 13’, the X value at that x is determined noting that VI’ = (A - X)/(A PM). (A and PM are known from the experiment.) Then, substituting this X value into eq 15 and neglecting the term vZ0’/M,x1 corresponding to the former arbitrary x1 is obtained. If both XI values agree with each other, this consistent value is the correct XI value a t that 5. From this XI, K = K’ exp(1 XI) can also be calculated. We can examine the applicability of theory I1 to a system by checking the constancy of X I and K over the whole range of relative vapor pressure. For the present nitrocellulose-acetone system, according to the Langmuir adsorption isotherm x/X = 0.382 3.58X obtained by applying the least-square method to the x/X values at x = 0.02,0.03, and 0.05, B = 0.279 (3), and K’ = 0.937 (0). Theresultsof cal-
0.8 0
+
12
c
y
K‘ exp(1
1.0
W
+
+
OA
Q
f.0
0.6 X
Figure 2. Interaction parameter (a),x,, and theoretical isotherm (-) calculated by theory I for egg albumin-water system (26.5’); 0,experimental isotherm.
theory I still cannot be neglected. (For nitrocellulose of nitrogen content 11.45% and acetone system a t 22O, XI was given as 0.265 by osmotic prewure determination.) In the case of the protein-water system, X was small compared with P M . In other words, the volume fraction of water molecule, VI, was given by ( A X)/ (A - X PM)with an error of less than 501,. In the nitrocellulose-acetone system, the corresponding volume fraction of acetone is smaller than (A X)/ (A - X P M ) with an error of more than 10%. We have to use as the volume fraction (A - X)/ (A PM) instead of ( A X)/(A - X PM) in this case. This means that theory I1 of the preceding section is a better description of this system. The practical procedure of applying theory 11 is somewhat intricate. At first, applying eq 17 to the experimental A v8. x relation in the low-pressure region and neglecting the minor decrease of vt‘and urn’, we XI). can obtain the values of B and K’ = K/exp(l Then eq 13is rewritten as
-
+
-
+
+
-
+
+
”
0
Q.2
44
0.6
0.8
1.0
X
Figure 3. Interaction parameter, xI,and constant K calculated by theory I1 vs. relative pressure relations and separation of experimental isotherm into “adsorption” and “polymer solution” by the theory (cellulose nitrakacetone system, 20’9. (15) H. A. Stuart, “Die Physik der Hochpolymeren, Zweiter Band: Daa Makromolekul in Lasungen,” Springer-Verlag, Berlin, 1953, p 152.
Volume 71, Number 6 May 1967
E. P. GUYMON AND J. T. SPENCE
1616
culations of x1 and K at various z values are shown in Figure 3. From Figure 3, it is seen that XI and K show almost constant values over the whole range of pressure and the average value of XI (0.290) is almost equal to that (0.265) obtained from osmotic pressure measurement, in spite of the several assumptions involved in the theory. Thus, as it is clear that theory I1 is adequate to describe the experimental sorption isotherm of the nitrocellulose-acetone system,
we can divide the isotherm into an “adsorption” part and a “polymer solution” part as is shown in Figure 3.
Acknowledgment. Part of this work was done during the stay of the author in Wayne State University, Detroit, Mich., and the author expresses his gratitude to Professor W. Heller of the university for continuous encouragement and fruitful discussions about the problem.
The Oxidation of Molybdenum(V) by Iodine and Oxygen’
by E. P. Guymon2 and J. T. Spence Chemistry Department, Utah State University, Logan, Utah (Received August 4, 1966)
~
~
~~
The oxidation of MoV2 by 13- and O2 has been studied over the pH range 1.65-7.20 in phosphate buffer. In all cases the reaction is first order in MoVz and zero order in oxidant. From pH 1.65 to 3.92 the rate of the reaction is dependent on H+, while above pH 5.50, the rate is pH independent. The activation energy and the entropy of activation for the reaction have been measured, and three possible mechanisms have been considered. The epr studies have eliminated Mov monomer as an intermediate in the reaction.
As part of a study of reactions of molybdenum(V) as possible models for molybdenum enzymes, the oxidation of MoV2 by NOz- and NO,- has recently been To obtain more information about the mechanisms of molybdenum(V) reactions, this study has been extended to other oxidants and the reaction of Mov2with Is- and 0 2 is reported here. Aqueous solutions of MoV2,unless stabilized by strong acid or certain chelating agent^,^ are easily oxidized by atmospheric 0% and must be handled under anaerobic conditions. No quantitative studies of the oxidation have been reported, however.
Experimental Section Stock solutions of Mov2 were prepared by quantitative reduction of Na2Mo04 by shaking over Hg in 3 M HCI; they were allowed to stand over Hg in 3 M HC11 week before use. The NazMo04 was standardized as previously described.6 Standard solutions of 13The Journal of Physied Chemistry
were prepared by the method of Kolthoff and Sandell.’ K3Mo(CN)s was prepared by oxidation of K4Mo(CN)8 with Ce(1V) in 1 M H2S04. K4Mo(CN)8 was prepared and standardized as described by Audrieth.s All buffers were made from reagent grade chemicals, using HzO that had been passed through a mixed-bed ion exchanger to remove any metal ions. Helium, (1) Journal Paper No. 603, Utah State Agricultural Experiment Station. (2) Abstracted from the Ph.D. thesis of E. P. Guymon, Utah State University, 1966. (3) J. A. Frank and J. T. Spence, J . Phys. Chem., 68, 2131 (1964). (4) E.P.Guymon and J. T. Spence, ibid., 70, 1964 (1966). (5) J. T.Spence and E. R. Peterson, Inorg. Chem., 1, 277 (1962). (6) J. T.Spence and G . Kallos, ibid., 2, 710 (1963). (7) I. M. Kolthoff and E. B. Sandell, “Textbook of Quantitative Inorganic Analysis,” The Macmillan Co., New York, N. Y., 1952, p 592. (8) L.F. Audrieth, Inorg. Sun., 3, 160 (1950).
