Oct., 1963
2233
,hOTES
adsorption.l--3 At some stage in the theoretical treatment it has been found necessary to calculate the average number of adsorbed segments per polymer molecule a t the solid-liquid interface. Previously, this number was approxirnated for large molecular weight species by utilizing a random walk model a t a reflecting and/or absorbing w a ~ l l . This ~ required the use of the unnecessary restriction that the end segment of each polymer chain be located a t the surface. A statistical thermodynamic development is now presented which makes use of the matrix method6-? for calculating the desired quantities. Consider a plane surface a t a solid-liquid interface. The volume above the surface has been divided into a suitable lattiiee for the purpose of calculating the thermodynamic functions of the bulk polymer solution.* The successive layem of the lattice above the surface are now numbered 1, 2 , 3, ..., i, ..., t, ..... For consistency, the segments of an adsorbed molecule are permitted to be only a t these distances from the surface. It is seen that a t-mer (where t is the number of statistical segments in a polymer chain) must have an end segment in any one of the first t-layers in order to be capable of being adsorbed. Thus t is the number of components in the statistical weight vector as well as the order of the square matrix used in the calculation. Let one coiisider tlie end segment of a polymer chain which is to be adsorbed (the ends are considered to be distinguishable). It may be a t any of the first t-levels with an unnormalized probability exp( -E,/kT), where E i is the potential energy, e . g . , a 3-9 potential, a t level i, IC is the Boltmann constant, and T is the absolute temperature. I n this model the distance of the first level has been chosen to correspond to the minimum of the potential energy curve. The matrix elements Mmnare determined in a similar manner, but because of certain physical requirements not all of the elements are nonzero. For the purpose of illustration, t = 3 has been chosen. The statistical weight (row) vector for the first segment is 21:
=
(wa, b, c)
;
1 &.
+
+
Q (a,b, c) = uMzI
(3)
where I is a (column) vector with all of its components unity (used for summing). If one were to write out Q (a,b, c) explicitly, it would be seen that two of the terms do not contain an adsorbed segment and therefore are not to be included in the sample (state) space of adsorbed molecules. They may be eliminated by setting a equal to zero and subtracting the resulting expression from Q (a, b, c). Thus
Q
=
Q (a,b, C) - Q
(0, b,
C)
(4)
which is the desired partition function. The average number of adsorbed segments m and the average number of adsorbed sequences a are in general given by lii = a/& X b&/ba = w/Q X bQ/dw (8 with w set equal to unity. It is emphsized that the method is flexible and may be generalized to allow for steric effects and other miscellaneous interactions. It may be seen that in general the stronger the attractive force, the greater the average number of segments that will be adsorbed. Acknowledgment.-It is acknowledged that part of this work was conceived while the author was on a postdoctorate fellowship a t Brown University with Prof. J. H. Gibbs. The author wishes to thank Dr. William S. Magee for many enlightening discussions on the matrix method.
(1)
where a, b, and c are the Boltzmann factors for levels 1, 2, and 3, respectively, and w is an indicator (tagging parameter) used for counting the number of adsorbed sequences. The matrix M is given by state of lrt ,) a e of segment j segment j i 2 M = 1 b 2 0 3 b
where the symbols have their previous meaning. The zeros appear because of the requirement that segment j a t level i will permit segment j 1 to be a t either i - 1 or i 1 in an analogy with the random walk. The one exception to this, however, is a t the first level where (arbitrarily) successive segments are permitted to adsorb. The partition function for a trimer is seen to be
+1
THE IONIZATION OF WATER AT HIGH PRESSURES BY S. D. HAMANN Division of Physical Chemistry, Australian CommonwGalth Scientific and Industrial Research Organization, Fisherman’s B e n d , Melbourne, Australia Received April 11, 1963
3 0 C
0
(2)
(1) R. Simha, H. L. Frisch, a n d F. R. Eirich, J . P h y s . Chem., 57, 684 (1953). (2) H. L. Frisch a n d R. Simha, J . Chem. Phys., 27, 702 (1957). (3) A. Silberberg, J . P h y s . C’hem., 66, 1872 (1962). (4) S.Chandrasekhar, Rev. M o d . Phys., 15, 1 (1943). ( 5 ) G. F. Kewell and E. W.Montroll, zbzd., 26, 353 (1953). (6) B. H. Zimm a n d J. K. Bragg, J . Chem. Phys., 81, 526 (1959). (7) N. Davidson, “Sts tistical Mechanics,” McGraw-Hill Book Co., Inc., Xew York. N. Y., 1962, Chapter XVI. 18) P. J. Flory, “Principlea of. Polymer Chemistry, ’ Cornell Univ. Press, Ithaca. N. Y 1853 Chapter XII.
A large number of measurements have been made of the influence of pressure on the degree of ionization of weak electrolytes dissolved in water, but, rather surprisingly, there have been no direct measurements of the pressure-dependence of the ionization product K , of water itself a t normal temperatures. The main reason lies in the fact that the conductivity methods commonly used in high pressure work are not applicable to water unless it is extremely pure or unless its ionization is greatly enhanced (as it is under the combined (1) For reviews of the results, see S. D. Hamann, “Physico-Chemical Effects of Pressure,’’ Butterworths, London, 1957; S. D. Hamann, “Chemical Equilibria: Condensed Systems,“ in “High Pressure Physics a n d Chemistry,” ed. by R. 8.Bradley, Academic Press, London, 1963.
KOTES
2231 40
0.001 and 0.02 mole/kg. The measurements were made a t 25' and at pressures bctweeii 1 and 2000 atm. at intervals of 250 atm.
30
T
>
E
20
L
F u
,o u 10
influence of high temperatures and high pressures). 2-4 The problem demands the use of a rather less convenient potentiometric method. Ideally, this should be based on a cell with a hydrogen electrodelS but unfortunately there are many practical difficulties in using hydrogen electrodes at high pressures and there is also the serious theoretical problem of properly allowing for the large change of free energy which occurs when gaseous hydrogen is compressed over water.'j Glass electrodes, although of more dubious thermodynamic standing, suffer from neither of these disadvantages. Di~t&che'-~ has used glass electrodes to measure the effect of pressure on the ionization constants of some weak acids in water and his results agree quite well with those obtained by conductivity methods. In two experimentsQ he obtained evidence that K , is considerably greater a t 1000 atm. than a t 1 atm., but the evidence was indirect in the sense that it depended on prior knowledge of the way in which the ionization constants of ammonia and acetic acid change with pressure. The present note describes some rather more direct measurements of IC, based 011 the use of the cell
'I
Experimental The experimental arrangement was similar to the one used by Distbche.s The HC1 solution and the Agl AgCl electrode, const,ituting the left-hand half of the cell ( I ) , were contained in an open-topped commercial glass electrode mride from B rare emth glass ahich had a negligible nlkaline error at p H 13. The right-hand half of KOH solution and another AgCl the cell comprising the KC1 /Ag electrode WAS held in a surrounding beaker. Pressure was transmitted equally to the two halves by a light hydraulic fluid (dried transformer oil) which surrounded the whole cell and partially filled the glass elect.rode and The Agl AgC1 electrodes were made according to the method of Noyes and Ellis.lo The pot,entials of the two electrodes in 0.1 wi. HC1 differed by only 0.4 mv., and this ciifcrence was found to be unaffected by presssure. The asymmetry potentid of the glass membrane was several millivolts and, although it changed slightly with age, in any particular experiment it reached a nearly steady value after one or two pressure cycles and was then independent of the pressure t,o within 0.5 mv. betmecn 1 and 2000 atm. It was established that the membrane had the correct theorctica.1 response of 59 mv. per pH unit, a t 25", bot.h a t atmospheric: pressure and a t high prcssnrcs. The cell was mounted in :i steel pressure vessel of the kind used by I3uch:~nan and Hamam'' for conductivit,y work. E1ectric:tl leads were taken int,o the pressure vessel through silica-insulated senls12 and the complete system of connections, including the oilinsulilted leads inside the pressure vessel, had a shunt leakage resistance of not less than 1011 ohms. The temperature of the vessel n-as controlled to within f 0 . 0 2 " of 25" and the internal pressure to within f 1 0 atm. of the required value. The e.m.f. of the cell was measured on a Tinsley vernier potentiometer, using a Keithley 610A elcctromcter as a null-point, indicator. The reagents were of analytical grade and the NOH contained no more than 1% of carbonate.
+
U
z
Vol. 67
aqueous
aqut~olls
.2g AgCl1 solution 1 glass 1 solution of ~
of HCl
KC1
1
+ KOHl
I
1
AgCl -2g (1) I
in which the molalities of €TCl and of MC1 n-ere both 0.1 mole/kg. and that of KO13 was varied between
Results The followiiig are values of the e.m.f. of the cell (1) measured a t t,he concentrations mHcl = mKc1 = 0.1 mole/kg. mid rn1to11 = 0.01 mole/kg. The results show Pressure, atm. E.m.f., m v . during cornpreasion during expan.eion
{
631.9 631.0
500
622.1 622.1
1000 612.8 012.1
1500 604.4 603.5
2000
j96'6
that t.here was only a slight hysteresis between the compression and expansion phases of the measurements. Ten other sets of measurements were made 011 cells in which the concentrations of I-IC1 and KCl were kept the same (0.1 mole/kg.) but that of KOH was varied between 0.001 and 0.02 moleikg. These cells had, of course, different absolute e.m.f. values from those listed above but in every case the change of e.m.f. for a particular change of pressure was the same within t,he reproducibility of the measuremcnts. The mean values of all the results are plotted in Fig. 1. Discussion The e.m.f. of the cell (1) is related to the molalities (m)and activity coefficients (7)of the various ionic species by RT RT E = - -In F K, F In r n ~ r n , o ~ '
+
( 2 ) 1: U l'ranck, 2 pk?/?ik Chem. (I'ranhfurt), 8 , 192 (1933) ( 3 ) I1 G David and S. D Hamann ?'inns Parndnv Soc , 65, 72 (Iq5rJ) (4) If David and S D IIatriann, zbzd , 66, 1043 (1960) (-5) €I S IIarned a n d 15- J IIarncr, J Am Clicin Soc , 66, 2194 (10.13) (6) F.R IIainsaorth, If J. R o n l e ) , and 1) 4 lIurInnes, ?hid 46, 1-137
(1924) (7) A. Distdcho, Rev. Sci r n s t r , SO, $74 ( I q X ) ( 8 ) A. l)i?.tdche a n d 31 Uubuisson, H u l l rrrst Oceanour. ( A l o n a ~ o ) 67, , X o 1174 (1960) ( 8 ) A . Dintilnhm, J. lilarlrarham S o < , 109. 1084 (1Q62)1
1
+
(10) A . A. Noles and .J. 11. Ellis, J . Am. Chem. Sor., 39, 2532 ( 1 0 1 i ) . (11) J. Huchanan and S. D. IIarnunn, T r a n s . Furndnii Sor., 49, I125 (1953). (12) A . hlichels ant1 C. ;\Siohala, I'hrl. l'rona. R o y . S n c . (Lonilon), A231, 400
(ioaa),
NOTES
Oct., 1963 where the unprimed quantities refer to the left-hand solutions and the primed quantities to the right-hand solutions. The Debye-Huckel theory suggests that the final term in (2) is unlikely to alter by more than 1mv. between 1 and 20010 atm. If we ignore this small change,13then
2235
concentrations of the two solutions were nearly the same (0.1 mole/kg.), the quantity ( 5 ) should be close to the change of molar volume A P for the hypothetical ionization of pure water into H+ and OH- ions, both a t the molality m = 0. 1. Bodanszky and Kauzmann’s analysis of density data shows that the volume change for the process HzO (pure) --+ H+(m)
where the subscripts P and 1 denote the pressures P atm. and 1 atm. Table I lists values of the ratio (Kw)p/(K,v)l derived from the present measurements. TABLE I THE INFLUENCE OF PRESSURE ON THE IONIZAT~ON OF WATER AT
25”
Pressure, atm.
1 250 500 750 1000 1250 1500 1750 2000
1 .oo 1.23 1.49 1.78 2.14 2.58 2.97 3.53 4.01
7.000 1.261 1.574 1.943 2.380 2.893 3.492 4.189 5.000
I n 1941, Owen and BrinkleyI5predicted the change of
Kw with pressure from the thermodynamic relation
+ OH-(m)
is A v / ~ m mole-’ .~
=
-21.28
+ 2.571/m
- 0.84 m
so that when m = 0.1, A P = -20.55 ~m.~/rnole.The value of the derivative (4) calculated from the present measurements is -20.4 i 0.5 ~m.~/rnole at 1atm. To summarize, this note describes the first, nearly direct, measurements of the influence of pressure on K,. The results disagree with Owen and Brinkley’s prediction of the change but confirm Bodanszky and Kauzmann’s revised estimate. Acknowledgment.-The author is indebted to Mr. P. Goldberg for his help in the experimental work.
ISOTOPE EFFECTS IN THE MERCURYPHOTOSEPI’SITIZED DECOMPOSITION AND THE DIRECT RADIOLYSIS OF THE SYSTEMS METHANE, METHANE-& AND BUTANE, BUTANE-&’ BY E. G. SPITTLER,S.J., P. JORDAN, LEONM. DORFXAN, AND MYRANC. SAUER,JR.
Arm
where denotes the difference between the molar volume of pure water and the partial molar volumes of OH- a t infinite dilution. They integrated (3) H+ using the best available data for a t 1 atm. and for its change with pressure. Their results are shown in Table I and in Fig. 1,where it is apparent that there is a significant discrepancy between the predicted values and those found in this work. It is almost certain that the disagreement arises from a, then, unavoidable error in Owen and Brinkley’s calculations. In a recent note E:odansxky and KauzmannlGhave shown convincingly that the value 23.4 cm. 3/mole vhich Owen and Brinkley took for - A 17” a t 25’ and 1 atm. is too large by about 2.1 ~m.~/rnole. I n this connection it is interesting to examine the pressure dependence of the e.m.f. a t atmospheric pressure. Making the justifiable assumption that the ratio y C l / y c l ’ is unaltered by pressure, we find that
In connection with our investigations of the radiolysis2n3and vacuum ultraviolet photolysis4 of gaseous hydrocarbons, we have examined the mercury-photosensitized decomposition of CHrCD4 and C4Hl0-C4DI0. The initial purpose of these experiments was the determination of the isotopic distribution of the hydrogen formed in the four abstraction reactions involving H and D atoms and the two isotopic hydrocarbon molecules for comparison with radiation chemical data. The sensitization seemed a reasonable approach since it has been shown in an extensive investigation of butane16and more recently of methane16that the hydrogen is formed entirely from hydrogen atom precursors. The most striking result of our experiments is the observation of very large isotope effects in the mercuryphotosensitized decompositions.
where the terms in parentheses denote the differences between the partial molar volumes of the ions in their actual solutions and a t infinite dilution. Since the
Experimental The C ~ D I O was from Merck, Sharp and Dohme of Canada, Ltd., and the C4H10was Phillips research grade. The CaHlowas purified by gas chromatography. The C~DIO was purified isotopically by mercury-sensitized photolysis t o about 207, decomposition and was then separated from the products by a gas chromatographic purification. (Because of the isotope effects involved, the photolysis, followed by the chromatographic purification, served to improve the isotopic purity so that the ratio of
(13) It should be mentioned that in the present experiments it was impossible to follow the usual procedure of working over a wide range of ionic strengths and extrapolating the results t o zero ionw strength to obtain K w . Glass electrodes only function properly over a rather restricted range of conoentrations.’4 (14) See, e.g., R. G. Bates, in “Reference Electrodes,” ed. by D . J. G. Ives and G. J. Janz, Academic Presa, New York, N. Y.,1961, p. 258. (16) B.B. Owen and S. R. Brinkley, Chem. Rev., !4Q, 461 (1941). (16) A. Bodanezky and W. Kauemann, J . Phye. Chem., 66, 177 (1962).
(1) Based on work performed under the auspices of the U. S. Atomic Energy Commission. (2) L. M. Dorfman, J . Phys. Chem., 6S, 29 (1958). (3) M. C. Sauer, Jr., and L. Ill. Dorfman, ibid., 66, 322 (1962). (4) AX. C. Sauer, Jr., and L. M.Dorfman, J . Chem. Phys., 85, 497 (1961). (5) R.J. Cvetanovik, W. E. Falconer, and K. R. Jennings, ibid., 86, 1225 (1961). ( 6 ) R. A. Back and D,Van der Auwera, Con, J , Chem,, 40, 2389 (1062).
+
Arm
Argonne National Laboratory, Argonne, Illznois Received A p r i l 13. 1963
