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A novel osmotic pressure route to the activity coefficient of a molecule in a solution
27 October 1995
CHEMICAL PHYSICS LETTERS ELSEVIER
Chemical PhysicsLetters 245 (1995) 178-182
A novel osmotic pressure route to the activity coefficient of a molecule in a solution J.G. Powles a, S. Murad b, B. Holtz a a Physics Laboratory, University of Kent, Canterbury, Kent CT2 7NR, UK b Chemical Engineering Department, University of Illinois at Chicago, Chicago, IL 6060Z USA
Received 18 May 1995; in final form 21 August 1995
Abstract It is proposed that a powerful and convenient method of measuring the activity coefficient of a molecular species in a liquid mixture is to do an osmosis experiment. The only information required is the equation of state of the pure solvent. The procedure has been tested and verified by a modified GEMC simulation. The results are compared with van der Waals one-fluid theory, which is found to be remarkably good.
1. Introduction The chemical potential of molecules of type A in a mixture or solution, /xA, is defined by 0A
(1)
Several different meanings are given to the term activity coefficient. We consider the most useful to be that chosen by Rowlinson and Swinton [1] namely, /zA(P M, T, XA) = / ~ ° ( P M, T ) + k T In(XATA),
(2)
~'£A "-~ - ~ A T.V.Nx ~ ^ '
where A is the Helmholtz free energy and NA the number of molecules of A in the volume V at temperature T. It is an important quantity in the study and explanation of the properties of solutions and of pure liquids. However, in chemical and engineering applications one often uses instead a dimensionless quantity, TA, which is the activity coefficient of the A component in the mixture - it is defined below. It is therefore of considerable interest to be able to measure a n d / o r to calculate the value of 3'A for a given mixture under given conditions. We give below a method of measuring YA, and hence /~A, which is direct and straightforward and only requires some previous knowledge of the p u r e solvent, the liquid composed of component A only.
where/x A is the chemical potential of molecule A in the mixture at pressure PM and mole fraction x A. /z°A is the chemical potential of A molecules in a pure fluid of A molecules at the same pressure as that of the mixture. Among the advantages of this choice is that 3' is dimensionless. Moreover 3' = 1 for ideal mixtures the most important of which are mixtures of perfect gases and mixtures in which all molecules interact in exactly the same way but some of them, the solute molecules, are identified by an innocuous label. We shall encounter the latter type of mixture shortly and we just call it ideal. It is also easy to show that for any mixture if the molecule in question, A say, is in major abundance,
0009-2614/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0009-2614(95)00977-9
J.G. Powles et aL / Chemical Physics Letters 245 (1995) 178-182
i.e. x A ~ 1, then 7A ~ l. In particular, for any pure liquid A then 3'a - 1. It is to be noted that 3'A is a function of temperature, pressure on the mixture PM, and mole fraction of A in the mixture, x A, as indicated in Eq. (2).
2. Osmosis for y We propose that TA be determined by an osmosis and we shall illustrate the proposed procedure by a computer simulation of the experiment and the analysis. In the osmosis experiment we have two compartments, one containing the mixture and the other the pure solvent separated by a semi-permeable membrane, which allows the passage of the solvent molecules only. In osmotic equilibrium the chemical potential of A in the two compartments is equal in value but as a result of the osmotic flow, if any, the pressure in the pure A compartment, P, is different from that in the mixture compartment, PM, i.e.,
experiment
/XA( P•, T, x A) = / z ° ( P, T).
(3)
Combining (2) and (3) we have
1 3'A = - - e x p { - [
~z°( PM, T)
xA
-/z°( P, T)]/kr}.
(4)
/7 = PM -- P-
dP' _ H p(p,) Pp (l - ½HXT),
(9)
(5)
The quantity [/x°(PM, T) - / x ° ( P , T)] in (4) can be written
~o(/7+p)_~O(p)=f;
..~]rdP,"
where Xr is the isothermal compressibility of the solvent. Even more simply, and even more approximately,
(6)
3"A=
But for the pure liquid
NdI~=VdP-SdT,
We advocate the use of this relation and measurement a n d / o r simulation for the determination of 3'. It is attractive because, once P and /7 have been determined by experiment or simulation, the evaluation of 3"A requires only a knowledge of the appropriate part of the equation of state for the pure liquid A. A is usually the solvent (the major component of the mixture) and few different solvents are used in scientific and industrial processes, On the other hand one usually requires to know the value of /xA for mixtures of the solvent A with a large variety of solutes over a wide range of composition x A. Determining /XA(PM, T, x A) in (3) by any of the conventional methods is tedious and expensive, whereas the hardware for osmotic pressure is cheap and widely available. Even this direct method for y demands a knowledge of the pure solvent because of pO in (8). The exact relation (8) has not been used in this way as far as we are aware. The relation is no doubt well known as an intermediate result. But theoreticlans would feel obliged to obtain estimates of P and /7, not to mention the equation of state of the solvent, pO(p, T), which are all non-trivial undertakings. It is easy to show explicitly that (8) gives 3' -= 1 or 3'--* 1 for the special cases mentioned above, as required. For dilute mixtures a n d / o r rather incompressible solvents the integral in (8) simplifies,
f;+n
The osmotic pressure, /-/, is defined as
179
so
r
=
N
~
p
, (7)
so that (4) becomes
( °) 1
kTo(e)
/xA,
(10)
which corresponds essentially to van 't Hoff's approximation for the osmotic pressure [2], i.e.
/7=1%kr/V,
(11)
3 ' A ( P M , T, XA)
=
exp - ~ - ~ g p
p~(P',T)
"
(8)
where N a is the number of solute molecules in the volume V. We have tested and illustrated our proposed procedure, using Eq. (8), by Gibbs ensemble
J.G. Powles et al. /Chemical Physics Letters 245 (1995) 178-182
180
M o n t e C a r l o ( G E M C ) s i m u l a t i o n s and also u s i n g van d e r W a a l s o n e - f l u i d t h e o r y . O u r m o d i f i c a t i o n o f
e q u i l i b r i u m is d e s c r i b e d in detail e l s e w h e r e [4], It d i f f e r s f r o m the m e t h o d u s e d in Ref. [5], In o u r
conventional GEMC
s i m u l a t i o n s w e kept the t e m p e r a t u r e c o n s t a n t s u c h
[3] for d e t e r m i n i n g o s m o t i c
Table I Results of the simulations and the corresponding van der Waals one-fluid theory GEMC simulation xA
P
Hvn /I
~/^
vdW 1 fluid theory x^
P
/I
),^
(I) tra/trA = 0.981 0.961 0.943 0.924 0.888 0.852 0.817 0.783 0.751 0.658 0.517 0.391
1.0, eB/~A = 1.0 0.413 0.013 0.406 0.035 0.400 0.045 0.391 0.063 0.380 0.095 0.369 0.127 0.360 0.148 0.352 0.175 0.339 0.217 0.315 0.323 0.284 0.528 0.259 0.782
1.003 0,994 1,000 0.996 0.995 0.994 1,009 1.018 1,006 1.009 1.009 1.002
0.015 0.029 0.044 0.059 0.088 0.117 0.146 0.176 0.205 0.293 0.439 0.586
0.981 0.961 0.943 0.924 0.888 0.852 0.818 0.784 0.752 0.658 0.517 0.390
0.413 0.406 0.400 0.393 0.38 ! 0.369 0.358 0.348 0.339 0.314 0.283 0.260
0.015 0.029 0.044 0.059 0.090 0.120 0.152 0.184 0.217 0.32 ! 0.522 0.770
(2) trn/o"A = 0.988 0.980 0.961 0.922 0.883 0.844 0.766 0.683 0.600 0.493
1.25, Ca/e^ = 1.0 0.417 0.011 0.415 0.020 0.421 0.026 0.421 0.060 0.417 0.106 0.419 0.135 0.421 0.226 0,435 0.341 0.445 0.480 0.420 0.742
0.998 0.994 1.007 1.005 0.991 1.002 0.994 0.983 0.963 0.938
0.009 0.015 0.029 0.059 0.088 0.117 0.176 0.234 0.293 0.381
0.988 0.980 0.961 0,922 0.883 0.845 0.766 0.685 0.602 0.474
0.419 0.419 0.417 0.415 0.415 0,416 0.421 0.429 0.440 0.459
0.009 0.015 0.030 0.063 0.099 0.137 0.227 0.335 0.467 0.726
1.000 1.000 1.000 0.998 0.996 0.994 0.988 0,978 0.966 0.942
(3) tra/tr^ = 1.5, ~B/~A = 1.0 0.980 0.427 0.013 0.960 0.431 0.034 0.919 0.454 0.065 0.831 0.516 0.179 0.732 0.601 0.353 0.618 0.722 0.584 0.489 0.872 0.910 0.264 I. 143 1.786
1.004 0.997 1.004 0.978 0.926 0.872 0.831 0.727
0.015 0.029 0.059 0.117 0.176 0.234 0.293 0.38 !
0.980 0.960 0.920 0.833 0.736 0.627 0.503 0.291
0.425 0.432 0.449 0.499 0.575 0.677 0.806 1.037
0.015 0.032 0.072 0.178 0.333 0.562 0.908 1.867
0.999 0.998 0.993 0.971 0.934 0.878 0.798 0.612
(4) o'a/o"k = 1.5, EB/~A = 4.0 0,981 0.378 0.005 0,964 0.357 0.016 0,930 0.328 -0.005 0.861 0.324 - 0.047 0.784 0.348 - 0.008 0.692 0.405 - 0.060 0.582 0.501 0.430 0.355 0.802 1.300
1.014 1.015 1.081 1.251 1.289 1.571 1.059 0.793
0.015 0.029 0.059 0. I 17 0.176 0.234 0.293 0.381
0.981 0.964 0.931 0.864 0.790 0.705 0.604 0.450
0.376 0.348 0.319 0.310 0.328 0.367 0.434 0.516
0.006 0.001 -0.022 - 0.068 - 0.079 - 0.002 0.614 0.538
1.010 1.035 1.113 1.304 1.447 1.422 0.823
P, H and H,,n are pressures in units of ~A/O'~. X and y are dimensionless.
J.G. Powles et al. / Chemical Physics Letters 245 (1995) 178-182 1.6
.
.
.
. ~ t
1.5
.
/ \
..... f. 1.4
/"
/
1 - ideal ¢aae, ¢ g / a A = cb./e,i = I
\ ~,,
2 - os/#A= t.~ ee/eA= 1.0 \
3 - rts/aA = 1.5. ~a/e,4 = 1.0
1.2 1.1
"rA
1.0 ~
~
-
-
-
-
41
0.9 0.8
"''-.
0.7 0.6
d.1
o
d.2
d.3
dA
d.s
d.s
3 "'""
d.7
.s
XB
Fig. I. The activity coefficient, as defined in Eq. (2), as a function of the solute molar concentration, x n, for four different binary mixtures, one ideal and three of increasing non-ideality, for Lennard-Jones 12-6 intermolecular interactions with LorentzBerthelot combining rules. The points, <>, are for modified GEMC simulations and the points + are for vdWl-fluid theory of mixtures. The temperature is fixed at 1.5e A / k . The Johnson et al. [6] equation of state is used in both methods.
that kT/E A = 1.5. The intermolecular interactions were of Lennard-Jones 12-6 form,
[/oral2
~(r)=4e[tr)
-(-~)
61 .
(12)
For the solvent (cr, e) is (trA, CA), for the solute it is (tr B, e B) and for the interactions between them it is (½(trA + c%), (eAeB)l/2). We give here results for four cases and they are shown in Table 1 and Fig. 1. All quantities are in dimensionless form using the parameters trA and eA. We compare our virtually exact simulation results, where for the evaluation of the integral in (8) we use the virtually exact equation of state of Johnson et al. [6], with the simplest mixture theory, van der Waais one-fluid theory [7] in which the mixture is approximated by an effective pure fluid, x, with
o~3= E E x , xjcri~ and i i
with the vdW1 theory. The small differences from unity in Table 1 and Fig. 1 are due to statistical fluctuations in the simulation and small uncertainties in the equation of state. This is true even for the most concentrated mixture, for which x B ( = 1 - x A) = 0.609. Notice that the van 't Hoff equation (11) is only good at low concentrations of solute even for ideal mixtures (see Ref. [4] for details). In example 2 the diameter ratio is changed to 1.25, but this is a substantial departure from ideality, since the effect surely depends more on the relative 'volumes', i.e. on (trB/trA) 3, which is = 2 here. In this case 3'A falls below unity with increasing x B but the v d W l values are remarkably good. Example 3 is for trB/crA = 1.5, i.e. (trB/o'A) 3 = 3.4, which is surely a very non-ideal mixture. Indeed 3' falls even further below unity than for example 2. But vdWl is quite good except for the most concentrated mixture we investigated ( x a = 0.736!). In example 4 we made the solute both much larger (O'B/tr A = 1.5) and much more self-attractive ( e B / e A = 4 ) , which is a grossly non-ideal mixture, in fact, the analogue of such a mixture could probably not be made in a real experiment. Nevertheless we were able to get a stable simulation and the required equality of the chemical potentials in the two phases. Generally for ~B/~A > 1, TA increases (see Table 1 and elsewhere [4]) but, as we have seen already, for trB/O"A > 1, 3'A decreases. The competition between the two effects is seen in the figure when at low concentration 3' increases but eventually falls back through unity. Once again the vdWl theory does remarkably well in this case in which it is severely tested. Notice that for low concentrations of solute 3' always tends to unity, as expected, the more slowly the less ideal the mixture.
3. Conclusion
j
e.~cr~3 = E Exix)eij~;~ •
181
(13)
j
We report on four mixtures. The first is an ideal mixture, o-B/o-a = EB/~ a = 1, where the innocuous label is the index number of the particle in the computer. In this case 3' = 1, as it is indeed even
We have demonstrated that the osmotic pressure method of obtaining the activity coefficient, as specified in Eq. (8), is a convenient and accurate route in the usual situation that the equation of state of the pure solvent is available. We hope our proposal commends itself to the experimentalist.
182
J.G. Powles et al. / Chemical Physics Letters 245 (1995) 178-182
Acknowledgement BH was supported by EPSRC Research Grant GR/K/06471. SM acknowledges grants from the US Department of Energy No. FG 02-87ER-13769 and the Petroleum Research Fund. Computing services were provided by the University of Illinois and by the Physics Department of the University of Kent. Discussions were facilitated by grants from NATO (CRG910040) and the US National Science Foundation (INT-9123242). The use of Eq. (8) for getting 3' was also suggested independently in correspondence with Professor J.M. Halle (Clemson University, USA). We thank Dr. W.A.B. Evans for helpful discussions.
References [1] J.S. Rowlinson and F.L. Swinton, Liquids and liquidixtures, 3th Ed. (Oxford Science Publication, Butterworth, 1982) s. 4.4. [2] J.H. van 't Hoff, Z. Physik. Chem. I (1887) 481. [3] A.Z. Panagiotopouios, Mol. Phys. 61 (1987) 813. [4] S. Murad, J.G. Powles and B. Holtz, Mol. Phys., in press. [5] A.Z. Panagiotopouios, N. Quirke, M. Stapelton and D.J. Tildesley, Mol. Phys. 63 (1988) 527. [6] J.K. Johnson, J.A. Zollweg and K.E. Gubbins, Mol. Phys. 78 (1993) 591. [7] T.W. Leland, J.S. Rowlinson and G.A. Sather, Trans. Faraday Soc. 64 (1968) 1447.
CHEMICAL PHYSICS LETTERS ELSEVIER
Chemical PhysicsLetters 245 (1995) 178-182
A novel osmotic pressure route to the activity coefficient of a molecule in a solution J.G. Powles a, S. Murad b, B. Holtz a a Physics Laboratory, University of Kent, Canterbury, Kent CT2 7NR, UK b Chemical Engineering Department, University of Illinois at Chicago, Chicago, IL 6060Z USA
Received 18 May 1995; in final form 21 August 1995
Abstract It is proposed that a powerful and convenient method of measuring the activity coefficient of a molecular species in a liquid mixture is to do an osmosis experiment. The only information required is the equation of state of the pure solvent. The procedure has been tested and verified by a modified GEMC simulation. The results are compared with van der Waals one-fluid theory, which is found to be remarkably good.
1. Introduction The chemical potential of molecules of type A in a mixture or solution, /xA, is defined by 0A
(1)
Several different meanings are given to the term activity coefficient. We consider the most useful to be that chosen by Rowlinson and Swinton [1] namely, /zA(P M, T, XA) = / ~ ° ( P M, T ) + k T In(XATA),
(2)
~'£A "-~ - ~ A T.V.Nx ~ ^ '
where A is the Helmholtz free energy and NA the number of molecules of A in the volume V at temperature T. It is an important quantity in the study and explanation of the properties of solutions and of pure liquids. However, in chemical and engineering applications one often uses instead a dimensionless quantity, TA, which is the activity coefficient of the A component in the mixture - it is defined below. It is therefore of considerable interest to be able to measure a n d / o r to calculate the value of 3'A for a given mixture under given conditions. We give below a method of measuring YA, and hence /~A, which is direct and straightforward and only requires some previous knowledge of the p u r e solvent, the liquid composed of component A only.
where/x A is the chemical potential of molecule A in the mixture at pressure PM and mole fraction x A. /z°A is the chemical potential of A molecules in a pure fluid of A molecules at the same pressure as that of the mixture. Among the advantages of this choice is that 3' is dimensionless. Moreover 3' = 1 for ideal mixtures the most important of which are mixtures of perfect gases and mixtures in which all molecules interact in exactly the same way but some of them, the solute molecules, are identified by an innocuous label. We shall encounter the latter type of mixture shortly and we just call it ideal. It is also easy to show that for any mixture if the molecule in question, A say, is in major abundance,
0009-2614/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0009-2614(95)00977-9
J.G. Powles et aL / Chemical Physics Letters 245 (1995) 178-182
i.e. x A ~ 1, then 7A ~ l. In particular, for any pure liquid A then 3'a - 1. It is to be noted that 3'A is a function of temperature, pressure on the mixture PM, and mole fraction of A in the mixture, x A, as indicated in Eq. (2).
2. Osmosis for y We propose that TA be determined by an osmosis and we shall illustrate the proposed procedure by a computer simulation of the experiment and the analysis. In the osmosis experiment we have two compartments, one containing the mixture and the other the pure solvent separated by a semi-permeable membrane, which allows the passage of the solvent molecules only. In osmotic equilibrium the chemical potential of A in the two compartments is equal in value but as a result of the osmotic flow, if any, the pressure in the pure A compartment, P, is different from that in the mixture compartment, PM, i.e.,
experiment
/XA( P•, T, x A) = / z ° ( P, T).
(3)
Combining (2) and (3) we have
1 3'A = - - e x p { - [
~z°( PM, T)
xA
-/z°( P, T)]/kr}.
(4)
/7 = PM -- P-
dP' _ H p(p,) Pp (l - ½HXT),
(9)
(5)
The quantity [/x°(PM, T) - / x ° ( P , T)] in (4) can be written
~o(/7+p)_~O(p)=f;
..~]rdP,"
where Xr is the isothermal compressibility of the solvent. Even more simply, and even more approximately,
(6)
3"A=
But for the pure liquid
NdI~=VdP-SdT,
We advocate the use of this relation and measurement a n d / o r simulation for the determination of 3'. It is attractive because, once P and /7 have been determined by experiment or simulation, the evaluation of 3"A requires only a knowledge of the appropriate part of the equation of state for the pure liquid A. A is usually the solvent (the major component of the mixture) and few different solvents are used in scientific and industrial processes, On the other hand one usually requires to know the value of /xA for mixtures of the solvent A with a large variety of solutes over a wide range of composition x A. Determining /XA(PM, T, x A) in (3) by any of the conventional methods is tedious and expensive, whereas the hardware for osmotic pressure is cheap and widely available. Even this direct method for y demands a knowledge of the pure solvent because of pO in (8). The exact relation (8) has not been used in this way as far as we are aware. The relation is no doubt well known as an intermediate result. But theoreticlans would feel obliged to obtain estimates of P and /7, not to mention the equation of state of the solvent, pO(p, T), which are all non-trivial undertakings. It is easy to show explicitly that (8) gives 3' -= 1 or 3'--* 1 for the special cases mentioned above, as required. For dilute mixtures a n d / o r rather incompressible solvents the integral in (8) simplifies,
f;+n
The osmotic pressure, /-/, is defined as
179
so
r
=
N
~
p
, (7)
so that (4) becomes
( °) 1
kTo(e)
/xA,
(10)
which corresponds essentially to van 't Hoff's approximation for the osmotic pressure [2], i.e.
/7=1%kr/V,
(11)
3 ' A ( P M , T, XA)
=
exp - ~ - ~ g p
p~(P',T)
"
(8)
where N a is the number of solute molecules in the volume V. We have tested and illustrated our proposed procedure, using Eq. (8), by Gibbs ensemble
J.G. Powles et al. /Chemical Physics Letters 245 (1995) 178-182
180
M o n t e C a r l o ( G E M C ) s i m u l a t i o n s and also u s i n g van d e r W a a l s o n e - f l u i d t h e o r y . O u r m o d i f i c a t i o n o f
e q u i l i b r i u m is d e s c r i b e d in detail e l s e w h e r e [4], It d i f f e r s f r o m the m e t h o d u s e d in Ref. [5], In o u r
conventional GEMC
s i m u l a t i o n s w e kept the t e m p e r a t u r e c o n s t a n t s u c h
[3] for d e t e r m i n i n g o s m o t i c
Table I Results of the simulations and the corresponding van der Waals one-fluid theory GEMC simulation xA
P
Hvn /I
~/^
vdW 1 fluid theory x^
P
/I
),^
(I) tra/trA = 0.981 0.961 0.943 0.924 0.888 0.852 0.817 0.783 0.751 0.658 0.517 0.391
1.0, eB/~A = 1.0 0.413 0.013 0.406 0.035 0.400 0.045 0.391 0.063 0.380 0.095 0.369 0.127 0.360 0.148 0.352 0.175 0.339 0.217 0.315 0.323 0.284 0.528 0.259 0.782
1.003 0,994 1,000 0.996 0.995 0.994 1,009 1.018 1,006 1.009 1.009 1.002
0.015 0.029 0.044 0.059 0.088 0.117 0.146 0.176 0.205 0.293 0.439 0.586
0.981 0.961 0.943 0.924 0.888 0.852 0.818 0.784 0.752 0.658 0.517 0.390
0.413 0.406 0.400 0.393 0.38 ! 0.369 0.358 0.348 0.339 0.314 0.283 0.260
0.015 0.029 0.044 0.059 0.090 0.120 0.152 0.184 0.217 0.32 ! 0.522 0.770
(2) trn/o"A = 0.988 0.980 0.961 0.922 0.883 0.844 0.766 0.683 0.600 0.493
1.25, Ca/e^ = 1.0 0.417 0.011 0.415 0.020 0.421 0.026 0.421 0.060 0.417 0.106 0.419 0.135 0.421 0.226 0,435 0.341 0.445 0.480 0.420 0.742
0.998 0.994 1.007 1.005 0.991 1.002 0.994 0.983 0.963 0.938
0.009 0.015 0.029 0.059 0.088 0.117 0.176 0.234 0.293 0.381
0.988 0.980 0.961 0,922 0.883 0.845 0.766 0.685 0.602 0.474
0.419 0.419 0.417 0.415 0.415 0,416 0.421 0.429 0.440 0.459
0.009 0.015 0.030 0.063 0.099 0.137 0.227 0.335 0.467 0.726
1.000 1.000 1.000 0.998 0.996 0.994 0.988 0,978 0.966 0.942
(3) tra/tr^ = 1.5, ~B/~A = 1.0 0.980 0.427 0.013 0.960 0.431 0.034 0.919 0.454 0.065 0.831 0.516 0.179 0.732 0.601 0.353 0.618 0.722 0.584 0.489 0.872 0.910 0.264 I. 143 1.786
1.004 0.997 1.004 0.978 0.926 0.872 0.831 0.727
0.015 0.029 0.059 0.117 0.176 0.234 0.293 0.38 !
0.980 0.960 0.920 0.833 0.736 0.627 0.503 0.291
0.425 0.432 0.449 0.499 0.575 0.677 0.806 1.037
0.015 0.032 0.072 0.178 0.333 0.562 0.908 1.867
0.999 0.998 0.993 0.971 0.934 0.878 0.798 0.612
(4) o'a/o"k = 1.5, EB/~A = 4.0 0,981 0.378 0.005 0,964 0.357 0.016 0,930 0.328 -0.005 0.861 0.324 - 0.047 0.784 0.348 - 0.008 0.692 0.405 - 0.060 0.582 0.501 0.430 0.355 0.802 1.300
1.014 1.015 1.081 1.251 1.289 1.571 1.059 0.793
0.015 0.029 0.059 0. I 17 0.176 0.234 0.293 0.381
0.981 0.964 0.931 0.864 0.790 0.705 0.604 0.450
0.376 0.348 0.319 0.310 0.328 0.367 0.434 0.516
0.006 0.001 -0.022 - 0.068 - 0.079 - 0.002 0.614 0.538
1.010 1.035 1.113 1.304 1.447 1.422 0.823
P, H and H,,n are pressures in units of ~A/O'~. X and y are dimensionless.
J.G. Powles et al. / Chemical Physics Letters 245 (1995) 178-182 1.6
.
.
.
. ~ t
1.5
.
/ \
..... f. 1.4
/"
/
1 - ideal ¢aae, ¢ g / a A = cb./e,i = I
\ ~,,
2 - os/#A= t.~ ee/eA= 1.0 \
3 - rts/aA = 1.5. ~a/e,4 = 1.0
1.2 1.1
"rA
1.0 ~
~
-
-
-
-
41
0.9 0.8
"''-.
0.7 0.6
d.1
o
d.2
d.3
dA
d.s
d.s
3 "'""
d.7
.s
XB
Fig. I. The activity coefficient, as defined in Eq. (2), as a function of the solute molar concentration, x n, for four different binary mixtures, one ideal and three of increasing non-ideality, for Lennard-Jones 12-6 intermolecular interactions with LorentzBerthelot combining rules. The points, <>, are for modified GEMC simulations and the points + are for vdWl-fluid theory of mixtures. The temperature is fixed at 1.5e A / k . The Johnson et al. [6] equation of state is used in both methods.
that kT/E A = 1.5. The intermolecular interactions were of Lennard-Jones 12-6 form,
[/oral2
~(r)=4e[tr)
-(-~)
61 .
(12)
For the solvent (cr, e) is (trA, CA), for the solute it is (tr B, e B) and for the interactions between them it is (½(trA + c%), (eAeB)l/2). We give here results for four cases and they are shown in Table 1 and Fig. 1. All quantities are in dimensionless form using the parameters trA and eA. We compare our virtually exact simulation results, where for the evaluation of the integral in (8) we use the virtually exact equation of state of Johnson et al. [6], with the simplest mixture theory, van der Waais one-fluid theory [7] in which the mixture is approximated by an effective pure fluid, x, with
o~3= E E x , xjcri~ and i i
with the vdW1 theory. The small differences from unity in Table 1 and Fig. 1 are due to statistical fluctuations in the simulation and small uncertainties in the equation of state. This is true even for the most concentrated mixture, for which x B ( = 1 - x A) = 0.609. Notice that the van 't Hoff equation (11) is only good at low concentrations of solute even for ideal mixtures (see Ref. [4] for details). In example 2 the diameter ratio is changed to 1.25, but this is a substantial departure from ideality, since the effect surely depends more on the relative 'volumes', i.e. on (trB/trA) 3, which is = 2 here. In this case 3'A falls below unity with increasing x B but the v d W l values are remarkably good. Example 3 is for trB/crA = 1.5, i.e. (trB/o'A) 3 = 3.4, which is surely a very non-ideal mixture. Indeed 3' falls even further below unity than for example 2. But vdWl is quite good except for the most concentrated mixture we investigated ( x a = 0.736!). In example 4 we made the solute both much larger (O'B/tr A = 1.5) and much more self-attractive ( e B / e A = 4 ) , which is a grossly non-ideal mixture, in fact, the analogue of such a mixture could probably not be made in a real experiment. Nevertheless we were able to get a stable simulation and the required equality of the chemical potentials in the two phases. Generally for ~B/~A > 1, TA increases (see Table 1 and elsewhere [4]) but, as we have seen already, for trB/O"A > 1, 3'A decreases. The competition between the two effects is seen in the figure when at low concentration 3' increases but eventually falls back through unity. Once again the vdWl theory does remarkably well in this case in which it is severely tested. Notice that for low concentrations of solute 3' always tends to unity, as expected, the more slowly the less ideal the mixture.
3. Conclusion
j
e.~cr~3 = E Exix)eij~;~ •
181
(13)
j
We report on four mixtures. The first is an ideal mixture, o-B/o-a = EB/~ a = 1, where the innocuous label is the index number of the particle in the computer. In this case 3' = 1, as it is indeed even
We have demonstrated that the osmotic pressure method of obtaining the activity coefficient, as specified in Eq. (8), is a convenient and accurate route in the usual situation that the equation of state of the pure solvent is available. We hope our proposal commends itself to the experimentalist.
182
J.G. Powles et al. / Chemical Physics Letters 245 (1995) 178-182
Acknowledgement BH was supported by EPSRC Research Grant GR/K/06471. SM acknowledges grants from the US Department of Energy No. FG 02-87ER-13769 and the Petroleum Research Fund. Computing services were provided by the University of Illinois and by the Physics Department of the University of Kent. Discussions were facilitated by grants from NATO (CRG910040) and the US National Science Foundation (INT-9123242). The use of Eq. (8) for getting 3' was also suggested independently in correspondence with Professor J.M. Halle (Clemson University, USA). We thank Dr. W.A.B. Evans for helpful discussions.
References [1] J.S. Rowlinson and F.L. Swinton, Liquids and liquidixtures, 3th Ed. (Oxford Science Publication, Butterworth, 1982) s. 4.4. [2] J.H. van 't Hoff, Z. Physik. Chem. I (1887) 481. [3] A.Z. Panagiotopouios, Mol. Phys. 61 (1987) 813. [4] S. Murad, J.G. Powles and B. Holtz, Mol. Phys., in press. [5] A.Z. Panagiotopouios, N. Quirke, M. Stapelton and D.J. Tildesley, Mol. Phys. 63 (1988) 527. [6] J.K. Johnson, J.A. Zollweg and K.E. Gubbins, Mol. Phys. 78 (1993) 591. [7] T.W. Leland, J.S. Rowlinson and G.A. Sather, Trans. Faraday Soc. 64 (1968) 1447.