B. A. Cosgrove M.Sc. Thesis 1977

AN EXAMINATION OF THE SCALED PARTICLE
THEORY OF GAS SOLUBILITIES
Bruce A. Cosgrove
B.Sc., Simon Fraser University, 1977
A THESIS SUBMITTED I N PARTIAL FULFILLMENT OF
THE REQUIREMENTS FOR THE DEGREE OF
MASTER OF SCIENCE
i n t h e Department
0f
Chemistry
Bruce A. Cosgrove 1980
SIMON FRASER UNIVERSITY
April 1980
A l l r i g h t s reserved. This t h e s i s may not be
reproduced i n whole or i n part, by photocopy
or other means, without permission of t h e author.

Name
APPROVAL
Bruce Anthony Cosgrove
Degree Master of Science
T i t l e of Thesis: An Examination of t h e Scaled P a r t i c l e
Theory of Gas S o l u b i l i t i e s .
Examining C o dttee :
Chairperson: T.N. Bell
- > - - -, -
J. Walk&.y+,-- . -
Senior Supervisor
- L
I.D. Gay
J.M. DIAuria
-
G.L. Malli,
I n t e r n a l Examiner
Date Approved: // nSd a d .

PARTIAL COPYRIGHT LICENSE
I hereby grant t o Simon Fraser University the r i g h t t o lend
my thesis, p r o j e c t o r extended essay (the t i t l e o f which i s shown elo ow)
t o users of the Simon Fraser University Library, and t o make p a r t i a l o r
single copies only f o r such users o r i n response t o a request from the
l i b r a r y of any other university, o r other educational i n s t i t u t i o n , on
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f o r m u l t i p l e copying o f t h i s work f o r scholarly purposes may be granted
by me o r the Dean o f Graduate Studies. I t i s understood t h a t copying
or- publication of t h i s work f o r financial gain shall not be allowed
without my w r i t t e n permission.
T i t l e o f Thesi s/Project/€xtended Essay
" ~ nExamination of the Scaled Particle Theory of Gas
Solubilities"
Bruce A. COSGROVE
( name
April 14, 1980
(date)

ABSTRACT
The extension of Scaled Pasticle Theory (s.P.T.) t o t h e predictions
of t h e thermodynamic properties of dissolved gases was f i r s t demonstrated
i n 1963. Over t h e intervening years t h i s theory has been extensively
used and has always been shown t o provide excellent agreement with
eqerimental data. No modifications of t h e original theory have been made
despite the f a c t t h a t , since its conception, progress i n &her theoretical
studies has indicated basic inconsistencies i n i t s formulation.
The purpose of t h i s research was twofold, t o t e s t t h e theory against
experimentally determined solubility data f o r a range of gases i n H 0 and
2
D 0 solvents, and t o incorporate certain improvements into t h e structure
2
of t h e theory i t s e l f and predict thermodynamic properties of dissolved
gases i n various organic solvents.
Much data already e x i s t s f o r t h e solubility of gases i n H20.
Suprisingly enough, t h e S.P.T. i s able t o predict quite accurately t h i s
data. It i s known t h a t H 0 exhibits properties t h a t a r e unlike those of
2
simple organic solvents and t h i s difference i s reflected i n gas solubility
data. No adequate theory f o r t h e description of t h e properties of pure
H20 exists and yet it would appear t h a t t h e S.P.T., despite i t s simplicity,
'can predict t h e properties of gases dissolved i n t h i s solvent. D20 i s a
solvent matching t h e complexity of H 0 and yet known t o show a 5 t o 10%
2
difference i n i t s gas solubility behaviour. No application of t h e S.P.T. t o
gases dissolved i n D20 has been made, primarily due t o insufficient data.
Such a comparison was considered a good t e s t of t h e S.P.T., so solubility
data was obtained f o r a range of gases over a 40 OC temperature range
i n t h i s l a t t e r solvent.
The technique developed f o r these measurements i s novel i n t h a t it
iii

allows direct determination of a gas dissolved i n a 1 / 2 ec sample of
D 0 and simultaneous determination of the same dissolved gas i n a
2
1/2 cc sample of H20, t h e l a t t e r providing an internal reference. The
gas chromatographic method developed compare% directly t h e stripped gas
from t h e 1/2 cc solvent sample t o a pure gas calibration curve obtained
under identical conditions, hence eliminating t h e need f o r r e l a t i v e
measurements by comparison t o known amounts of 'another' dissolved gas
i n t h e same solvent.
The Scaled P a r t i c l e Theory of gas s o l u b i l i t i e s i s rigorously developed
incorporating certain features only appreciated as necessary t o the theory
from theoretical studies performed since i t s original formulation. From
t h i s development t h e approximations intrinsic t o t h e original theory a r e
seen and t h e i r v a l i d i t y i s tested. Consideration of t h e limiting (hard
sphere) conditions of t h e theory show t h e original development t o be
inconsistent. It i s found that t h e more rigorous theory cannot predict
t h e properties of gases dissolved i n H 0 and D 0 solvents and has only
2 2
limited success i n t h e common organic solvents. The reason f o r the
predictive success of t h e original Scaled Particle Theory i s revealed by
the subsequent examination of a hard sphere version of t h e S.P.T. It
i s shown t o be rather fortuitous.

DEDICATION
To Katherine Cosgrove

ACKNQWLEDGMENTS
I would l i k e t o express my sincere thanks t o D r . John Walkley for his
intellectual leadership, interest and more importantly h i s approachability
during the author's graduate career.
I would also l i k e t o thank D r . Ian Gay f o r t h e discussions and ideas
regarding t h e experimental solubility apparatus.
Thanks are also due t o M r . Peter Hatch and h i s associates in the
glassblowing shop for the many ideas i n the design of the solubility apparatus
and the s k i l l f u l construction.
Thanks are due t o Mr. Bob Lyall f o r t h e design of the computer program
used i n the S.P.T. calculations. I wish t o thank Daniel Say of the S.F.U.
science library for h i s assistance i n locating many of the required parameters
used i n the S.P .T. calculations.
Last, but not least, I would l i k e t o thank Margaret Fankboner for
t
typing t h i s t h e s i s which, I ' m sure, a t times appeared t o be a formidable
task.
Simon Fraser University
Burnaby, B.C ., CANADA
April 1980
Bruce A.. Cosgrove

TABLE OF CONTENTS
T i t l e Page
Approval
Abstract
Dedication
Acknowledgments /
Table of Contents
List of Tables
L i s t of Figures
Section I
Introduction
CHAPTER I: Solubility of Gases i n H20 and D20
Experimental
Results and Discussion
References
Appendix I
Section I1
Introduction
CHAPTER 11: Scaled P a r t i c l e Theory of Gas Solubilities
Theory and Development
Hard Sphere Limits
Scaled Particle Theory
CHAPTER I11
Discussion
Conclusions
References
Appendix I
Page
i
ii
iii
v
v i
v i i
v i i i
X
v i i

L I S T OF TABLES
T a b l e NO
Section I
I: MOLE FRACTION $ OF GASES DISSOLVED I N H z 0 AM) D20
11: COEFFICIENTS I N THE EQUATION R l n x 2 = A + B/T + c l n ( ~ / O ~ )
111: ENTHALPY OF SOLUTION OF GASES I N H 2 0 AND D 2 0
IV: ENTHALPY OF SOLUTION OF GASES I N H 2 0 AND D 2 0 (REF. 1)
V: THERMODYNAMIC PROPEBTIES I N D20 A T 298.15 OK
V I : THERMODYNAMIC PROPERTIES I N H20 A T 298.15 OK
Section I1
I: COEF'FICIENTS I N THE EQUATION l n K ~= A + B a + ca2 + ~a~
11: SOLVENT EFFECTIVE HARD SPHERE DLAMETERS
111: EXTRAPOLATED SOLVENT DIAMETER TEMPERATURE DEPENDENCE
IV: EXTRAPOLATED TKERMODYNAMIC HARD SPHERE DATA A T 298.15 OK
V: PROPERTIES OF THE SOLUTES AT 298.15 OK
V I : PROPERTIES OF THE SOLVENTS AT 298.15 OK
THERMODYNAMIC DATA CASE I
V I I : H20 CASE I
V I I I : CqH6 CASE I
IX: c-C6H12 CASE I
X: n-C6%4 CASE I
X I : AS H20 CASE I
X I I : A s C6H6 CASE 1
X I I I : AS c-C6H12 CASE I
' XIV: AS n-C6H14 CASE I
v i ii

THERMODYNAMIC DATA CASE I1
XV: H20 CASE TI
XVI: C6H6 CASE 11
XVII: C!-C6H12 CASE 11
X V I I I : n-C6H14 CASE 11
XIX: HARD SPHERE PROPERTIES AT 298.l5 OK
THERMODYNAMIC DATA CASE I11 AND IV
XX: H20 CASE I11 AND I V
XXI: C6H6 CASE 111AND I V
XXII: C!-C6H12 CASE 111AND IV
X X I I I : n-C6$4 CASE I11 AND I V
XXIV: SOLVENT EE'FECTIVE HARD SPHERE DIAMETERS AT 298.15 OK
XXV: H20 AT 298. l 5 OK INTERICTION TEBMS FOR CASE I AND 11
XXVI: C6H6 AT 298.15 OK INTWACTION TERMS FOR CASE I AND 11
XXVII: H20 AT 298.15 OK INTERACTION TERMS FOR CASE I11 AND IV
XX.111: C6H6 AT 298.15 OK INTERACTION TERMS FOR CASE 111AND IV

FIGURE 1
FIGURE 2
FIGURE 3
FIGURE 1
FIGURE 2
FIGURE 3
FIGURE 4
FIGURE 5
FIGURE 6
FIGURE 7
FIGURE 8
FIGURE g
LIST OF FIGURES
S e c t i o n I--
GAS STRIPPING LINE
0
FIGURE 1 0 AH-R~ FOR HARD SPHERE (2.55 A) IN BENZENE
AT 298.15 OK VS. BENZENE HARD SPHERE DIAMETER
FIGURE 11 FOR HARD SPHEBE IN BENZENE AT 298.15OK VS
BENZENE SOLVENT HARD SPHERE DIAMETER
FIGURE 1 2 THERMAL EXPANSION COEFFICIENT AS A FUNCTION OF a1 VS
BENZENE SOLVENT HARD SPHERE DIAMETER
FIGURE 13 INWT GASES IN BENZENE A H ~ ~ ~ ~ - A H ~ ~ ~vs
SOLUTE H.S. DIAMEm
STRIPPING CELL
LNx2 VS 1 / T (OK-') ARGON IN H20 AND D20
S e c t i o n I1
m T GASES HARDSPHERE DIAMETER (a)VS
POLARIZABILITY (CM~/MOLECULE)
L% VS a (POLARIZABILITY) GASES I N Hz0
AT 1 0 , 20, 30 AND 45OC
LNKH VS a (POLARIZABILITY) GASES IN D20
AT 10, 25, 35 AND 45Oc
AS $'OR INERT GASES IN C6H6 VS GAS POI&RIZABILITY
AS FOR INERT GASES IN H20 VS GAS POLARIZABILITY
LNK~-?&/RT FOR INERT GASES IN BENZENE VS
SOLUTE EFFECTIVE HARD SPHERE DIAMETER (8)
LNK~-?!~/RT FOR INERT GASES I N H20 VS
SOLUTE EFFECTIVE HARD SPHERE DIAMETER
AH-^& FOR INERT GASES I N H20 VS.
AH-Ri FOR INERT GASES IN C6H6 VS
Page

INTRODUCTION
This thesis i s divided into two separate sections. The
outline summarizes t h e topics covered i n each section.
following
Section I details the experimental gas chromatographic system which was
designed for the determination of gas s o l u b i l i t i e s i n H20 and D20. ' From
the solubility values so obtained the enthalpies associated with t h e
dissolution process were calculated for both H 0 and D20 and compared t o
2
other experimental data where available. The data for gases dissolved i n
D20 were used t o t e s t the Scaled Particle Theory (S.P.T.) which had been
previously shown t o give goo'd agreement f o r gases dissolved i n H20 Further
examination of the S.P.T. using transfer coefficients revealed that the
enthalpy expression was a strong function of the solvent experimental thermal
expansion coefficient and represented a rather simplistic approach for the
calculation of the enthalpy.
I n section I1 the S.P.T. for gas solubility i s rigorously developed.
It was noticed that i n the original theory t h e temperature dependence of
the effective hard sphere diameter of the solute and solvent molecules had
been ignored. Use was made of the (extrapolated) properties of a solute
molecule of zero polarizability t o determine the need for inclusion of such
a temperature dependence. It was unambiguously shown t h a t the dependence
must be included i n the theory. Rederiving the theory t o include the
temperature dependence, it was shown that agreement between theory and
experiment (for a wide range of solvents) was now very poor and, i n an attempt
t o further understand the theory, the theory was now derived i n a purely
hard sphere formulation.
The approximations necessary t o produce the equations of the original
S.P.T. were then examined. This showed that the originally observed

agreement between theory and experiment was purely fortuitous. Indeed
it i s shown that the theory can only be applied t o systems wherein the
behaviour of the solute molecules i s sufficiently similar t o that of a
hard sphere.
The theory is thus shown t o be a very simple model of a perturbation
theory in which the reference system is that of a system of hard spheres.

Section I - Introduction
The thermodynamic properties of solutions of gases i n water display
anomalies i n contrast t o t h e properties observed i n organic solvents.
These a r e revealed i n t h e unusually low molar entropies of solution
and large negative molar heats of solution. Correlations of gas
solubilities have been attempted using such properties a s p a r t i a l vapour
pressures, surface tension and p a r t i a l molar volumes i n an attempt t o
formulate a theory f o r dissolved gases, but lack of r e l i a b l e solubility
data has hindered these attempts.
~ a t t i n o ' l ) has tabulated and c w e d f i t t e d solubility values for
dissolved gases i n water t o obtain t h e temperature dependence of t h e
solubility according t o t h e equation
where A, B and C a r e coefficients
for selecting t h e solubility data
experimental method employed, the
and T is Temperature (OK). The c r i t e r i a
were based on the r e l i a b i l i t y of the
reproducibility of t h e worker's own
data and from comparison with data from other sources. Consequently data
for many of t h e temperature dependent studies consists of data obtained by
( 2various workers using many different techniques .
This research has developed a gas chromatographic technique which,
unlike other techniques, allows an absolute measurement of a dissolved
gas i n a solvent. The use of a gas chromatographic
technique t o determine t h e amount of a gas dissolved i n a known

4
volume of solvent is widely reported(3,4 95) . With appropriate
chromatographic conditions, small solvent samples can be analysed and
amounts of dissolved gas down t o moles can be accurately
measured. Most workers have used t h e so-called ' s t r i p p i n g ' technique
wherein t h e chromatograph c a r r i e r gas is bubbled r a p i d l y through
t h e sample under i n v e s t i g a t i o n , t h e gas is then dried o f solvent by
passage through an appropriate absorbent, and f i n a l l y t h e amount o f
s o l u t e gas stripped from solution i s measured using a g a s c h r o 1 1 i a t o ~ r a ~ h ( ~ , 7 ) .
The s t r i p p i n g techique has caused most workers t o adopt a r e l a t i v e
r a t h e r than an absolute measurement o f gas s o l ~ b i l i t ~ ( ~ , ~ ) .Thus, t h e
amount o f g a s dissolved i n a solvent is measured r e l a t i v e t o t h e amount
o f 'another1 g a s dissolved i n t h e same solvent under conditions wherein
t h e s a t u r a t i o n s o l u b i l i t y is w e l l established.
I n t h i s research we r e p o r t a gas chromaographic method which
allows a dissolved gas sample t o be measured a g a i n s t a c a l i b r a t i o n
response curve obtained using pure gas samples, y e t allowing t h e
c a l i b r a t i o n curve t o be obtained under conditions matching those used
i n s t r i p p i n g t h e dissolved gas.

Solubility of Gases i n H20 and D,OL
" ~ e tus l e w n t o dream, gentlemen, then perhaps we shall
find the truth. But l e t us beware of publishing our
dreams till they have been tested by the waking understanding ,"
Friedrich August ~ e k u l 6
Translated by F.R .Japp
Journal of Chemical Education
35, 21 (1958)

Distilled H20 o r D20 (99.8% Stohler Isotope chemicals) i s degassed
by a sublimation technique. Twenty mls of the solvent i s introduced
into a degassing c e l l through a Rotaflo valve. Liquid nitrogen i s placed
i n t h e cold finger above t h e degassing c e l l and t h e system i s evacuated t oI
t o r r causing t h e solvent t o freeze. Sublimation i s achieved by
placing a heating mantle around the solvent container and subliming
t h e frozen solvent onto t h e cold finger. Once degassing i s completed
the saturation c e l l which i s connected t o the degassing c e l l i s
-4evacuated (10 t o r r ) and the frozen solvent i s allowed t o melt
under vacuum. Approximately 20 mls of t h e solvent i s then transferred
t o the saturation c e l l under an atmosphere of t h e gas under study.
Teflon stopcocks a r e used t o prevent grease contamination. No
detectable amounts of gas have been observed i n a G.C. analysis (on
maximum sensitivity) upon analysis of t h e degassed solvent.
Saturation Cell
The saturation c e l l i s maintained i n an insulated water bath at
0
T'C + .O1 C. Upon transfer of the solvent from t h e degassing c e l l t o
the saturation c e l l t h e gas under study i s dispersed through t h e solvent
using a f r i t t e d A t t h e same time the solution i s constantly s t i r r e d
by a magnetic s t i r r e r (controlled from outside the water bath). The gas
pressure above t h e sample i s maintained a t atmospheric pressure, thus
preventing supersaturation. Complete saturation i s normally achieved within
2.5 hrs. This was determined by monitoring gas uptake a s a function of
time for several gases. Before samples are withdrawn t h e
bubbling of t h e gas through t h e solvent i s halted and t h e

-
r
solution is l e f t s t i r r i n g f o r 1h under one atmosphere o f g a s t o ensure
7
Sample Transfer
The saturated sample is transferred t o t h e g a s s t r i p p i n g l i n e
using a g r e a s e l e s s g a s t i g h t 2.5000 _+ .0001 m l Gilmont micrometer syringe.
The syringe is i n i t i a l l y flushed with t h e gas under study t o prevent
a i r contamination o f t h e sample. The saturated sample is withdrawn from t h e
s a t u r a t i o n c e l l by i n s e r t i n g t h e needle of t h e syringe through a rubber serum
cap f i t t e d on t h e s a t u r a t i o n cell. The syringe is designed such t h a t
t h e b a r r e l can be f i l l e d with 2.5 m l s . of solvent extremely slowly, thus
preventing t h e sample from being placed under a reduced pressure. The
syringe is then withdrawn from the saturation cell, containing
approximatley 2.5 m l of t h e saturated solvent.
Gas Stripping Line
The g a s s t r i p p i n g l i n e shown diagrammatically i n FIG. 1 is
constructed of 4mm I.D. Duran 50 g l a s s .
The l i n e i s previously evacuated ( t o r r ) through stopcocks 12
and 13 and then permanently maintained under Helium c a r r i e r gas pressure.
I n i t i a l l y , .25 m l s of t h e sample a r e i n j e c t e d i n t o t h e gas s t r i p p i n g
c e l l (Fig. 2) t o "wetw t h e frit since it would appear t h a t some
absorption of dissolved gas occurs on t h e f r i t . Then four .500 m l
samples a r e i n j e c t e d sequentially (allowing t h e previous sample s u f f i c i e n t
time t o ' b e s t r i p p e d ) and t h e stripped gas analysed on t h e G.C. These
samples a r e i n j e c t e d without t h e syringe being withdrawn from the
s t r i p p i n g c e l l , ensuring t h a t no leaks o r a i r contamination occurs. For

BULBS
9-Q
FIG.1
GASSTRIPPINGLlNE
He-TANK
SAMPLE
LOOP
-LINEBRECORDER
Y
G.C.
COLUMN
1
1HELIUM
A'TANK
UDIGITAL
INTEGRATOR

i n j e c t i o n t h e dissolved gas is r a p i d l y s t r i p p e d from t h e
solution by Helium c a r r i e r gas dispersed through t h e sample by a
#2 porosity g l a s s frit placed a t t h e base o f t h e s t r i p p i n g c e l l ( 9 ) .
The Helium c a r r i e r g a s flow can be p r e f e r e n t i a l l y d i r e c t e d
i n t o s t r i p p i n g c e l l 81 o r c e l l #2 using stopcock 7 (3-way).
Closing stopcocks 9 and 10 or stopcocks 8 and 11 allows each
s t r i p p i n g cell t o be used individually. The s t r i p p i n g c e l l s were
i n i t i a l l y evacuated t o r r ) through stopcocks 12 and 13, and
t h e l i n e purged with Helium through stopcock 16.
The s t r i p p e d g a s is d r i e d by passage through a 50% CaC12 and
50% CaS04 drying t u b e before entering t h e chromatographic column;
appropriate columns a r e used f o r each g a s ( s e e APPENDIX I). The
stripped g a s is then analysed i n a Varian (Model gOP) d u a l filament
thermal conductivity d e t e c t o r . The s i g n a l from t h e d e t e c t o r is
i n t e g r a t e d on a C.M.C. d i g i t a l readout (Model 707 BN) frequency
counter and t h e response is compared d i r e c t l y t o a c a l i b r a t i o n p l o t .
The v a r i a t i o n i n response being no more than 1.0% amongst t h e four
i n j e c t e d samples.
GAS CALIBRATION
The s o l u b i l i t y value is obtained by a d i r e c t comparison with a
c a l i b r a t i o n p l o t o f moles o f g a s vs. chromatographic response a s
recorded by an on l i n e i n t e g r a t o r . The c a l i b r a t i o n curve is obtained
by observing t h e chromatogramls response t o a known number o f moles o f
g a s contained i n t h e g a s sampling loop(lO) ( ~ i ~ :1 ) .
The g a s sampling loop (Volume .7076 I+_ .0001 mls) and l i n e a r e
evacuated t o r r ) through stopcock 1 while stopcocks 3, 4, 5 and
6 a r e open. The l i n e i s then i s o l a t e d from t h e r e s t o f t h e vacuum
i

GAS DETECTION
The gas is detected by a dual filament thermal conductivity
detector with one filament being used as a reference and t h e o t h e r
filament a s the gas d e t e c t o r . A flow r a t e o f 75 mls/min f o r both
reference l i n e and sample l i n e is maintained. A c u r r e n t o f 175 >mA
is used f o r detection and t h e d e t e c t o r block maintained a t 1 1 0 ~ ~ .
The s i g n a l from t h e d e t e c t o r is recorded on a 1 mV f u l l s c a l e
Varian c h a r t recorder (Model 9176) and t h e s i g n a l is i n t e g r a t e d by an
e l e c t r o n i c C.M.C. (Model 707BN) d i g i t a l readout frequency counter.
,ystem by shutting stopcock 1, Shutting stopcock 5 enables one t o
u t i l i z e t h e U-tube a s a closed end manometer. The g a s is introduced
i n t o t h e sampling loop a t room temperature from one o f t h e gas bulbs.
After a t t a i n i n g equilibrium, t h e temperature o f t h e g a s is read and
the pressure accurately measured on t h e Hg-manometer. The number of
moles of gas i n t h e sampling loop is then obtained using t h e i d e a l
gas law. To t r a n s f e r t h e g a s t o t h e detector f o r a n a l y s i s t h e sampling
loop s t o ~ c o c kis r o t a t e d 90'. The c a r r i e r gas t r a n s f e r s t h e sample
v i a l i n e A through a s t r i p p i n g cell, drying tube, and chromatographic
column t o t h e d e t e c t o r . The loop stopcock is then r o t a t e d back go0
t o t h e o r i g i n a l position. Stopcocks 1 and 5 are opened and t h e system
is then evacuated t o t o r r . The process is repeated f o r various
I
1
gas pressures u n t i l t h e d e s i r e d number' of c a l i b r a t i o n points are
obtained.
Using t h i s method, a complete c a l i b r a t i o n curve can be obtained f o r
t h e gas under study from '1o ' ~moles.

The gas flow i n t o t h e d e t e c t o r is divided i n t o an i n t e r n a l
flow and a n e x t e r n a l c a r r i e r flow both maintained a t t h e
same flow rate. The c a r r i e r gas always flows through t h e l i n e and
then t o t h e d e t e c t o r v i a l i n e B with stopcocks 18, 19 and 16
closed and stopcock 17 open.
I
Before t h e sample is i n j e c t e d i n t o t h e s t r i p p i n g c e l l t h a t p a r t
13. Stopcocks 7, 8, 9, 10, 11, 14 and 15 a r e open, t h i s enables t h e
s t r i p p i n g l i n e and t h e sampling loop t o be completely evacuated.
Stopcocks 12 and 13 are s h u t o f f and t h e l i n e is thoroughly purged
with Helium through stopcock 16. This process is repeated and t h e
l i n e i s maintained under He carrier gas pressure by opening stopcock
18 while stopcocks 17 and 19 are closed. Stopcock 16 is closed ( t o t h e
Helium tank) and d i r e c t s t h e He c a r r i e r gas through t h e
chromatographic column t o t h e d e t e c t o r . Stopcocks 8 and 9 are then
closed, while stopcock 7 (3-way) remains open. This enables t h e
c a r r i e r gas t o flow through t h e g a s sampling loop, and through each of
t h e two s t r i p p i n g c e l l s . The g a s flow is then d i r e c t e d through t h e
drying column by a d j u s t i n g stopcocks 14 and 15, and subsequently
through t h e chromatographic column t o t h e d e t e c t o r . Each s t r i p p i n g
c e l l may be used independently by d i r e c t i n g t h e c a r r i e r g a s with
stopcock 7.
Stripping c e l l #I can be used by d i r e c t i n g t h e He c a r r i e r g a s
through stopcock 7 i n t h e d i r e c t i o n o f cell #1 with stopcock 10 open
and s h u t t i n g stopcock 11, t h u s i s o l a t i n g c e l l #2. S i m i l a r l y c e l l #2
can be used by t u r n i n g stopcock 7 180•‹ and closing stopcock 10 while
stopcock 11 i s open, t h u s i s o l a t i n g c e l l #1 and maintaining t h e
c a r r i e r gas flow through c e l l #2. By p r e f e r e n t i a l l y i s o l a t i n g each

c e l l one can rapidly perform sample a n a l y s i s on two d i f f e r e n t
solvents. I n p r a c t i c e , two saturation c e l l s a r e used simultaneously
thus allowing one g a s t o be studied i n two d i f f e r e n t solvents (eg.
H20 and D20) a t i d e n t i c a l temperatures and pressures. One can
thus determine t h e amount of g a s dissolved i n H 0 i n c e l l # 1 and t h e
2
same g a s dissolved i n D20 i n c e l l #2. The g a s c a l i b r a t i o n curve is
obtained a s previously described.
I n p r a c t i c e a c a l i b r a t i o n response curve is obtained p r i o r t o t h e
analysis of t h e dissolved gas samples ( ' d r y t c a l i b r a t i o n ) and then
repeated a f t e r t h e analysis. These c a l i b r a t i o n points obtained after
the 'stripping' o f t h e dissolved gas samples r e q u i r e s t h e dry gas t o be
passed (with Helium c a r r i e r gas) through t h e 2.5 m l s o f solvent
remaining i n t h e s t r i p p i n g c e l l s . We c a l l these 'wet' c a l i b r a t i o n
points, serving both t o show t h a t no instrumental f a c t o r s have
a l t e r e d during t h e time taken f o r sample a n a l y s i s and t h a t t h e
presence of t h e solvent on t o p o f t h e f i t t e d d i s k s i n t h e s t r i p p i n g
c e l l s h a s n o t caused changes i n chromatographic response through a
change i n c a r r i e r gas flow r a t e .
Our experiments show t h a t t h e c a l i b r a t i o n p l o t i s independent
of t h e c e l l used. Our experiments have a l s o shown t h a t t h e
introduction o f approximately 2.5 m l s o f sample i n t o t h e s t r i p p i n g
c e l l is not s u f f i c i e n t t o r e t a r d t h e c a r r i e r gas flow r a t e , and hence
change the response o f t h e detector. Consequently "dry c a l i b r a t i o n n
i s i d e n t i c a l t o t h e "wet c a l i b r a t i o n w . By d i r e c t comparison o f the
i n t e g r a t o r response of t h e four stripped samples t o t h e c a l i b r a t i o n
Curve, one o b t a i n s t h e number of moles of dissolved gas. For low
S o l u b i l i t y gases t h e i n j e c t e d sample s i z e can be increased from
1/2 c c t o 1 cc and t h e amount o f gas can be accurately determined.

Again, no change in detector response i s observed f o r the 1 cc
injections. This experimental technique has been applied t o various
dissolved gases i n H 0 and D20 with an estimated accuracy of 1%
2
for most gases and 2% f o r t h e l e a s t soluble gases.
RESULTS AND DISCUSSION
The saturation of t h e gas i n t o each solvent was carried out
simultaneously and t h e analysis of t h e amount of gas dissolved was
made using one sample c e l l f o r H20 and one sample c e l l f o r D20.
Two s e t s of 'four 0.5 m l injections' were performed on each solvent.
A comparison of t h e data obtained with t h a t reported i n the l i t e r a t u r e
i s made i n Table 1. The comparison i s seen t o be excellent.
There i s l i t t l e data reported f o r gases dissolved i n D 0,
2
perhaps due t o t h e quantities of solvent required f o r experiments.
One of t h e advantages of our technique i s that l e s s than 20 mls of
solvent i s required f o r saturation and l e s s than .5 mls is required
for analysis f o r any one temperature, a l l of which is recovered.
Using t h e s o l u b i l i t y data i n Table I, t h e gas solubility values
were f i t t e d t o equation (1)with a standard deviation reported as a %
difference i n the 1nX f i t compared t o t h e experimental lnX2 a t
2
298.15 OK. Table I1 summarizes these r e s u l t s and Figure 3 i s a typical
plot of Argon gas s o l u b i l i t y i n H 0 and D 0 expressed a s InX versus
2 2 2
I/T (OK-' ) .
Using t h e coefficients so obtained from t h e curve f i t , the
enthalpies associated with t h e dissolution process i n H20 and D 0
2
were computed from equation (1) using t h e following expression:

Table I 15
. Mole Fraction X2 of Gases Dissolved i n H20 and D20
4 '
Argon ( ~ * 1 0)
H2•‹
- D20
To (K) /Ref. Present Present
hi) 0 63) (1) study study (3;)
.3769 .3788 - .3787 .3785 .4271 .4270
,3360 .3364 .3352 .3367 .3333 .3731 .3750
.3019 .3024 .2g53 .3025 .2988 .3333 i3341
.2742 .2756 ,2697 .2746 ,2722 .3048 .3003
.2517 .2530 .2482 .2516 .2520 .2680 --.2724
.2332 - .2284 .2326 .2316 .2497
Present
(a) study
~ 5 2 6 ~ 5 . 3 3
.64g8 .6577
.5680 .5740
.5025 .5113
.44g4 .4526
.4062 .4180
.3708 -
.3417 .3469
Fresent
Study
.8843
.6847
,5652
D2•‹
Present
stuay
.8624
.7264
.6472
.5544
.4803
.4451
.3809
Present
Study

Table I (continued)
4Nitrogen (x-10 )
Present
2 (17) ' ( 1 Study
Oxygen
H2•‹
Present
(li) 6-51 (1) Study
CHI,
Present
(16) (17.1 (1) study
Present
Study
Present
stuay
Present
Study (11

Table I (continued)
Present
(18) (19) (1) Study
H2•‹
Present
(-1> Study
*
I interpolated value
Present
Study h)
Present
Study
- D20
Present
Study
( ) omitted value, clearly i n error

FIG. 3
LN x, vs 1/T (OK-') ARGON in H20and D,O
@ PRESENT STUDY
0 REFERENCE
A REFERENCE

Table I1
Coefficients i n the Equation
A
-1 0 1
CalMole K-
Rlnxp = A + B/T + c l n (T/KO)
B C
C a l ole-I -1 0 1
CalMole K-
16452.2 45.7
14317.4 36.9
16005.4 41.3 .
28350.6 82.3
20059.8 57.9
18752.4 53.0
18544.7 54.2
846.8 -6.2
24361.5 71.0
14919.8 37.8
27749.5 77.4
12680.5 24: 4
36142.5 104.8
56244.3 172.6
* Standard Deviation of F i t In X2 as a percentage at 298.l5 OK

Tables I11 and I V summarize these r e s u l t s f o r various gases i n both
H20 and D20. The experimental enthalpies of solution compare
q u i t e favorably with those from Reference ( 1 ) . One must remember t h a t
t h e experimental values o f t h e Henry's law constants from t h i s reference
were f i t t e d t o equation (1) using data from many workers incorporating
d i f f e r e n t techniques. ,
These data can be considered i d e a l since any values which deviated
from the f i t by g r e a t e r than one standard deviation were r e j e c t e d ; hence
only d a t a conforming t o t h e curve f i t of equation (1) were used.
From the above Tables o f enthalpy values, one can see d i s t i n c t t r e n d s
f o r both H20 and D20: as t h e temperature increases t h e enthalpy
decreases. Also one observes a g r e a t e r enthalpy of s o l u t i o n f o r t h e
dissolved gases i n D20 than i n H20. To draw comparison t o t h e
experimentally determined s o l u b i l i t y thermodynamic p r o p e r t i e s , t h e
generally accepted p r a c t i c e h a s been t o use t h e Scaled P a r t i c l e Theory
(S.P.T.) of gas s o l u b i l i t i e s , although t o d a t e no such comparison f o r
t h e s o l u b i l i t y of t h e i n e r t gases i n D20 has been made. The next
section of t h i s t h e s i s d e a l s i n g r e a t depth with t h e S.P.T. of gas
s o l u b i l i t y . To present t h e predicted gas s o l u b i l i t y data f o r D20
one must introduce t h e required equations of t h e theory. Here, only
a b r i e f description o f t h e theory w i l l be presented t o introduce t h e
equations whilst t h e reader is referred t o t h e following section f o r a
detailed discussion.
The dissolution o f a g a s i n t o a l i q u i d is considered a s a two s t e p
process. The first being t h e creation of a c a v i t y of s u i t a b l e s i z e t o
accomodate t h e s o l u t e molecule. The second s t e p i s t h e introduction of
t h e s o l u t e i n t o t h e c a v i t y created. The thermodynamic functions

TableI11
EnthalpyofSolutionofGasesinH20andD20
-1*H20AH(~alMol)
-1*D,OAH(CalMol)
*ErrorexpressedasafunctionofAHfromthederivativeoflnxat298.15OK
2

TableIV
EnthalpyofSolutionofGasesinH20andD20(~ef.1)
D2•‹
AH(CalMol-l)

associated with these( s t e p s a r e t h e reversible work o r t h e Gibbs f r e e
energy i n c r e a t i n g a hard sphere cavity and, a f t e r i n s e r t i o n of t h e
s o l u t e , t h e Gibbs f r e e energy of solute-solvent i n t e r a c t i o n . These
s t e p s were described i n a series of papers by Reiss e t a l . ( * O ~ ~ ~ )f o r
a system of hard spheres, and l a t e r extended by P i e r o t t i f o r t h e s o l u b i l i t y
f o r t h e process is given by
-
where cc and Gi describe t h e above two s t e p dissolution process
and a r e defined i n equations (7) and (17) respectively i n t h e following
section, V i s the solvent molar volume. The Henry's law constant KH i s given by
1
The corresponding enthalpy of solution is written a s
where R i s t h e universal gas constant, T is t h e temperature and ap is
-t h e thermal expansion c o e f f i c i e n t o f the solvent. Hc and Hi a r e t h e
enthalpies associated with c a v i t y formation and i n t e r a c t i o n respectively.
Tables V and VI give t h e predicted Henry's law constant ( h K H )
and t h e enthalpy associated with the dissolution o f s e l e c t e d gases i n
both H20 and D20 and compares them t o my experimentally obtained
values.

I
Table V
Thermodynamic Properties in D20 at 298.15 OK
Gas In KH In KH AH (cal ole-l) AH (cal ole-l)
exP S.P.T. eQ S.P .T.
Table VI
Thermodynamic Properties in H20 at 298.15 OK
Gas In KH In KH AH (cal ole-l) AH (cal ole-l)
exp . S.P.T. exp. S.P.T.

One is amazed a t t h e predictive a b i l i t i e s of t h e Scaled P a r t i c l e
Theory: not only does it p r e d i c t reasonably good AH values, but it a l s o
p r e d i c t s a higher enthalpy of solution (%$) f o r the gases dissolved i n
D20, which compares favorably with t h e experimental difference. One
notes t h a t t h e Henry's l a w constants (expressed a s lnKH) a r e predicted
well f o r both H20 and D20, b u t one recognizes t h a t these p r e d i c t i o n s
could be rather f o r t u i t o u s s i n c e there is l i t t l e d i f f e r e n c e i n t h e
H20 and D20 lnKH values.
O f i n t e r e s t are t h e values o f the enthalpy ! t r a n s f e r t c o e f f i c i e n t s
f o r t h e i n e r t gases from H20 t o D20, t h a t is t h e enthalpy associated
with t h e t r a n s f e r of one mole of s o l u t e from H20 t o D20. These
a r e o f p a r t i c u l a r i n t e r e s t s i n c e t h e values of t h e hard iphere diameter,
t h e dipole moment and t h e p o l a r i z a b i l i t y of H20 and D20 are t h e
same. This e f f e c t i v e l y allows t h e cancellation o f t h e i n t e r a c t i o n term
i n t h e enthalpy expression (eqn. (5)) shown above f o r t h e d i s s o l u t i o n process.
The enthalpy change accompanying t h e t r a n s f e r of a s o l u t e from H20
t o D20 is
which, with t h e c a n c e l l a t i o n o f l i k e terms i n equation ( 5 ) , g i v e s
where a p is the experimental thermal expansion c o e f f i c i e n t o f t h e
solvent and y is t h e !compactnesst f a c t o r ( a function o f t h e molar volume
of t h e solvent). Table VII summarizes the experimental enthalpy t r a n s f e r
c o e f f i c i e n t s f o r s e l e c t e d gases and compares these t o t h e values
predicted by the S. P.T.

Table V I I
( c a l ole-l) at 298.15 OK
Gas A%?
Present Study Ref. 1 S.P.T.
(') Reference (25)

One observes from Table V I I t h a t t h e experimental enthalpy
t r a n s f e r c o e f f i c i e n t s decrease f o r t h e noble gases from Argon t o Xenon,
whilst t h e S.P.T. p r e d i c t s an increase. Generally poor agreement is
obtained f o r t h e experimental t r a n s f e r c o e f f i c i e n t s when
compared t o t h e Scaled P a r t i c l e Theory prediction.
One notes t h a t a small difference i n e i t h e r t h e H20 o r D20
enthalpy values f o r t h e gases contributes t o a very l a r g e d i f f e r e n c e
i n t h e experimental t r a n s f e r c o e f f i c i e n t s . It is a l s o i n t e r e s t i n g t o
note t h a t t h e t r a n s f e r e n t h a l p i e s predicted by t h e S.P.T. a t other
temperatures a c t u a l l y increase with increasing temperature while t h e
experimental values decrease with increasing temperature. One is
surprised a t t h e predicted enthalpies of s o l u t i o n using
t h e S.P.T. considering t h e simplicity o f equation ( 7 ) .
The t r a n s f e r enthalpy c o e f f i c i e n t s a c t u a l l y give one an
i n s i g h t i n t o t h e Scaled P a r t i c l e Theory's method o f d i f f e r e n t i a t i n g
between H20 and D20, a s shown i n equation (7). Upon examination
of t h e bracketed term i n equation ( 7 ) , one sees t h a t , f o r t h e gas
Argon, t h i s term represents approximately a 1%d i f f e r e n c e , while t h e
difference i n t h e solvent thermal expansion c o e f f i c i e n t s represents
H20= 2.57x OK-'; <zO = 1 . 9 1 8 ~10
-4 oK-l)
approximately 25% ( (+
a t 298.15 OK. From Tables V and V I one can s e e approximately a 6%
difference i n t h e predicted enthalpies f o r t h e various gases i n H20
and D20. This d i f f e r e n c e i n predicted enthalpies appears t o be a r e f l e c t i o n
of t h e differences i n t h e experimental thermal expansion c o e f f i c i e n t s
of H20 and D20, which a l s o suggests why there is no apparent difference
i n t h e predicted lnKH values, s i n c e Cc is not a function o f t h e
expansion c o e f f i c i e n t s of t h e solvents. This then appears t o be a
r a t h e r s i m p l i s t i c approach t o t h e prediction .of t h e thermodynamic

properties associated with t h e dissolution process f o r these two
complex systems. It suggests t h a t t h i s theory is i n f a c t overspecified
since both y (compactness f a c t o r ) and ap a r e functions o f solvent molar
volume, and a l s o s i n c e t h e temperature and pressure o f t h e system is
specified. A t t h i s point it is i n t e r e s t i n g t o examine t h e S.P.T. i n
depth t o determine what i n f a c t allows one t o predict with such apparent
success t h e thermodynamic
of a gas i n t o a l i q u i d .
p r o p e r t i e s associated with t h e dissolution

' REFERENCES SECTION I
E. Wilhelm, R. Battino and R . J . Wilcock,
Chem. Reviews 77: 219 (1977).
R. Battino and H.L. Clever.
Chem. Reviews 66: 395 (1966).
S.K. Shoor, R.D. Walker, Jr. and K.E. Gubbins.
J. Phys. Chem. 73: 312 (1969).
H.P. Kollig, J.W. Falco and F.E. S t a n c i l , Jr.
Environmental Science and Technology 9: 957 (1975).
R.F. Weiss and H. Craig.
Deep Sea Research 20: 291 (1973).
J.W. Swinnerton and V . J . Linnenbom.
J. of Gas Chromatography 5: 570 (1967). ,
K.E. Gubbins, S.N. Carden and R.D. Walker, Jr.
J. of Gas Chromatography Oct.: 330 (1965).
K.E. Gubbins, S.N. Carden and R.D. Walker, Jr.
J. of Gas Chromaography March: 98 (1965).
J.W. Swinnerton, V . J . Linnenbom and C.H. Cheek.
Anal. Chem. 34: 483 (1962).
B.A. Cosgrove and I.D. Gay.
J. of Chromatography 136: 306 (1977).
E. Douglas.
J. Phys. Chem. 68: 169 (1964).
C.E. Klots and B.B. Benson.
J. of Marine Research 21: 48 (1963).
T.J. Morrison and N.B. Johnstone.
J . Chem. Soc. 3441 (1954).
A. Lannung.
J. Am. Chem. Soc. 52: 68 (1930).
C.N. Murray and J.P. Riley.
Deep Sea Research 16: 311 (1969).
T.J. Morrison and F. Billett. .
J . Chem. Soc. 3819 (1952).
W.-Y. Wen and J.H. Hung.
J. Phys. Chem. 74: 170 (1970).

,
(18) H.L. Friedman.
J..Am. Chem. Soc. 76: 3294 (1954).
(19) J.T. Ashton, R.A. Dawe, K.W. Miller, E.B. Smith and B.J. Stickings.
J. Chem. Soc. A 1793 (1968).
(20) H. Reiss, H.L. Frisch and J.L. Lebowitz. '
J. Chem. Phys. 31: 369 (1959).
(21) H. Reiss, H,L. Frisch, E. Helfand and J.L. Lebowitz.
J. Chem. Phys. 32: 119 (1960).
(22) R.A. P i e r o t t i .
J. Chem. Phys. 67: 1840 (1963).
(23) R.A. P i e r o t t i .
J. Chem. Phys. 69: 281 (1965).
(24) R.A. P i e r o t t i .
Chem. Revs. 76: 717 (1976).
(25) W.-Y. Wen and J.A. Muccitelli.
J. Sol. Chem. 8: 225 (1979).

Gas
APPENDIX I
Column
6'x1/4" S i l i c a l Gel
30160 Mesh
30/60 Mesh
6'x1/4" Molecular Sieve 5A
40160 Mesh
6'x1/4" Molecular Sieve 5A
40160 Mesh
40/60 Mesh
30/60 Mesh
Temperature
8'x1/4" Molecular Sieve 5A' -28'~
30/60 Mesh
30"x1/4" Molecular Sieve 5 A
30/60 Mesh

Section 11- Introduction
Over t h e years much work has been done t o develop
32
a theory i n an attempt
t o predict gas solubility data. Sisskind and ~asarnowsky('), studying Argon
solubility i n various solvents, showed t h a t t h e energy of solution consisted
of two terms. The f i r s t term was described by t h e increase i n surface area of
t h e cavity created (upon insertion of t h e solute) times the surface tension of
t h e solvent. The second term being t h e molecular solute-solvent interaction.
~ i g ' ~ )derived thermodynamic equations which expressed the solubility -
of a gas a s a function of i t s molecular radius and t h e surface tension of
the solvent . Unfortunately , solute-solvent interactions a r e rather complex
and a t t h a t time not well understood, hence only a qualitative representation
could be made. Also, t h e concept of using macroscopic parameters t o describe
microscopic phenomena is a poor assumption. Uhlig had t h e r i g h t idea i n
considering t h e dissolution process as a two step process: t h e energy of
cavity formation and t h e complex interaction energy. Lack of accurate
solubility data and of an understanding of solute-solvent interactions made
t h e developnent of a theory a formidable task. Even so, these calculations
yielded an insight into the anomolous behaviour of water when compared t o
organic solvents .
~ l e ~ ( ~ )considered a model i n which t h e number of gas molecules
dissolved w a s interchangeable with t h e same number of water molecules
i n a quasi-latt ice. Eley concluded t h a t t h i s was energetically
unfavourable and that t h e available points must i n f a c t be cavities which
could easily be enlarged t o accomodate a gas molecule. An approximate
partition function was constructed which took i n t o account the effect
of t h e gas upon dissolution. Again, gross approximations were made

regarding solute-solvent interactions.
In a series of papers,Reiss, Frisch, Leb0wi-t~'~)and Helfand( 5 )
developed a statistical mechanical theory of rigid spheres, based upon an
exact radial distribution function, which could be applied to real
fluids. From this theory it emerged that, for hard sphere particles,
the only part of the radial distribution function which contributed to
the chemical potentfal of the fluid was the part which determined the
number density of particles in contact with the hard sphere particle.
Hence, the calculation of the entire radial distribution function was
not required. The theory yields an approximate expression for the
reversible work required to introduce a spherical particle into a
fluid of spherical particles. The particles in solution obey a
pairwise additive potential and, upon addition of anotherparticlGtc8this
system, the particle obeys the same potential by a procedure of distance
scaling. A necessary condition for this Scaled Particle Theory (S.P.T.)
is that the solvent molecules be approximately spherical with effectively
rigid cores for both the solvent and solute. Assuming the solvent-solute
interactions are described by a Lennard-Jones 6-12 potential, Reiss et al.Ck
calculated Henry's law constants for systems which satisfied the above
criteria for the theory. Satisfactory agreement was found for Helium in
Benzene and Helium in Argon.
The development of this theory provided the foundation for Pierotti
to formulate a theory of gas solubility using the soft potential as a
perturbation to the treatment of hard spheres. Using the expressions of
Reiss et al. for the reversible work required to introduce a spherical
particle into a fluid of spherical particles and by a method of
extrapolation using experimental data, Eerbttit:as able to predict

thermodynamic properties f o r t h e dissolution of gases i n organic solvents( 6
and water (798). The major drawbacks i n P i e r o t t i 's Scaled P a r t i c l e Theory
a r e t h e basic assumptions inherent i n t h e treatment, which w i l l be
discussed later. I n l i g h t of t h e assumptions made, it seemed appropriate
t o redo t h e Scaled P a r t i c l e Theory extending t h e hard sphere treatnient a s
was o r i g i n a l l y developed by Reiss e t a l . (4'5) incorporating t h e concept of
temperature dependent hard sphere diameters. An extension of t h e Scaled
P a r t i c l e Theory i s presented incorporating a more comprehensive hard sphere
treatment f o r t h e dissolution of a gas i n t o a l i q u i d .

e
CHAPTER I1
Scaled Particle Theory of Gas
Solubilities

Theory and Development
I f one considers a system of N spherically symmetrical molecules
possessing an effective r i g i d core of diameter G1 with t h e necessary
pressure t o maintain t h e system a t a constant volume and, if one excludes
the centres of a l l N molecules from a spherical region of space o f r a d i u s
r i n the volume, then one has created a cavity i n the fluid. It i s - , '
convenient t o consider t h e process of introducing the solute molecule i n t o
the r e a l solvent a s consisting of two steps. The f i r s t step being the
. creation of a cavity i n the solvent of suitable s i z e t o accommodate t h e
solute molecule, assuming the reversible work o r p a r t i a l molecular
Gibb's f r e e energy i s identical with t h a t of charging t h e hard sphere
of the same radius a s t h e cavity into the solution.
The second step i s t h e introduction into t h e cavity of a solute
molecule which i n t e r a c t s with the solvent according t o some potential
law. The reversible work i s identical with t h a t of charging t h e
hard sphere or cavity introduced i n step one t o the required potential.
If the cavity s i z e chosen i n step one was suitable, there should be no
change i n s i z e upon charging.
The standard equation for the chemical potential of a nondisso-
ciative solute i n a very d i l u t e solution i s given by (4)
-X2 is t h e potential of t h e solute molecules i n the solution relative t o
-
i n f i n i t e separation, P i s the pressure, v the p a r t i a l molecular volume .
2

37
3of' the solute, X and j are the partition functions per molecule for
2 2
the translational and internal degrees of freedome for the solute, x2
is the number of solute molecuL2s in solution, and V is the volume of
solution. The sum of the first two terms on the right represents the
reversible work required to introduce a solute molecule into a solution
of volume N2/V These terms can be replaced by ( i + Ec ) for the two
i
step process, assuming that the solution is sufficiently dilute to ignore
solute-solute interactions. Replacing N2/v by x2/v1 yields the chemical
potential of the solute in the liquid phase
The chemical potential of the solute in the gas phase (assumed ideal)
is given by
Assuming the internal degrees of freedom of the solute are not affected
by the solution process, equating equations (2) and (3) one~~obtains
-
lnPB = g./KT + /KT + ln(~T/v~)+ lnX2
1 C
(4)
Using Henry's law (defined as P p = %X2, where 5 is the Henry's law
constant'),equation (4) becomes, for partial molar quantities,

The molar heat of solution i s
where % i s t h e thema%
The P a r t i a l Molar
by Reiss e t a l .--
( W f o r a
expansion coefficient.
Gibbs f r e e energy of cavity creation was derived
system of hard spheres. They obtained
where t h e K ' s were evaluated t o be
The K's a r e functions of density, molar volume and diameter of t h e solvent
3Val N
through t h e compactness factor y , where y = ---
6v
. The radius of t h e
cavity created i s equivalent t o inserting a hard sphere of radius
a = (al + a ) / 2 (excluding t h e centers of solvent molecules), wheres,a2
12 2
i s t h e diameter of t h e cavity t o be created.
-
( 6The Gibbs Free Energy of interaction ( G ~ )was derived by P i e r o t t i in
the following fashion. The reversible work for t h e charging process i s

where e i s t h e molecular
i
volume and entropy. The
energy and ? 3. a r e t h e p a r t i a l molecular
i 1
P;. term f o r t h e change of s t a t e of t h e solute
1
at normal pressures i s considered negligible compared t o t h e energy of
t h e system and t h e entropy w i l l be small and negative i n t h e charging
process. This term w i l l then be assumed small, hence
If t h e i n t e r a c t i o n energy of a solute molecule with a given solvent mole-'
cule i s e , . ( r ) , then t h e i n t e r a c t i o n p e r mole of solute w i l l be E . Since
1 i
Ei
i s approximately H from equation 8, hence one can determine g. .i 1
Approximating t h e repulsive and dispersive interactions by a Lennard-
( 91J m e s 6-12 pairwise additive p o t e n t i a l - u(rIdi; -k12[(yr-(y)12]and
t h e inductive i n t e r a c t i o n f o r a nonpolar s o l u t e i n a polar solvent by
2 6~ ( r ) ~ ~ ~= U a /r , t h e i n t e r a c t i o n energy per solute molecule described
1 2
by inductive, dispersive and repulsive forces i s given by
e = -C
-6 6 -12 +6
i dis8 r p -5,rp ) - Cin&
P P
where r i s t h e distance from t h e center of t h e solute molecule t o t h e
P
center of t h e pth solvent molecule and a i s t h e distance of closest
1 2
approach of t h e solute and solvent molecule hard sphere diameters. The
Lennard-Jones force constant i s approximated f o r t h e interaction of two
molecules, v i z E = ( E ~ E ~ ) ' . U i s t h e dipole moment of t h e solvent
12 1
and a2 t h e p o l a r i z a b i l i t y of t h e solute. Evaluation of equation (10)
requires t h a t t h e solute molecule be immersed i n t h e solventand compIetely
surrounded by t h e solvent which i s assuined t o be i n f i n i t e i n extent.

40
The number of solvent molecules i n a spherical s h e l l between r and
2
( r + d r ) i s equal' t o g(r)41'rr pdr, where g ( r ) i s t h e r a d i a l distribution
function and p i s t h e number density of the solvent. Replacing t h e
with an integration i n ( l o ) , t h e - t o t a l interactxion energy yields
,
-6- 0 r -12) -c $ ,.r-614r;~pg (r)dr
1 2 9 ind vol p
The integration of t h i s ,energy-.requires information regarding t h e r a d i a l di'stril
bution function, which i s generally not known. A reasonable value can be
approximated from x-ray studies on liquids. Generally cdie assumes t h a t
g(r) = 1 outside t h e radius r , recognizing t h a t common solvents behave
" --s.
a s - ~ a gder Waals type f l u i d s . S u b s t ~ t u t i o nf o r t h e various constants
w-
yields
wher6 R .is the distance from t h e center of t h e solute molecule t o t h e centre
of the nearest solvent molecule, t h e integration yields
*
Introducing ci = Cirp/6012 and R' = ~ / o ~ ~
where R' i s found by d i f f e r e n t i a t i n g equation (14) and equating t o zero
t o obtain t h e minimum value of E. with variation of R'. The r e s u l t being
1

41
B *
Since E i s small compared t o
ind 'dis
then R' i s approximately unity.
Substitution f o r R' i n t o equation (14) and noting from equation (8) t h a t
-
Gi :E. yields t h e required equation
1
This equation i s i n exact agreementwith t h a t of P i e r o t t i ( 8 ) , noting t h a t
t h e dipolar interactions have been omitted since they a r e not required i n
2
t h i s t h e s i s . Also, C = U a f o r a nonpolar solute and a polar solvent
ind 1 2
6and Cdis = 4 ~ ~ ~ 0 ~ ~ ~ h ~ ~ ~ C S ~and 0 a r e t h e solute and solvent Lennard-Jones
2
parameters and o12 = (GI+ 02) . These values a r e defihed when t h e L. J.
2
6-12 potenkial i s zero. One then assumes that-Ol.and 02 'can be replaced
"
by alVand a2 .
The enthalpy of i n t e r a c t i o n defined i n equation (16) can be further
reduced by d i r e c t s u b s t i t u t i o n of t h e appropriate constants, hence
-16where K i s t h e Boltzman constant = 1.3805 x 10 erg OK-', R i s t h e gas
-1 0 -1
constant = 1.987 c a l mole K and p i s t h e number density of t h e solvent
which i s t h e only term t h a t i m p l i c i t l y accounts f o r ' d i f f e r e n t solvent
structures. U1 i s t h e dipole moment of t h e solvent i n esu-cm,
0 as previously defined has u n i t s of cm, t h e polarzability of t h e
12
solvent a has u n i t s cm3 molecule-l, and cl2/K, a s previously defined,
0
has u n i t s K.

42
The enthalpy of solution was defined i n equation (6) as t h e
derivative of in%, which e n t a i l s the derivatives of z. and the
1 c
Gibbs f r e e energy of interaction and cavity formation. It was assumed i n
-equation ( 9 ) t h a t ci = Hi, hence only the temperature derivative of a s
c '
-defined by equation ( 7 ) i s required. I n t h i s present derivation, the hard
sphere diameter of the solvent i s assumed t o be temperature dependent. A s
t h e temperature of t h e solvent molecules increases t h e root mean square
( r . m . s . ) kinetic energy increases, which allows the molecules t o penetrate.
It w i l l be shown l a t e r t h a t obtaining
from Henry's law coefficient shows a
1
the solvent hard sphere diameter al
t o decrease with increasing
temperature.
An effective hard sphere o r r i g i d sphere i s defined by t h e
potential energy of interaction with another molecule.
where r i s t h e separation between the centers of a solute and solvent
molecule of diameter al and a
2 '

- 43
To obtain B t h e derivative of i s required, !!%.&T = fi , which
C ) C a( ~ / R T ) C
e n t a i l s d i f f e r e n t i a t i n g a l l t h e K' s , equations (7a-7d). One can
s i m p l i e t h e problem with t h e following procedures. The substitution of
-~T/RT* f o r B ( ~ / R T ) . Hence
Next divide by RT and group t h e first s e t of terms i n K K and K2
C 0 , 1
together, then group t h e l a s t term i n K K and K with K and evaluate
0' 1 2 3
these groups separately.
Hence
3
4 'rfPNa12 - N'rfPal 3 2 ~ . r r ~ 2
znd Term 2 - + "12"1 -3
2mala12
Then l e t Z = 16y/(1-y)] + 18[y/(l-y)]2 and a = a + a
1 2 1 2 •÷
2
A = - - -a2 . The f i r s t and second term reduce t o
"1

The expression f o r t h e f i r s t term can now be further simplified
Resubstitution f o r Z y i e l d s
Inclusion of t h e second term y i e l d s t h e simplified expression f o r /RTC
The derivative of G involves the derivative of A and since both ?and a2
C
a r e assumed t o be temperature dependent and t h e derivative of A
yields A(k2-k1), where i?i i s introduced as t h e hard sphere temperature
dependence 1 / ~ i(dai/dT ) , then t h e following assumption

dA
i s made: --- 0. Typical values of -2 f o r organic solvents a r e(10)
dT 1
.13 t o .16 x ~ O - ~ S O K - ~and a t y p i c a l value f o r -!L2 f o r Argon (13) is
.16 x so , using t h e above values, t h e assumption i s valid.
Hence
Further simplification y i e l d s
3rial N
Since y = 7
The thermal expansion coefficient % i s defined a s .
Since one i s dealing with a hard sphere system, it seem;- appropriate t o

H.S.
introduce a a s a hard sphere (H.s. ) expansion c o e f f i c i e n t (a
P P 1*
~ u b s t i t u t i nboth t h e hard sphere temperature dependence coefficient !Ll
and t h e H.S. thermal expansion coefficient y i e l d s
Substitution f o r i n t o equation (28) y i e l d s
dT
~ % / R T - H.S. 2 3
- y(3g1-g 2 N~rPa2d~
) ( 1 - ~ ) - ~ 1 3 ( 1 + 2 y ) ~+ 3(1-y)A + ( I - ~ ) I + -(3Tk2-1 )
6 ~ ~ 2
Since = -RT
C
2d'c/RT
, one obtains t h e enthalpy of cavity formation a s
dT
The thermal expansion coefficient substituted i n t h e equation f o r t h e
enthalpy of cavity formation, Rc, was previously defined a s a hard sphere
term and, hence, should be derived from an equation of s t a t e f o r a system
of hard spheres. The equation of s t a t e f o r a hard sphere system derived
by Carnahan and S t a r l i n g(12,131 serves a s a s t a r t i n g point

Differentiating equation (32) f o r V yields
RSubstitution of equation (32) f o r p and RT- followed by division with V
P
yields

Further simplification yields
Substitution of equation (30) for *followed by suitable rearrangement
dT
yields
With sufficient manipulation, one obtains an expression for the thermal
expansion coefficient for a system of hard spheres
This is 'similar. to the expression obtained by Wilhelm(IL)presuming
that a typographic error exists in the latter half of equation (17)in
Wilhelm's(14)paper.
A good test lor any theory is the ability to predict the partial molar
volumes of the solute(15), which is defined as

Hence, from (2)
-therefore,
v2 = v.1 + 7C + RT (-L-)V d P T
where t h e isothermal compressibility i s defined as B = -
E)T V a p T ?
substitution f o r t h i s
-0, -
This equation c l e a r l y
term yields
shows t h a t t h e p a r t i a l molar volume of t h e solute i s
both a function of t h e cavity formation and solute solvent interaction upon
charging. Using equation (17), one can deduce q u i t e readily t h e ti term,
ap
. Since z.i s a function of t h e density o r t h e p a r t i a l molar
1
volume of t h e s o l u t e i n both terms of t h e equation, then
- - -
Gi
- (constants A - constants B) and
v
Substitution f o r B y i e l d s f o r a system of hard spheres
T
The determination of Vn involves t h e derivative of t h e more complex
C -
function . A good s t a r t i n g point i s t h e simplified version of GC'RT
C

from equation (27) where i s defined as
C
Hence
Substituting f o r = y(BT + 3 ~ ~ )and introducing Q a s ,t h e
1 a dl?1
solvent effective H.S. diameter pressure dependenee, one obtains with suitable
manipulation and collection of terms
Finally,
The isothermal compressibility term i s t h a t f o r a s y ~ t e m o f ~ ~ d s p h e r e
(11)
andas suchmus't bederived from t h e Carnahan-Starling equation o f . s t a t e . Using
H.S.
equation (32) as a s t a r t i n g point and t h e definition of B
T
one obtains
H.S. - -
-VBT - dP

, . Then
4
Rearrangement y i e l d s
Finally, t h e isothermal compressibility i s given by
where one recognizes t h a t t h e pressure P i s now f o r a system of hard spheres
and i s calculated from equation ( 3 2 ) : Typically for organic solvents t h e
pressure required t o maintain t h e syscem of hard spheres a t a constant

temperature and volume resembling t h a t of t h e r e a l f l u i d system i s
The p a r t i a l molar volume of t h e solute f o r t h e dissolution process
52
3
10 atm.
i s given by
One notes t h e s i m i l a r i t y between t h e equations (47) f o r vc and equation (31)
f o r fic' . Substitution for t h e bracketed term i n t with t h e equivalence of
C
t h a t term i n gc, y i e l d s f o r P2
Using t h e o r i g i n a l concept of t h e Scaled P a r t i c l e Theory, a s presented by
R e i s s e t a l . ( ) , t h e thermodynamic equations f o r t h e dissolution process and
t h e corresponding hard sphere parameters were derived incorporating an
equation of s t a t e f o r a system of hard spheres. Using these equations t h e
P i e r o t t i S.P.T. w i l l be examined and a pure hard sphere theory w i l l be
presented and considered a s an a l t e r n a t e method f o r t h e calculation of
t h e thermodynamic properties f o r t h e dissolution process.

Hard Sphere Limits 53
The dissolution process o f a gas i n t o a solvent i s now considered as a
four s t e p process: ( 1 ) t h e g a s s o l u t e is discharged t o t h a t o f a hard sphere
with no a t t r a c t i v e p o t e n t i a l ; (2) a hard sphere c a v i t y i s then made i n
t h e solvent o f s u i t a b l e s i z e t o accommodate t h e s o l u t e ; (3) t h e hard
sphere solute is placed i n t h e c a v i t y a t a constant pressure and
temperature; and (4) t h e s o l u t e is then recharged, thus allowing
solute-solvent i n t e r a c t i o n s . The thekmodynamic s t e p s considered i n
t h i s model a r e t h e r e v e r s i b l e work i n creating a c a v i t y i n a real
-
f l u i d a s given by t h e S.P.T. G c t e r m . The second is t h e solute-solvent
interaction i n t h e recharging process; t h i s can be w r i t t e n i n terms
of some potential of i n t e r a c t i o n and the r a d i a l d i s t r i b u t i o n function
o f t h e solvent and expressed a s Gi.
In t h i s section four cases o f t h e above model f o r t h e d i s s o l u t i o n
o f a gas i n a l i q u i d w i l l be considered. For Case I, only t h e
s o l u t e has been discharged t o t h a t of a hard sphere and t h e solvent
always r e t a i n s its normal p o t e n t i a l of i n t e r a c t i o n , even s o t h e solute-
solvent interaction i n forming t h e c a v i t y is s t i l l t h a t o f a p a i r of
hard spheres. A consequence o f l e t t i n g t h e solvent r e t a i n i t s r e a l
p o t e n t i a l of i n t e r a c t i o n is t h a t t h e up and BT values a r e those o f
t h e r e a l solvent and a temperature dependence on t h e solvent e f f e c t i v e
hard sphere diameter must be incorporated. Case I1 is a s
above, but excluding t h e temperature dependence on t h e solvent and s o l u t e
e f f e c t i v e hard sphere diameters, a s was assumed i n t h e o r i g i n a l model
used by P i e r o t t i .
For case I11 n o t only is t h e s o l u t e discharged t o t h a t of a hard
sphere, but s o is t h e solvent. A consequence of this is that one must use
pressures calculated from an equation o f s t a t e f o r a system of hard

5
spheres (typically 10 atm. for organic solvents) t o maintain the
system a t the volume and temperature of the r e a l system. Hence,
the a and B values w i l l now be those values calculated from an
P T
equation of s t a t e for a system of hard spheres. In t h i s particular
case the temperature dependence of the effective hard sphere diameter
of the solvent and solute w i l l be included. Finally, i n Case IV,
both the solvent and solute temperature dependence of the effective hard
sphere diameter w i l l be excluded. Hence, Case IV i s the hard sphere
equivalence of the original Pierotti Theory.
For t h e inert gases dissolved i n a given solvent, one i s able t o plot
any solubility thermodynamic property a s a function of the gas polarizability
and obtain a smooth curve. Extrapolation of t h i s curve t o zero
polarizability yields the value of that thermodynamic property for an
inert gas type hard sphere dissolved i n that particular solvent. Figure 1
i s a plot of the effective hard sphere diameter of the inert gases versus
their respective polarizabilities. A smooth extrapolation yields a value
of 2.55 8 for t h e effective hard sphere diameter of the inert gas type
hard sphere having no attraction potential, E /K=O, which defines the solute
2
hard sphere limiting case.
I f a similar-plot of the Henry's law constant (for the inert gases i n
a given solvent) against the solute polarizability i s extrapolated t o zero
polarizability, one obtains a value for the Henry's law constant 1"HO of
0
the hard sphere gas (a1=2. 55 A, E/K=O) i n that solvent. The extrapolated
ln%O value can be equated t o equation (5) t o obtain
l i m %= l n q = Z,/RT + l n (RT/V)
WO

FIG.1
INERTGASES
HARDSPHEREDIAMETER(A)vsPOLARIZABILITY(cm3/molec)

56
One n o t e s t h a t ei becomes zero when a =O. This is e a s i l y seen since
t h e inductive constant Cind, which is a function o f t h e s o l u t e
p o l a r i z a b i l i t y becomes zero. Also, s i n c e t h e a t t r a c t i v e w e l l depth
p o t e n t i a l f o r a hard sphere is zero (E2/K = 0 ) , then t h e dispersive
constant Cdis is zero. This hard sphere l i m i t value of lnKH was
o r i g i n a l l y used by P i e r o t t i t o e s t a b l i s h t h e e f f e c t i v e hard sphere
diameter value f o r various solvents a t 298.15 OK. This concept
was l a t e r used by ~ i l h e l m ( ~ O )a t other temperatures, thus
e s t a b l i s h i n g a f i n i t e temperature dependence f o r t h e e f f e c t i v e hard
-
sphere diameter of a wide range of solvents. Since t h e i n t e r a c t i o n
term i s zero f o r a hard sphere, one simply equates equation(57)to
t h e extrapolated lnKHO values obtained f o r t h e i n e r t gases i n a
given solvent over a temperature range, and solves f o r a a t t h e various
- 1
temperatures (incorporating t h e equation f o r G,). Using our data we
i l l u s t r a t e i n Figures 2 and 3 f o r H20 and D20 t h e method o f graphical
extrapolation t o o b t a i n lnKHO values over a temperature range. For
these graphs my experimental s o l u b i l i t y data was supplemented with Helium
D20. The curves were f i t t e d using a cubic polynomial, Table I
iglves t h e c o e f f i c i e n t s .

Table I
Coefficients i n t h e equation h K H = A + ~ c r+ ~ a 2+ m3
(1) Percentage standard deviation from experimental h K H values
f o r helium.
S e t t i n g t& p o l a r i z a b i l i t y t o zero i n t h e cubic polynomial f o r lnKH
y i e l d s t h e values a t a given temperature. Hence
Substituting t h e l n ~ ~ Ovalues i n equation (57) one o b t a i n s

Recognizing f o r t h e P i e r o t t i treatment t h a t P = 1 atm
0
a2 = 2.55 A , one is a b l e t o solve t h i s equation f o r a
e f f e c t i v e hard sphere diameter of the solvent.
and using
t h e solvent
Table I1 summarizes t h e extrapolated lnKHOvalues f o r a hard
sphere s o l u t e o f 2.55 1, and t h e respective hard sphere diameters
obtained from solving equation (59) f o r H20 and D20.
Solven
Table I1
~tEffective Hard Sphere Diameters
Using t h i s data one o b t a i n s a gl-value ( 4 = l / a l dal/dT) of
-.12 x 10-
3 oK-1
f o r H20 and -.42 x 1 0-3 oK-1 f o r D20.
It is d i f f i c u l t t o make a quantitative assessment of a l l t h e possible
sources of e r r o r involved with t h e determination of &-values. Wilhelm
estimated t h e e r r o r t o be '30% using the graphical extrapolation

b v a l u e obtained f o r D20 appears unusually high with respect t o
water and other solvents. This is d i r e c t l y a t t r i b u t a b l e t o t h e
I
lnKH values o f Helium and Neon suggesting t h a t t h e d a t a o f Abrosimov
(Ref. 17) is unreliable. A s can be seen from Figure 3, t h e s e values
deviate considerably from t h e smooth curve which is expected when compared
t o t h e H20 curves. It is i n t e r e s t i n g t o note, however, t h a t t h i s
method o f e x t r a p o l a t i i o n y i e l d s an e f f e c t i v e hard sphere diameter f o r
0
both H20 and D20 of 2.78 A a t 298.15 OK, which is t h e average
value used by most workers.
If one p l o t s t h e enthalpies of solution f o r t h e i n e r t gases i n a given
solvent versus t h e gases' p o l a r i z a b i l i t y and e x t r a p o l a t e s t o t h e hard
sphere l i m i t , one o b t a i n s AH0 f o r a hard sphere i n t h a t solvent.
-
Equating these hard sphere enthalpies and recognizing t h a t t h e Hi term
i n equation 6 is z e r o , one obtains
where is defined i n equation (31). Since y i s a function of
a l , which h a s previously been obtained from the lnKHO values,
equation (60) can be solved f o r !?, a t t h e hard sphere l i m i t . The
R-value obtained with t h e AH' extrapolation can be compared t o t h e
1-value obtained from t h e lnKHOvalues, thus producing a s e l f
consistent check f o r el. Using t h e o r i g i n a l concept o f t h e scaled
p a r t i c l e theory as i n Case I, apHbS*i n equation (30) i s replaced by
the experimental value of (+f o r a given solvent, and leaving k1
a s an undetermined v a r i a b l e , equation (60) can be solved t o obtain
t h e temperature dependence l / a da/dT of t h e solvent e f f e c t i v e hard

$ 3
sphere diameter. The following t a b l e
t h e el values obtained by 1 n ~ ~ Oaniimo
compares f o r selected solvents
extrapolations.
Table 111
Extrapolated Solvent Diameter Temperature Dependence
Included i n t h e above t a b l e a r e t h e AHO values obtained from equation (60)
with R1 having a value of zero a s i n t h e case of t h e o r i g i n a l P i e r o t t i
formulation, it is seen t h a t these values a r e nowhere near t h e values
obtained from t h e experimental extrapolations. One is amazed a t t h e
remarkable agreement between t h e temperature dependence o f t h e solvent
e f f e c t i v e hard sphere diameter obtained from t h e m o and l n ~ ~ Oextra-
polations. Whilst t h e r e was never any doubt as t o t h e temperature
dependence, t h i s points out unambiguously t h a t t h e o r i g i n a l P i e r o t t i
formulation should i n f a c t contain a term r e f l e c t i n g t h e temperature
dependence o f t h e solvent e f f e c t i v e hard sphere diameter.
An extrapolation of a p l o t o f the entropy of s o l u t i o n f o r t h e i n e r t
gases i n given solvent against t h e gasest respective p o l a r i z a b i l i t y
t o t h e hard sphere l i m i t w i l l y i e l d AS0 values. The values s o
obtained can be compared t o t h e t h e o r e t i c a l entropy given by
As = AH - AG = (As0 f o r a s o l u t e o f 2.55 8) (61)
T

The method o f obtaining As0 from equation (61) is r a t h e r f o r t u i t o u s
since from the assumptions i n t h e theory i t s e l f one is not
required t o d e l i b e r a t e l y s e t t h e Ei and Ei terms equal t o zero. he
enthalpy of s o l u t i o n AH is given by equation (6) and t h e Gibbs free
energy of solution AG i s given by
Hence t h e entropy of s o l u t i o n is
- -
So, using our previous assumption t h a t Hi=Gi, then
T
which is i d e n t i c a l t o equation (61). The ASO values obtained from
equation (65), when compared t o t h e hard sphere extrapolated values,
provide a good s e l f c o n s i s t e n t check f o r both t h e k1 and a, values
used f o r c a l c u l a t i o n s of t h e thermodynamic properties of the dissolution
process. The following t a b l e gives a summary of extrapolated
thermodynamic data using t h e hard sphere s o l u t e a s t h e l i m i t i n g case

for selected solvents.
Table I V
Extrapolated Thermodynamic Hard Sphere ~ a t aa t 298.15 OK
ASO AS' m0 mHO
( c a l ~ o l - IOK-') ( ~ s l ~ o l-1oK-1) - ( c a l ~ o l-1 oK-1) .,
SOLVENT Experimental Eqn. (65) , Case I Experimental ~ q n .(59)

Scaled P a r t f c l e Theory
The S.P.T., although severely c r i t i c i s e d , y i e l d s remarkably good Henry's
law constants f o r both organic solvents and water. This is r a t h e r
unexpected s i n c e t h i s theory t r e a t s H20 l i k e any o t h e r solvent and
does not take i n t o account any fundamental d i f f e r e n c e s (such a s those
due to H-bonding). The hard sphere equations derived i n t h e theory
s e c t i o n above (taking i n t o account temperature and pressure d e r i v a t i v e s
of t h e solvent e f f e c t i v e diameters) w i l l now b e shown equivalent t o t h e
equations obtained by P i e r o t t i .
The Henry's law constant f o r t h e d i s s o l u t i o n process can b e calculated
For Case I and 11, where P = 1 atm., t h e l a s t term i n t h e expression i s
n e g l i g i b l e ( 2 5 x f o r a t y p i c a l gas, Argon, a t 298 OK). I n
c a l c u l a t i n g t h e enthalpy of s o l u t i o n , A H, equation (31) describes t h e
contribution due t o c a v i t y formation as
.- 2 H.S. 2
Bc = yRT (% --3!L1)(1-y)-3~3(1+2y)~2+ 3(1-y)~+ (1-y) 1
Again f o r Case I and 11, the l a s t term i s small ( z .3 c a l mol-l f o r
argon a t 298 OK) and may b e assumed negligible. I n order t o obtain
-
t h e expression f o r Bc given by P i e r o t t i ,
a p hard sphere by t h e experimental value
2 EXP
8 = yRT ($
c - 3~,) 0-y)-~13(1+ 2 y b 2
i t i s necessary t o replace
-
of ap. We w i l l w r i t e Hc a s

and regard t h e o r i g i n a l P i e r o t t i formulation of t o be t h a t case f o r
which R1=O (though, a s w e have shown above, t h i s i s a poor approximation).
The partial molar volume f o r t h e dissolved s o l u t e is derived i n equation (56)
using hard sphere diameter temperature and pressure d e r i v a t i v e s . For
Case I, replacing hard sphere ap and BT values with experimental
-
values, V2 a t 1 atm. reduces t o
where Q1 was previously introduced a s t h e pressure
dependence on t h e e f f e c t i v e hard sphere diameter Q, = l / a l
and has been estimated t o be of the order 10-4 atm'l, an order
(18 >of magnitude less than t h e R1-values ..
I n order t o a s s i s t i n t h e computations of the thermodynamic properties
f o r t h e d i s s o l u t i o n process, a PLI program labeled 'Hard Sphere1 (see
Appendix I) was developed incorporating a l l t h e required equations f o r
Cases I and I11 where Cases I1 and I V a r e obtained by s e t t i n g
k1 = e2 = 0. Computer c a l c u l a t i o n s f o r t h e systems: i n e r t gases 02,
*2
CH4 and SF6 i n H20, D20, C6H6, CC14, n-C H
7 16'
C-C6~12, and ~ - c ~ H , ~over t h e temperature range 273.15 OK t o
323.15 OK, were c a r r i e d out t o find t h e thermodynamic p r o p e r t i e s
f o r t h e d i s s o l u t i o n process. Tables V and V I summarize a l l s o l u t e and
solvent parameters a t 298.15 OK required f o r t h e c a l c u l a t i o n s . For
convenience, only t h e 298.15 OK values a r e given f o r a p , BT and
d e n s i t i e s with t h e appropriate references containing o t h e r temperature
values.

Table V
P r o p e r t i e s of t h e s o l u t e s a t 298.15 OK
H.S . ~ i a m e t e r ( ~ )~ o l a r i z a b i l i t ~ ( ~ )L-J Force const .(a)
Hard Sphere 2 55 0 0
H e 2.63 .204 6.03
N e 2.78 .393 34.9
A r 3.40 1.63 122
K r 3 60 2 -46 171(b)
(a) Ref. 19, (b) Ref. 7 , (c) Ref. 20
The following t a b l e s present thermodynamic d a t a a t 298.15 OK f o r t h e
d i s s o l u t i o n process incorporating t h e temperaure dependence Case I
(R1 = l/al d a l / d ~ ) of t h e solvents a s previously determined
a t t h e hard sphere l i m i t .

*
'd
Y
8b
Y
ct
I-'*
(D
m
1m
a
ct
0
I-'.
e3I-'.
0
ZP
P
ct
0
ct
r0
m
(D
0
Y
X
IU
0
u J
IU I
0 0
CR
X
I-'
C
10
-4
a
n
(D
u
I-'
Co
-r
Co
vl
W
*
I
IU
IU
*
I
0
03
*
J7
-4
0
O
n
Y
w
P
v3
P
Co
n
0
V
I-'
P
0
J7
J7
vl
h
rw
vl
O
IU
0
u
P
4
I
-4
4
I
P
W
I-'
ch
IU
-4
n
rw
P
W
0
h
Y
V
CR
UI
Ln
0
n
P.
V
0
vl
03
O
I
-4
w
I
I-'
W
I-'
P
IU
vl

~ a h l eVII
H20 Case I
lnK13
Calc .
Hard Sphere 284 ---
( a ) Reference 16

Table VIII
GAS
Hard Sphere
He
N e
Ar
Kr
Xe
O2
N2
CH4
SF6
(a) Reference 16

Table IX
r ---
GAS
Hard Sphere
c-.C6H12 Case I
R1 = -.14 x 10-3 of1

Table X
n-C6~14 Case I
-3 o f 1R1 = -.19 ,X 10
GAS
Hard Sphere
He
N e
A r
K r
X e
O2
N2
CH4
SF6
One can s e e
AH
( c a l ~ o l - l )
Calc .
2238
1913
1461
1052
751
672
from the above tables t h a t the theory p r e d i c t s q u i t e well
t h e Henry's law constants and r e l a t i v e l y poor agreement i s obtained f o r
t h e enthalpies of s o l u t i o n a s one deviates f u r t h e r from the i n e r t gas
type hard sphere. As previously shown, t h e entropies of solution
provide a good s e l f consistency check f o r t h e R1-values. The
following t a b l e s compare calculated entropies f o r t h e d i s s o l u t i o n process
of t h e i n e r t gases jn selected solvents a t 298.15 OK.

Table XI
~ S ( e a lole-' OK-')R~O Case 1
Calc . Eqn. (65) EXP
Hard Sphere -23.15 ----
Hard Sphere
He
I
Ne
Ar
Kr
Xe
Table XI1
A S (cal ole-l ) . c ~ R ~
Calc. ~ q n .( 6 5 )
-10.14
Case I
0 '1
K

Table XI11
hS(cal ole-I O r 1 ) c-C6H12 Case I
a, = -.14 10-3 oK-1
Calc. Eqn. (65) - Exp.
Hard Sphere -10.09 ----
Table XIV
~ s ( c a lole-l OK-') n-C6H14 Case I
-3 0- -1
R 1 = - . 1 9 x 1 0 K
Calc. Eqn. ( 6 5 )
Hard Sphere -9 50
He -9 .36
Ne -9 .08
Ar -7.78
K r -7.30
Xe -6.11
From the above t a b l e s it is c l e a r t h a t the entropies a r e predicted
r e l a t i v e l y well f o r H20, w h i l s t t h e organic solvent entropies a r e
generally low and a l s o decrease i n value while t h e experimental values
increase.
It is i n t e r e s t i n g t o note t h a t the hard sphere entropies calculated from
equation (65) a r e i n good agreement with t h e hard sphere (extrapolated)

l i m i t o f t h e experimental entropy values, which again supports t h e concept
75
o f a temperature dependent hard sphere diameter.
I n l i g h t o f t h e r a t h e r poor agreement obtained f o r t h e e n t h a l p i e s and
e n t r o p i e s o f s o l u t i o n , t h e s e thermodynamic p r o p e r t i e s were r e c a l c u l a t e d
excluding t h e temperature dependence o f t h e e f f e c t i v e hard sphere
diameters. The following t a b l e s summarize t h e s e r e s u l t s f o r t h e i n e r t
gases i n t h e same s o l v e n t s f o r Case 11.
Hard Sphere
He
Ne
A r
m?:
X e
Hard Sphere
He
Ne
Table XV
H20 Case I1
AS
( c a l Mol-1 oK-l)
Eqn. (61)
-24.58
-24.78
-25.49
-28.88
-30.03"'.
-33.07
(cal Mol
-1 0~4)
Eqn. (61)
-13.00
-12.94
-12.87
-12.55
-12.43
-12.17

Table X V I I
~ " 6 ~ 1 2
AH
( c a l MOI-l)
Eqn. (6)
Hard Sphere 1807
H e 1391
N e 791
AX - 22
K r -499
X e -897
Case I1
AH
( c a l ~ol-'1
E ~ P
----
2421
1461
-2 18
-831
----
(a) Ref;7, (b) Ref. 19, (c) Ref. 35, (d) Ref. 36
Table X V I I I
n-C6H14 Case I1
Hard Sphere
H e
N e
A r
K r
AS
Eqn. (61)
-12.64
-12.60
-12.55
-12.26
-12.14
-11.83
Eqn. (61)
-12.16

One can see from t h e above Tables t h a t , f o r Case I1 (excluding a
temperature dependence on t h e solvent e f f e c t i v e H.S, diameter), t h e
predicted e n t h a l p i e s o f solution f o r t h e organic solvents a r e generally
poor when compared t o t h e appropriate experimental enthalpies. It is
not surprising, however, t o s e e good agreement between t h e predicted and
experimental e n t h a l p i e s f o r H20 using t h i s P i e r o t t i formulation. In
t h i s formulation t h e Lennard Jones force constant E1/K f o r H20 was
obtained by equating t h e experimental lnKH values f o r t h e i n e r t
-
gases t o t h e expression containing Gi and solving f o r t h e variable
- - -parameter E,/K. Since di = Hi and Hi is t h e dominant term i n t h e
enthalpy expression (one notes f o r He and Ne t h e -RT term cancels
-
t h e icterm, leaving Hi a s t h e dominant term), one expects good
-
agreement f o r t h e predicted enthalpies. For Argon and Krypton t h e Hc
values a r e 707 c a l ole-l and 797 c a l ole-l r e s p e c t i v e l y , while t h e
respective gi values a r e -2585 c a l ole-l and -3394 c a l ole-l.
Upon examination o f t h e entropies, one observes a l l t h e calculated
organic entropies t o be generally low and have decreasing values from
Helium t o Xenon, whilst t h e experimental values a c t u a l l y increase.
Figure 4 is a p l o t o f t h e calculated e n t r o p i e s f o r Cases I, I1 and I11
and t h e experimental entropies of the noble gases i n Benzene solvent versus
t h e gas p o l a r i z a b i l i t y . It is c l e a r from t h i s graph t h a t the remarkable
difference i n t h e calculated entropies observed f o r cases I and I1 is
d i r e c t l y a t t r i b u t a b l e t o t h e solvent e f f e c t i v e H.S. diameter temperature
dependence. One notes, however, t h a t inclusion o f an Rvalue (Case I)
enables t h e c a l c u l a t e d entropy t o have t h e same hard sphere limiting case
interception point a s was obtained with t h e experimental entropy values.

CASE l
CASE II
-15.0
-16.0
-17.0
-18.0
EXPERIMENTAL

- * - ' l a
0 CASE Ill
-
I I I I
1.O 2.0 3.0 4.0

79
A t t h i s p o i n t it i s i n t e r e s t i n g t o compare Case I and I1 with t h e
p u r e hard sphere treatment which incorporates pure hard sphere pressures
and clp and BT v a l u e s calculated from t h e Carnahan S t a r l i n g equation
of state. Table XIX summarizes t h e pure hard sphere thermodynamic
parameters used i n t h e calculations.
Table XIX
Hard Sphere P r o p e r t i e s a t 298.15 OK
H2•‹ gH6 ~ " 6 ~ 1 2 n-C6H4
7 .841 3.887 3.426 2.368

1.085 .737 .717 .770
-923 .386 .387 .332
-413 .564 .624 .968
*413 .5 64 .624 .968
Tables XX t o XXIV compare t h e hard sphere treatment using t h e
appropriate c a l c u l a t e d values from t h e equation of state f o r
c a s e s I11 and I V f o r t h e i n e r t gases i n H20, C6H6, C-C6H12,
and n-C6H14 f o r t h e d i s s o l u t i o n process a t 298.15 OK-

Table XX
H20 Case I11 and I V
-1 ' -1 , -
(cal ~ o l - l ) (cal Mol .) (cal Mol oK-lj' (cal Mol-1
3 oK-LR=- .08xl0- R=O ~=-.08~lo-3O K - ~ &o
Table XXI
C6H6 C a s e I11 and IV

T a b l e XXII
T a b l e X X I I I

82
Examination of t h e enthalpies f o r t h e d i s s o l u t i o n process using the har
sphere theory r e v e a l s t h a t the expected trend, when compared t o experimental
values, e x i s t s . For organic solvents, although t h e e n t h a l p i e s a r e
generally low, t h e values decrease from Helium t o Xenon, which is t h e
experimental trend. The e f f e c t of including a solvent e f f e c t i v e hard spher
diameter temperature dependence f o r t h e pure hard sphere theory is not a s
pronounced. One notes t h a t the enthalpies f o r H20 a r e predicted a l l
-
p o s i t i v e , which i s due i n p a r t t o t h e l a r g e p r e s s u r e t e r m i n H,.
Subsequently with t h e inclusion of t h e pressure term t h e value of is
-
p o s i t i v e and approximately twice a s l a r g e a s t h e Hi t e r m . (For Argon,
- -
Hc is 6697 c a l mole-' and Hi is -3394 c a l mole-l.) One f u r t h e r
notes t h a t t h e predicted lnKH values f o r H20 i n c r e a s e from Helium t o
Xenon while t h e experimental values decrease, w h i l s t t h e h K H values
f o r organic s o l v e n t s , although high, display t h e c o r r e c t trend. Examination
of t h e entropies r e v e a l s t h a t these values a r e ~ r e d i c t e dhigh f o r organic
s o l v e n t s and low f o r H20 but, unlike the previous cases, they increase
i n value from Helium t o Xenon l i k e t h e experimental values. Although
generally poor agreement i s obtained using t h e hard sphere theory, i t
i s i n t e r e s t i n g t o s e e t h a t t h i s theory t r e a t s H20 l i k e organic solvents
b u t , u n l i k e t h e o r i g i n a l formulation, has no p r e d i c t i v e success.

DISCUSSION
Plots, o f t h e experimental entropy and calculated entropy f o r t h e i n e r t
gases i n H20 and C6H6 versus t h e gas p o l a r i z a b i l i t y f o r Cases I, 11
and I11 are shown i n Figures 4 and 5. One'notes t h a t t h e experimental
entropy curve and t h e Case I entropy curve (with an "I,., value) i n t e r s e c t
a t t h e hard sphere l i m i t . A s one deviates from t h e hard sphere l i m i t , it
becomes apparent t h a t a r a t h e r l a r g e entropy associated with t h e charging
process is required t o bring agreement between t h e entropies from Case I
and t h e experimental entropies.
A s previously mentioned, one o f t h e inherent assumptions of the S.P.T.
-
i s t h a t fii was assumed t o be Gi, a consequence o f t h i s being t h a t t h e
calculated entropy is i n f a c t not a function o f t h e i n t e r a c t i o n term.
Examination o f equation (64) reveals t h a t t h e major contributors t o
t h e entropy a r e t h e cc and & terms. For Xe i n Benzene Case 11,
-
Hc 4800 c a l ole-l, inclusion o f a temperature dependence a s i n
Case I y i e l d s a value f o r Ec of 9400 c a l ole-l, hence contributing
an extra 15 c a l ole-l O K - ~ t o t h e entropy, which c r e a t e s t h e
l a r g e difference i n t h e calculated entropy curves f o r t h e i n e r t gases i n
Benzene (Figure 4 ) f o r Cases I and 11.
It is i n t e r e s t i n g t o note t h a t , f o r t h e o r i g i n a l P i e r o t t i formulation,
i n Case I1 t h a t t h e 8, term is almost i d e n t i c a l t o t h e cc term f o r t h e
i n e r t gases i n organic solvents. Typically, f o r Argon i n Benzene,
- -
Hc and Gc, have values of 3612 and 3659 c a l ole-' respectively.
Remembering t h a t Bi equals Ei, then the entropy term f o r t h e P i e r o t t i
formulation is e s s e n t i a l l y a function o f constants with t h e major contri-
bution being R ~ ~ C R T / V ~ ) .A s a r e s u l t t h e c a l c u l a t e d entropies when
plotted a g a i n s t t h e i n e r t gases ' p o l a r i z a b i l i t y a r e almost a straight
-
l i n e f o r organic solvents. For H20, though, t h e Hc values a r e much

FIG. 5
A S FOR INERT GASES
INH20 vs.
GAS POLARlZABlLlTY
CASE l
CASE II
EXPERIMENTAL

- 86
less t h a t t h e Gc values, and hence one o b t a i n s a l a r g e negative
contribution t o t h e entropy term through Gc r e s u l t i n g i n a curve which
looks s i m i l a r t o t h e experimental plot shown i n Figure 5 .
For Case 111 (with !L, and !$ values) ,' t y p i c a l l y f o r Argon i n Benzene
t h e and terms have values of 3281 and 4828 o a l ole-I
respectively; t h u s t h e major contribution t o t h e entropy is from these
terms. This is t y p i c a l f o r a l l organic solvents and presents a more
r e a l i s t i c method f o r calculating the entropy associated ~ 5 t ht h e
dissolution process. From these graphs it is a l s o c l e a r t h a t a large
-
entropy (Si) contribution is required t o bring agreement between the
calculated and experimental entropy values.
A t t h i s point it is i n s t r u c t i v e t o examine t h e magnitude of the .inter-
action term f o r t h e dissolution process. p l o t s of t h e experimental lnKH
values f o r t h e i n e r t gases i n a solvent, with c a l c u l a t e d (lnKH-Gi/RT)
values, versus t h e i n e r t gas p o l a r i z a b i l i t y y i e l d i n s i g h t i n t o t h e
- -
magnitude of t h e Gi term. Exclusion of t h e Gi/RT term from lnKH
f o r the i n e r t gases presents a !'pure hard sphere" theory, a theory
whereby one can include s o l u t e and solvent values i n t h e Ec term,
which allows one t o examine t h e interaction term f o r t h e r e a l system by
comparison with experimental lnKH values f o r t h e d i s s o l u t i o n process.
Figures 6 and 7 a r e such p l o t s f o r t h e i n e r t gases i n Benzene and H20
respectively. One notes t h a t the experimental lnKH values a r e
i d e n t i c a l t o t h e ~ ~ K ~ - ~ ~ / R Tvalues a t t h e hard sphere s o l u t e value
0
of 2.5 A. A s one deviates from the hard sphere l i m i t , c l e a r l y t h e
i n t e r a c t i o n term becomes increasingly more s i g n i f i c a n t . Typically
-
f o r Argon i n Benzene t h e 'Gi value required is approximately -3970
c a l ole-l a s compared t o a calculated value o f -3088 c a l ole-' t o
.
bring agreement between the hard sphere theory and experimental values.

FIG. 6
LNKH- GIRTFOR INERT
GASES IN BENZENE vs.
SOLUTE EFFECTIVE HARD
/SPHERE DIAMETER (i)
LNKH- GjIRT HARD SPHERE 0
LNKH EXPERIMENTAL
H.S. DIAMETER (i)

F I G . 7
LNKH- GIRT FOR INERT .
GASES IN H20 vs.
SOLUTE EFFECTIVE HARD
SPHERE DIAMETER
I
'
'
'.
'

' .
LNKH- CJRTHARD SPHERE 0
-
LNKH EXPERIMENTAL 't
I I I I
1.O 2.0 3.0 4.0
H.S. DIAMETER (i)

- 89
If .one p l o t s &--Hi and AH experimental f o r t h e i n e r t gases i n a
given solvent versus t h e s o l u t e e f f e c t i v e hard sphere diameter, one can
c l e a r l y see t h e magnitude of the enthalpy of i n t e r a c t i o n required t o
bring agreement between t h e calculated hard sphere e n t h a l p i e s and t h e
experimental enthalpies. Figures 8 and 9 are p l o t s o f t h e calculated
and experimental e n t h a l p i e s f o r t h e i n e r t gases i n H20 and Benzene
respectively. It is clear from these graphs t h a t inclusion o f a
solvent e f f e c t i v e hard sphere diameter temperature dependence does not
produce as pronounced a n e f f e c t as it did f o r t h e c a s e o f t h e o r i g i n a l
P i e r o t t i formulation (Tables V I I and VIII), although a l a r g e i n t e r a c t i o n
enthalpy i s required t o bring agreement with t h e experimentally observed
values. It is i n t e r e s t i n g , however, t o note t h a t t h e r e are no parameters
-i n t h e Hi term which account f o r t h i s difference.
Another method f o r t h e examination of t h e i n t e r a c t i o n term e n t a i l s
varying t h e solvent e f f e c t i v e hard sphere.diameter and using a hard sphere
0
s o l u t e value of 2.55 A, thus eliminating t h e Hi term. One notes t h a t ,
-
i n t h e expression f o r lnKH, t h e p a r t i a l molar volume V2 o f t h e
- -
solvent carries t h e s o l v e n t temperature dependence through Gc t o Hc
when one d i f f e r e n t i a t e s lnKH t o obtain A H. AS a r e s u l t an i n t e r e s t i n g
0
2.55 A, calculated from t h e expression f o r A H , and A H obtained from
t h e derivative o f lnKH ( equation (6)), as
a function o f t h e solvent e f f e c t i v e hard sphere diameter. Figure 10
c l e a r l y shows f o r t h i s p l o t o f AH values f o r t h e hard sphere s o l u t e . i n
Benzene an i n t e r s e c t i o n point e x i s t s f o r t h e two curves a t a solvent diameter
0
of 4.13 A when R1=R2=0 (Case IV). Also included on t h i s graph is t h e
same plot f o r AH values obtained by the two methods o f c a l c u l a t i o n using

F I G - 9 I*
AH - Hi FOR INERT GASES
IN C6H6 vs.
SOLUTE EFFECTIVE HARD /
SPHERE DIAMETER /O /.

AH - Hi HARD SPHERE4=0 I
-3 0 -1
0 A H - Hi HARD SPHERE4=-.15 x 10 K I

A A H EXPERIMENTAL IA
1.O 2.0 3.0 4.0
H.S. DIAMETER (1)

93 I
Case I (2-values) and t h e o r i g i n a l P i e r o t t i formulation, Case TI. Remembering
0
a t t h e hard sphere l i m i t with aI2g8.5.26 A t h e extrapolated experimental
kIOvalue was only obtained f o r Case I (see Table 111). It is a l s o
seen t h a t i n t h e c a l c u l a t i o n of from the inKH values, t h e
extrapolated experimental AH0 value is only obtained if t h e lnKH values
a r e calculated recognizing t h a t al is a function o f temperature. Clearly,
t h i s temperature dependence is required t o r a i s e t h e AH-& curves of
Case I1 t o those o f Case I ( a s shown i n Figure 10) t o obtain t h e c o r r e c t
values. One observes t h a t these s e t s of curves f o r both cases a r e almost
i d e n t i c a l and never i n t e r s e c t ; t h i s could i n f a c t be due t o t h e experi-
mental e r r o r associated with Benzene solvent values. A change i n t h e
O$ value of approximately 3 t o 5% w i l l bring agreement between these two
curves. This agreement does not eliminate t h e p o s s i b i l i t y of a solvent
e f f e c t i v e hard sphere temperature dependence, nor does it a f f e c t t h e
extrapolated AH0 values used i n t h e determination o f a solvent H.S.
temperature dependence.
Both o f t h e preceeding trends are a l s o observed f o r a l l organic solvents.
Figure 11 is t h e same p l o t o f the AH values c a l c u l a t e d by t h e two
methods, but t h i s time f o r Case 111, with t h e inclusion o f a hard sphere
s o l u t e and Benzene solvent e f f e c t i v e hard sphere diameter temperature
dependence. Also included on t h i s graph a r e t h e same p l o t s , but t h i s time
with a d i f f e r e n t s o l u t e e f f e c t i v e hard sphere diameter, 3.55 i. Clearly
comparison o f these two p l o t s shows t h a t t h e s o l u t e s i z e has no e f f e c t
0
on t h e i n t e r s e c t i o n value of 4.55 A obtained using t h e two methods of
c a l c u l a t i n g a ~values. One notes t h a t t h i s Benzene solvent e f f e c t i v e hard
0
sphere diameter o f 4.55 is d i f f e r e n t from t h e value of 4.13 A obtained .
without a s o l v e n t hard sphere diameter temperature dependence (Figure 10).

AH-HiFORHARDSPHERE
H.S.DIAMETER(i)

95
The significance o f these hard sphere diameter values is shown by examining
p l o t s of t h e c a l c u l a t e d thermal expansion c o e f f i c i e n t (with t l = O and
$=-.15 x 10-3 O K - ~ , eqn. (37)) a s a function o f solvent e f f e c t i v e
H.S. diameter. It can be seen from ~ i g u r e12 t h a t a solvent hard sphere
value of 4.13 (obtained from the intersection of t h e two AH curves)
calculated f o r t h e case when +=112=0, gives an $ value of
1.23 x 10'3 f o r Benzene. This value i s i d e n t i c a l t o t h e
experimental thermal expansion c o e f f i c i e n t f o r Benzene a t . 298.15 OK.
For a hard sphere value of 4.5 8 (obtained from t h e AH curves using t h e
appropriate L1 v a l u e s ) , t h e 4 curve calculated with el= -.15 x OK-',
again gives an ap value equal t o t h e Benzene experimental thermal
expansion c o e f f i c i e n t . Obtaining t h e experimental value of up f o r Eenzene
c l e a r l y suggests t h a t t h e experimental thermal expansion c o e f f i c i e n t
values a r e required t o obtain self consistency between AH and lnKH.
This trend i s observed f o r a l l organic solvents and, a s shown i n Figure 11,
the al value obtained from t h e intersection o f t h e two AH curves is
independent o f t h e s o l u t e hard sphere diameter. In c a l c u l a t i n g lnKH,
experimental values of solvent molar volumes a t a given temperature a r e
used. Hence, one must input these same experimental values i n t h e
expression f o r t h e AH term which includes t h e expansion coefficient. This
explains why i n t h e o r i g i n a l P i e r o t t i formulation (Case 11) t h a t comparison
of the AH values calculated from both t h e ln% and the AH expression
yielded similar values. This is t h e case f o r a l l solvents considered and
is d i r e c t l y a t t r i b u t a b l e t o t h e use of the experimental expansion c o e f f i c i e n t .
For Cases I11 and I V , however, t h e calculated hard sphere AH values f o r H20
using t h e two methods never i n t e r s e c t a t a r e a l i s t i c a l value.

F I G . 12
THERMAL EXPANSION COEFF.
AS A FUNCTION OF al VS.
BENZENE SOLVENT HARD SPHERE DIAMETER-
H.S. DIAMETER (W)

97
Typically f o r t h e hard sphere solute i n H20, a s one decreases t h e a l value
0
from 2.78 A t o 2.38 A , t h e values decrease from 2400 t o 1100
c a l ole-l , w h i l s t t h e AH values1calculated from lnKH decrease from 150 t o
-270 c a l ole-' and one never obtains t h e experimental up value.
Hence, the hard sphere w i l l never work for H20 s i n c e one cannot
obtain self consistency through t h e thermal expansion c o e f f i c i e n t s and,
as such, H20 cannot be t r e a t e d l i k e a t y p i c a l organic solvent; thus t h e
theory has no predictive success with H20. This is how t h e hard sphere
7.
theory d i f f e r e n t i a t e s organic solvents from H20, s i n c e self consistency
can be obtained f o r organic solvents. The n e t r e s u l t o f t h i s being t h a t
a smaller solvent e f f e c t i v e hard sphere diameter is required f o r organic
solvents e f f e c t i v e l y t o make t h e up value calculated from t h e
equation o f s t a t e t o be equal t o t h e experimental value. The following
t a b l e summarizes solvent e f f e c t i v e hard sphere diameter values required
t o bring agreement between t h e calculated and experimental thermal
expansion c o e f f i c i e n t s f o r selected solvents obtained by t h e calculation
of AH by t h e two methods previously described.

Solvent
Table XXIV
Solvent Effective Hard Sphere Diameters a t 298.15 OK.
Pure Hard Sphere Experimental
calculated (1) ('2>
Since exact agreement between AH calculated by t h e two methods is
obtained f o r a hard sphere solute i n H20 f o r Case 11, it is surprising
t h a t , upon inclusion o f t h e interaction term and r e a l s o l u t e values,
b e t t e r agreement is not obtained f o r both t h e AH values and lnKH
values when compared t o experimental values. Also inclusion of the
i n t e r a c t i o n term generally gives poor agreement f o r t h e i n e r t gases i n
organic solvents. One must remember t h a t t h e only other variable i n these
c a l c u l a t i o n s of both t h e lnKH and AH values is t h e i n t e r a c t i o n term,
which a f f e c t s both of these calculations i d e n t i c a l l y . A t t h i s point it
i s i n s t r u c t i v e t o examine a c t u a l values of t h i s i n t e r a c t i o n term and t o
-
check the v a l i d i t y of t h e assumption t h a t Ri = G ~ .
It was previously shown t h a t the entropy expression was a good t e s t
f o r both t h e a l and a, values used. This same expression was
derived with t h e assumption t h a t <i was equal t o gi, but subsequent
p l o t s f o r t h e entropy (Figures 4 and 5) revealed t h a t a l a r g e interaction
term Si was required t o bring agreement between t h e calculated and

99
experimental entropies. It appears that, f o r some of t h e cases
-
considered, the assumption regarding ii and Gi might be
incorrect. The v a l i d i t y of t h i s assumption can be determined by a
s e r i e s of s e l f consistency checks incorporating experimental values and
the derived equations of lnKH and &If o r t h e d i s s o l u t i o n process.
-
These values thus obtained f o r t h e Hi and Gi terms can be compared'
t o t h e o r i g i n a l expression f o r Ei.
Using t h e i n e r t gases i n a given solvent, t h e exper'imental lnKH values
can be equated t o equation (5) t o obtain Ei f o r t h e s e gases. These
values can be compared t o t h e Ri values obtained by equating the
experimental enthalpies of t h e i n e r t gases t o equation (6) f o r t h e same
solvent. These values can then be compared t o t h e values obtained
4
from the expression f o r Ei (equation(17)), thus determining t h e v a l i d i t y
-
of t h e o r i g i n a l assumption t h a t ~ ~ . j i ~ . G ~ .The following t a b l e s
summarize these r e s u l t s f o r t h e i n e r t gases i n H20 and C6H6
calculated f o r a l l four cases considered.
Table XXV

Table XXVI
C 6 ~ 6a t 298.15 OK Interaction terms for Case I and Case 11
Table X X V I I
H20 a t 298.15 OK Interaction terms for Case I11 and Case I V
Eqn. (5) Eqn. (6) Eqn. (6) Eqn. (17)
R1=O Li=-.O8x10-3
He -1 383 -2669 -3010 - 399

B. A. Cosgrove M.Sc. Thesis 1977

B. A. Cosgrove M.Sc. Thesis 1977

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B. A. Cosgrove M.Sc. Thesis 1977