Liquid-liquid equilibria in binary
(2-methoxyethanol + alkane) systems
at pressures up to 4000 bar
H. B i j l
Liquid-liquid equilibria in binary
(2-methoxyethanol + alkane) systems
at pressures up to 4000 bar
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INJ 00 » - 'J1 13 j\ -J no -f-BIBLIOTHEEK TU Delft P 1774 4252Liquid-liquid equilibria in binary
(2-methoxyethanol + alkane) systems
at pressures up to 4000 bar
PROEFSCHRIFT ter verkrijging van
de graad van doctor in de
technische wetenschappen
aan de Technische Hogeschool Delft,
op gezag van de Rector Magnificus,
prof.ir B.P. Th. Veltman,
in het openbaar te verdedigen
ten overstaan van het College van Dekanen
op dinsdag 27 maart 1984
te 14.00 uur door
HUGO BIJL,
geboren te Delft,
scheikundig ingenieur
Dit proefschrift is goedgekeurd door de promotor
Prof.dr. R.N. Lichtenthaler
CONTENTS
SAMENVATTING 1 Chapter 1 PREFACE AND OBJECTIVES 5
Chapter 2 LIQUID-LIQUID EQUILIBRIA AT HIGH PRESSURES 7
1 Liquid-liquid demixing phenomena 7 2 Thermodynamic description of liquid-liquid equilibria
and critical points 10 3 Measuring methods 13
4 Literature 14
Chapter 3 THERMODYNAMIC MODELS FOR THE DESCRIPTION OF LIQUID-LIQUID
EQUILIBRIA 15
1 Introduction 15
2 Excess Qibbs energy models 16
2.1 The van Laar equation 16 2.2 The NRTL equation 17 2 . 3 The Uniquac equation 18 2.4 The Muniquac equation 22
3 Helmholtz energy model 24
4 Literature 27
Chapter 4 EXPERIMENTAL EQUIPMENT 29
1 The autoclave 29
2 The autoclave set up 29
2 . 2 Pressure c o n t r o l and measurement
3 Literature
32 33 Chapter 5 EXPERIMENTAL BINARY [2-METH0XYETHAN0L + ALKANE] EQUILIBRIA
AND CRITICAL CURVES 35
1 General remarks 35
2 Preparation of a binary 12-methoxyethanol + alkanel
mixture 35 3 Isopleths 38 4 Critical points 41 5 Isobario T,x-cross-sections 47 6 Literature 56 Chapter 6 CALCULATIONS 57
1 Excess Gibbs energy models 57
2 The lattice gas model 80
3 Unifac model 84 4 Literature 86
Chapter 7 EXCESS VOLUMES 87
1 Experiments and calculations 87
2 Literature 92
Appendix A EXPRESSIONS FOR ACTIVITY COEFFICIENTS AND CRITICAL POINTS
FOR EXCESS GIBBS ENERGY MODELS 93
1 The NRTL equation and the first, second and third
derivative with respect to x^ 95
2 The Uniquaa equation and the first, second and third
derivative with respect to x^ 96 3 The Muniquac equation and the first, second and third
derivative with respect to 99
Appendix B 1 Experimental isopleths in the systems Lx 2-methoxy-ethanol + (1-x) alkanel, demixing pressure p as a
2 Critical temperatures in Lx 2-ME + (1-x) alkane~\
systems as a function of mole fraction x. The
uncertainty is indicated 122 3 Excess volumes of some Lx 2-ME + (1-x) alkanel
systems at .1 MPa 123
SAMENVATTING
De l a a t s t e j a r e n neemt de i n t e r e s s e voor scheidingsmethoden b i j verhoogde of hoge druk t o e . D i t komt doordat b i j s t i j g e n d e e n e r g i e p r i j z e n i n v e s t e r i n g e n i n drukapparatuur steeds s n e l l e r rendabel worden. Een voorbeeld van een dergel i j k e scheidingsmethode i s c e n t r i f u g a a dergel e x t r a c t i e . H i e r b i j spedergelen v dergel o e i s t o f -v l o e i s t o f e-venwichten onder hoge druk een b e l a n g r i j k e r o l .
D i t p r o e f s c h r i f t h e e f t a l s onderwerp een onderzoek naar de i n v l o e d van de druk op ontmenging i n de v l o e i s t o f f a s e . Onderzocht z i j n een aantal b i n a i r e [2-methoxyethanol + a l kaan] systemen waarin v l o e i s t o f - v l o e i s t o f ontmenging o p t r e e d t . Experimentele r e s u l t a t e n worden voor een aantal van deze systemen gerapporteerd i n een temperatuurbereik van 250 t o t 430 K en b i j drukken van 1 t o t 4000 bar (.1 t o t 400 MPa). De onderzochte a l kanen z i j n : nhexaan, n -heptaan, n - o k t a a n , n-dodecaan, 2-methylpentaan, 2-methylhexaan, 2-methyl-heptaan, 2,4-dimethylhexaan en 2 , 2 , 4 - t r i m e t h y l p e n t a a n . Deze keuze maakt het m o g e l i j k de i n v l o e d van ketenlengte en vertakkingsgraad van het alkaan op de l i g g i n g van het evenwicht te onderzoeken.
Een o v e r z i c h t van m o g e l i j k e vormen van b i n a i r e v l o e i s t o f - v l o e i s t o f even-wichten en de i n v l o e d hierop van de temperatuur T, de druk p en de samenstell i n g x , wordt i n hoofdstuk 2 aan de hand van p , T , x f i g u r e n beschreven. V e r v o samenstell -gens worden een aantal mogelijkheden beschreven om deze evenwichten te meten. B i j de berekeningsmethoden, d i e i n hoofdstuk 3 behandeld worden, wordt i n g e -gaan op twee methoden.
De e e r s t e methode maakt gebruik van modellen voor de excess Gibbs energ i e , z o a l s de van L a a r , NRTL en U n i q u a c v e r energ e l i j k i n energ . De Uniquac v e r energ e l i j -king heeft a l s bezwaar dat deze v e r g e l i j k i n g i n het grensgeval van de verdunde o p l o s s i n g n i e t asymptotisch overgaat i n de v e r g e l i j k i n g voor de i d e a l e o p l o s -s i n g . Een m o d i f i c a t i e van de Uniquac v e r g e l i j k i n g , de Muniquac v e r g e l i j k i n g
d i e d i t bezwaar ondervangt, wordt g e ï n t r o d u c e e r d .
De tweede methode om v l o e i s t o f - v l o e i s t o f ontmenging te berekenen i s met behulp van een t o e s t a n d s v e r g e l i j k i n g . In d i t p r o e f s c h r i f t z i j n de m o g e l i j k h e -den van een recent door Koningsveld en K l e i n t j e n s ontwikkelde v e r g e l i j k i n g (het roostergasmodel) onderzocht.
De metingen z i j n uitgevoerd volgens de s y n t h e t i s c h e methode, waarvoor een a u t o c l a a f g e b r u i k t w o r d t , d i e voorzien i s van s a f f i e r e n v e n s t e r s . Het meet-v a a t j e met het te onderzoeken meet-v l o e i s t o f m e n g s e l wordt tussen de meet-vensters ge-p l a a t s t , zodat v i s u e l e waarneming van het fasegedrag b i j v e r s c h i l l e n d e drukken en temperaturen m o g e l i j k i s . De g e b r u i k t e o p s t e l l i n g wordt i n hoofdstuk 4 be-sproken.
In hoofdstuk 5 worden de experimentele r e s u l t a t e n van de onderzochte [2-methoxyethanol + a l kaan] systemen gepresenteerd. Deze meetresultaten z i j n verkregen door b i j een constante s a m e n s t e l l i n g en constante temperatuur de druk te v a r i ë r e n en d i e druk te b e p a l e n , w a a r b i j het mengsel op het punt s t a a t om te ontmengen.
In de onderzochte systemen z i j n b o v e n - k r i t i s c h e ontmengtemperaturen en o n d e r - k r i t i s c h e ontmengdrukken gevonden. De k r i t i s c h e s a m e n s t e l l i n g x£ hangt s l e c h t s i n geringe mate van de druk af ( a xc -v .015 i n het onderzochte drukge-b i e d ) . Gevonden i s dat met toenemend m o l e c u l a i r gewicht van het al kaan de k r i t i s c h e s a m e n s t e l l i n g naar l a g e r e a l k a a n c o n c e n t r a t i e s v e r s c h u i f t . Ketenverlenging van het alkaan met een { C h ^ g r o e p i n een l i n e a i r e of i s o a l k a a n v e r -oorzaakt b i j 1 bar (= 0.1 MPa) een toename van c a . 8 K en b i j 4000 bar (= 400 MPa) een toename van c a . 13 K van de b o v e n k r i t i s c h e ontmengtemperatuur.
Getracht i s met de i n hoofdstuk 3 behandelde thermodynamische modellen de experimenteel gevonden r e s u l t a t e n te b e s c h r i j v e n . In hoofdstuk 6 wordt u i t e e n -gezet hoe de parameters van de v e r s c h i l l e n d e modellen berekend z i j n . Van de modellen voor de excess Gibbs energie b l i j k t de Muniquac v e r g e l i j k i n g , na i n -t r o d u c -t i e van een i n de l i -t e r a -t u u r gevonden r e l a -t i e , d i e de druk- en - tempera-t u u r a f h a n k e l i j k h e i d van de aanpasbare parametempera-ters b e s c h r i j f tempera-t , r e l a tempera-t i e f hetempera-t bestempera-t i n s t a a t de experimenten te b e s c h r i j v e n . B i j een nadere beschouwing van de beide Muniquac parameters A l l ^ en A ü ^ b l i j k t de druk- en temperatuurafhanke-l i j k h e i d van deze parameters beter met de votemperatuurafhanke-lgende v e r g e temperatuurafhanke-l i j k i n g e n beschreven te kunnen worden:
A U1 2 = aQ + aj/T + a2( p - 1 )2 / 3
A U2 1 = bQ + bx/T + b2(p
-w a a r b i j a ^ , a ^ , a , , , b^, b^ en constanten z i j n .
Gebruik makend van enige aannames z i j n deze constanten voor i e d e r gemeten systeem, het systeem [2-methoxyethanol + n-dodecaan] u i t g e z o n d e r d , berekend. Het b l i j k t dat a^ en b1 o n a f h a n k e l i j k en aQ, a g , b^ en b^ a f h a n k e l i j k z i j n van de ketenlengte van de onderzochte al kanen. D i t s y s t e m a t i s c h gedrag van de costanten wordt g e b r u i k t om u i t de concostanten voor de systemen met hexaan, n-heptaan en n-oktaan de constanten voor het systeem met n-dodecaan te s c h a t t e n . Met behulp van deze constanten b l i j k t dat v l o e i s t o f - v l o e i s t o f evenwichten i n het systeem [2-methoxyethanol + n-dodecaan] zeer goed v o o r s p e l d kunnen worden.
Het b e s c h r i j v e n van de experimenten m . b . v . het roostergasmodel g e e f t a l -leen dan een bevredigend r e s u l t a a t a l s de b i n a i r e aanpasbare parameters van d i t model druk- en t e m p e r a t u u r a f h a n k e l i j k genomen worden en over het hele druk- en temperatuurbereik g e f i t worden. Hierdoor gaat het voordeel van het toepassen van een t o e s t a n d s v e r g e l i j k i n g t . o . v . een excess Gibbs energie model v e r l o r e n .
U i t de gevonden d r u k a f h a n k e l i j k h e i d van de excess Gibbs e n e r g i e , kunnen waarden voor het excess volume berekend worden. Deze berekende excess volumina z i j n i n hoofdstuk 7 vergeleken voor een tweetal systemen met experimentele excess volumina. Onder de 200 bar (= 20 MPa) was het r e s u l t a a t s l e c h t . Boven de 200 bar waren de berekende en experimentele excess volumina van d e z e l f d e orde van g r o o t t e . Het berekende excess volume i s e c h t e r een f a c t o r 2 t o t 3 te g r o o t . D i t maakt d u i d e l i j k dat men erg v o o r z i c h t i g moet z i j n i n het gebruik van een excess Gibbs energie model, waarvan de aanpasbare parameters bepaald z i j n u i t fasenevenwichten, voor het berekenen van andere thermodynamische excess grootheden.
Chapter 1 PREFACE AND OBJECTIVES
In a t y p i c a l chemical p l a n t 40 - 80% of the investment c o s t s are r e q u i r e d f o r s e p a r a t i o n operations such as d i s t i l l a t i o n , a b s o r p t i o n and e x t r a c t i o n . Nowadays with i n c r e a s i n g energy p r i c e s e x t r a c t i o n processes become more popular. The b a s i c p r i n c i p l e of an e x t r a c t i o n process i s to b r i n g an i n s o l u b l e l i q u i d i n t o d i r e c t c o n t a c t w i t h the mixture to be s e p a r a t e d , t o cause t r a n s f e r of substances (mixer) and a l l o w s e t t l i n g of the r e s u l t i n g d i s p e r s i o n by the f o r c e s of g r a v i t y ( s e t t l e r ) . These operations may be c a r r i e d out i n batch f a s h i o n or i n a continuous flow apparatus. Problems occur e s p e c i a l l y i n handling l i q u i d s with small d i f f e r e n c e s i n d e n s i t y , l i q u i d s with the tendency to form emulsions or i n cases where phase s e p a r a t i o n s have to be done r a p i d l y to avoid chemical decomposition of the products under the c o n d i t i o n s of e x t r a c t i o n . The f o r c e of g r a v i t y can be increased many thousandfold i f
d e s i r e d , by c e n t r i f u g e s . C e n t r i f u g a l e x t r a c t o r s are of great importance i n the pharmaceutical i n d u s t r y but nowadays they are a l s o used more and more i n petroleum p r o c e s s i n g . A problem on the use of c e n t r i f u g a l e x t r a c t o r s i s the lack of knowledge of the e f f e c t of pressure on l i q u i d - l i q u i d e q u i l i b r i a , both e x p e r i m e n t a l l y and t h e o r e t i c a l l y .
The o b j e c t i v e of t h i s work, t h e r e f o r e , i s to i n v e s t i g a t e and c o r r e l a t e l i q u i d - l i q u i d e q u i l i b r i a at various temperatures and pressures up t o 400 MPa f o r a number of b i n a r y systems. In a l l the b i n a r y systems i n v e s t i g a t e d 2-methoxyethanol i s one of the components. 2-Methoxyethanol i s a p o l a r
s o l v e n t , which i s e . g . w i d e l y used i n p a i n t i n d u s t r y . The other c o n s t i t u e n t was always a member o f the alkane " f a m i l y " . The alkanes were chosen i n order to determine the e f f e c t of molecular s i z e and shape on the e q u i l i b r i a . Since i t i s very d i f f i c u l t to measure the phase behaviour i n the dynamic process o c c u r i n g i n a c e n t r i f u g a l e x t r a c t o r , the experiments were performed i n a
s t a t i c way using an autoclave provided w i t h windows. The windows of the autoclave a l l o w to observe v i s u a l l y the phase behaviour of the mixture i n s i d e .
In the t h e o r e t i c a l part of t h i s t h e s i s some (semi-) t h e o r e t i c a l models f o r the Gibbs excess energy are d i s c u s s e d such as the van Laar model, NRTL model, Uniquac model and a new m o d i f i c a t i o n of the Uniquac model. A model f o r the Helmholtz energy, the s o - c a l l e d gas l a t t i c e treatment i s a l s o a p p l i e d . The a p p l i c a b i l i t y of a l l these models i s i n v e s t i g a t e d to d e s c r i b e the phase
e q u i l i b r i a determined e x p e r i m e n t a l l y . A l l the models of the Gibbs excess energy were o r i g i n a l l y developed f o r ambient p r e s s u r e . In order to be able to d e s c r i b e the e f f e c t of pressure on the l i q u i d - l i q u i d e q u i l i b r i a an emperical dependence on the pressure has to be introduced f o r the model parameters. The l a t t i c e gas model, however, has already a pressure dependence i n c o r p o r a t e d .
A data r e d u c t i o n method i s developed f o r c a l c u l a t i n g l i q u i d - l i q u i d e q u i l i b r i a o c c u r i n g i n t h i s kind of b i n a r y systems. In chapter 7 the attempt i s made to p r e d i c t excess volumes of some systems i n v e s t i g a t e d using only experimental data on l i q u i d - l i q u i d e q u i l i b r i a .
Chapter 2 LIQUID-LIQUID EQUILIBRIA AT HIGH PRESSURES
1 Liquid-liquid demixing phenomena
Most s t u d i e s of l i q u i d - l i q u i d ( L - L ) phase e q u i l i b r i a have been c a r r i e d out i n the temperature range 270 - 420 K and at pressures not exceeding 2 MPa [ r e f . 1 , 2 ] . Binary L L e q u i l i b r i a can be represented as s o c a l l e d T , x c r o s s -s e c t i o n -s , which give the mutual -s o l u b i l i t y (mole f r a c t i o n x) i n both pha-se-s a-s a f u n c t i o n of temperature T at constant pressure p. F i g . 2.1 shows some types of T , x - c r o s s - s e c t i o n s which are p o s s i b l e at ambient p r e s s u r e .
The l i n e s i n F i g . 2.1 which enclose the two phase regions are c a l l e d binodal c u r v e s . The compositions of two phases i n e q u i l i b r i u m (conjugated phases) are connected by h o r i z o n t a l l i n e s (conodes or t i e l i n e s ) . A b i n a r y system with demixing i n the l i q u i d phase might behave as i s shown i n F i g . 2 . 1 . a . This f i g u r e with a c l o s e d loop binodal shows t h a t at low temperatures only one l i q u i d phase e x c i s t s . I f the lower c r i t i c a l s o l u t i o n temperature "I"LCST i s reached with i n c r e a s i n g temperature the s i n g l e l i q u i d phase s t a r t s to s p l i t i n two l i q u i d phases. With the temperature s t i l l i n c r e a s i n g between 1"LCST and Tg the compositions o f the two l i q u i d phases become more and more d i f f e r e n t . At Tg t h i s behaviour reverses u n t i l the upper c r i t i c a l s o l u t i o n temperature T y ^ y i s reached. Above t h i s temperature again only one l i q u i d phase e x c i s t s . Such a behaviour i s found only f o r ^ 4% o f the systems reported i n l i t e r a t u r e [ r e f . 1 , 2 ] . The L C S T i n many cases i s metastable with r e s p e c t to the occurence of a s o l i d phase ( F i g . 2 . 1 . b ) and % 60 per cent o f the U C S T ' s are metastable with r e s p e c t to the formation of a gaseous phase ( F i g . 2 . 1 , c ) . Therefore i n more then ^ 50 per cent o f L - L demixing e i t h e r no L C S T o r no UCST i s found ( F i g . 2 . 1 . e ) . A h y p e r b o l i c type of demixing i s p o s s i b l e , where a L C S T occurs at a temperature which i s higher then the U C S T . But such behaviour was
'UCST 'UCST ' U C S T ' U C S T T L
11M
1-2 \ + VV
L2A
/ i • 1Figure 2.1 Effect of temperature on immisoibility phenomena in binary liquid systems.
metastable two phase region.
^UCST ~ VpPer Critical Solution Temperature,
^LCST ~ ^J<x>er Critical Solution Temperature3 L - liquid phase, G -- gas phase, S - solid phase.
found only f o r a few systems [ r e f . 3 ] .
F i g . 2.2 taken from Schneider [ r e f . 4] shows an extension of F i g . 2.1 i n which the e f f e c t of pressure on L-L e q u i l i b r i a i s i l l u s t r a t e d . This f i g u r e represents a l l types of l i q u i d - l i q u i d e q u i l i b r i a , which are p o s s i b l e i n b i n a r y systems. Examples e x i s t f o r a l l types except those marked with a q u e s t i o n -mark. On the l e f t - h a n d s i d e of the f i g u r e the already known T,x diagrams at constant pressure are given ( F i g . 2.2.a->-d). F i g . 2.2(e-*h) shows the p o s s i b l e
Figure 2.2 Effect of pressure and temperature on immiscibility phenomena in binary liquid systems (See text) Lref. 41.
types of p-x c r o s s - s e c t i o n s at constant temperature. Both upper and lower c r i t i c a l s o l u t i o n pressures (UCSP and LCSP r e s p e c t i v e l y ) are found. The f i r s t h o r i z o n t a l row ( F i g . 2.2.1+1} shows that the UCST can e i t h e r d e c l i n e ( F i g . 2 . 2 . i ) or r i s e ( F i g . 2 . 2 . j ) or run through a temperature minimum ( F i g . 2 . 2 . 1 ) with i n c r e a s i n g p r e s s u r e . The second h o r i z o n t a l row ( F i g . 2.2.m-»-p) shows a s i m i l a r behaviour f o r L C S T ' s . The pressure dependence of systems with c l o s e d loops ( F i g . 2 . 2 . q + t ) i s shown i n the t h i r d row. Closed loops may e i t h e r s h r i n k with i n c r e a s i n g pressure and disappear completely at a d e f i n i t e pressure i n the t h r e e - d i m e n s i o n a l T , p , x space (a hyper c r i t i c a l s o l u t i o n p o i n t ) ( F i g . 2 . 2 . q ) . They can appear only at h i g h e r pressures such as shown i n F i g . 2 . 2 . r ( s o - c a l l e d high pressure i m m i s c i b i l i t y ) or have a continued e x i s t e n c e as hyperboloids ( F i g . 2 . 2 . t ) . I m m i s c i b i l i t y behaviour of a s a d d l e - l i k e type such as those shown i n the f o u r t h row i n F i g . 2 . 2 . u and v have a l s o been found. Examples of the d i f f e r e n t types of l i q u i d - l i q u i d demixing are given i n a recent review by Schneider [ r e f . 4 ] .
The b i n a r y [2-methoxyethanol + alkane] systems i n v e s t i g a t e d i n t h i s work c o n t a i n a p o l a r and a n o n - p o l a r component. For t h i s type of systems ( p o l a r /
non-polar) u s u a l l y an UCST i s observed as reported by Schneider. In f a c t t h i s i s v e r i f i e d by the experimental r e s u l t s o b t a i n e d , which always show an UCST i n c r e a s i n g with i n c r e a s i n g p r e s s u r e , i . e . a behaviour as shown i n F i g . 2 . 2 . j . For some [2-ME + alkane] systems l i q u i d - l i q u i d e q u i l i b r i a data at .1 MPa are reported by Gladel [ r e f . 5] and Landauer [ r e f . 6 ] .
2 Thermodynamic description of liquid-liquid equilibria and critical points
A l i q u i d mixture i s i n i t s s t a b l e s t a t e i f at constant temperature and pressure i t s molar Gibbs energy G has a minimum. The mixture w i l l s p l i t i n t o two phases (denoted by 1 and ") i f upon doing so i t s Gm i s minimized. For a mixture o f two compounds 1 and 2 t h i s i s i l l u s t r a t e d i n F i g . 2 . 3 [ r e f . 7 , 8 ] which shows G as a f u n c t i o n o f composition x at constant T and p.
G
m
1 X X X 2 2 2 2
> x2
Figure 2.3 Molar Gibbs energy of a binary partially miscible liquid mixture at constant
temperature and pressure (See text).
If Gm i s given by the heavy l i n e , then Gm of the mixture w i t h composition x2 corresponds t o p o i n t a . I t i s e v i d e n t from F i g . 2 . 3 that p o i n t b represents a lower Gm than does p o i n t a , and as a r e s u l t the l i q u i d mixture having o v e r a l l composition x2 s p l i t s i n t o two l i q u i d phases having mole f r a c t i o n s x£ and x £ . P o i n t b represents the lowest p o s s i b l e Gm which the mixture may a t t a i n s u b j e c t to the r e s t r a i n t s of f i x e d temperature, pressure and o v e r a l l composition x2 < A decrease i n the Gm o f a b i n a r y l i q u i d mixture due to the formation of another l i q u i d phase can only occur i f a p l o t of Gm a g a i n s t mole f r a c t i o n i s , i n p a r t , concave downward. T h e r e f o r e , the c o n d i t i o n f o r i n s t a b i l i t y of a b i n a r y l i q u i d mixture i s
32 G \
—r) < o ( 2 . i )
In t h i s case the p l o t of G a g a i n s t mole f r a c t i o n has two p o i n t s of 2 2
i n f l e c t i o n 1 and 2 , where (3 Gm/3x )T = 0 . Upon a change i n temperature and/ or pressure those two p o i n t s can merge i n t o a s i n g l e p o i n t , which corresponds to a c r i t i c a l s t a t e , which i s the border between s t a b i l i t y and i n c i p i e n t i n s t a b i l i t y of the l i q u i d m i x t u r e . Therefore the c r i t i c a l s t a t e i s c h a r a c t e r i z e d by two equations
( 3 2 Gm/ 3 x 2> Tc, pc = 0 <2-2>
(
3V A
C,
P C=
0(
2-
3)
For a case represented i n F i g . 2 . 3 a l s o the f o l l o w i n g equation i s v a l i d [ r e f . 9]
(
3V
3 X 4)
T ,
P>
0(2-4)
From the demixing behaviour some knowledge on thermodynamic molar excess p r o p e r t i e s can be deduced. i s d e f i n e d as the d i f f e r e n c e o f the a c t u a l property W^e a^ o f a r e a l mixture and the property Wm of an i d e a l mixture a t the same p , T and x
WE = wr e a 1 - Wi d ( 2 . 5 )
m m m 1 >
the Helmholtz energy A, e t c . ) .
From the T , x - c r o s s - s e c t i o n of a b i n a r y mixture w i t h an UCST i t i s obvious that
( 32T / 9 x2)c < 0 (2.6)
Index c stands f o r the c r i t i c a l s o l u t i o n p o i n t . A T a y l o r expansion of 2 2
(3 Gm/3x ) about the c r i t i c a l p o i n t as a f u n c t i o n of temperature and composition r e s u l t s i n [see r e f . 10]
2 2 (
3V
3 X\
( 32T / 3 x2)c = - m g c . T (2.7)
< » V » * > c
Together w i t h eq. (2.4) and eq. (2.6) the f o l l o w i n g c o n d i t i o n s are obtained
( 32Hm/ 3 x2)c = ( 32H ^ / 3 x2)c < 0 (2.8)
( 32Sm/ a x2)c < 0 (2.9)
( s V / s xm c2) = 0 (2.10)
> Q
For b i n a r y systems w i t h a LCST s i m i l a r r e l a t i o n s [ r e f . 10] are o b t a i n e d . The pressure dependence of the c r i t i c a l s o l u t i o n temperature i s obtained from the t o t a l d i f f e r e n t i a l 9 ^G \ /8 \ /3^G \ / 9 \
ii/)-
fc^r'tajF
d T tb t '
d p ( 2-
u ) ^ O A ^ d X \ O A d I ƒ \ O A d p which leads to c(
3V
3 XV
Tc
=(^r)
dP (2-1 2>which can be rearranged to i/m \ / 32Vm. m / m m 2 2 . 2 E 3 1/1 | 3 X 3 1 dTn \ 3 X2" A \ 3x2 / „ \ 3 ? ' aZS \ / 3 \ \ C / 32H ^ \ C 12
With the simple assumptions that = A(p,J)x^x^ and HE = A ' ( p , T ) x1x2 t h i s leads to
3T . T
c = mc c ( 2 1 4 )
8p Ht
This l a s t equation can be used to o b t a i n i n f o r m a t i o n on the s i g n of the excess p r o p e r t i e s a t c r i t i c a l c o n d i t i c
This i s d i s c u s s e d i n chapter 7.
p r o p e r t i e s a t c r i t i c a l c o n d i t i o n s from (dTc/dp) determined e x p e r i m e n t a l l y .
3 Measuring methods
U n t i l now almost a l l research groups which i n v e s t i g a t e l i q u i d - l i q u i d e q u i l i b r i a at high pressures use an autoclave provided w i t h windows [ r e f . 11, 1 2 , 1 3 ] . The main p r i n c i p l e of t h i s method i s t h a t a sample of known
composition i s placed i n s i d e an autoclave and the onset of an eventual mixing or demixing process i s observed v i s u a l l y f o r v a r i o u s temperatures and
p r e s s u r e s . The two main disadvantages of t h i s method are 1 ° A new sample must be prepared f o r each c o m p o s i t i o n .
2 ° For systems w i t h more then two components the composition of the conjugated phases cannot be deduced from the measured phase diagram.
Nowadays a l o t of e f f o r t i s invested i n c o n s t r u c t i n g an autoclave from which d i r e c t l y samples o f both phases i n e q u i l i b r i u m can be taken and analysed by chromatografic or other methods [ r e f . 14 to 2 0 ] . Taking r e l i a b l e samples of both phases i s the main d i f f i c u l t y of t h i s method.
Other methods f o r measuring l i q u i d - l i q u i d e q u i l i b r i a are based on the f a c t t h a t w i t h any phase t r a n s i t i o n c h a r a c t e r i s t i c changes of p h y s i c a l p r o p e r t i e s o c c u r . Such a method i s the decametrical one which i s based on measuring the d i - e l e c t r i c a l constant e of a l i q u i d m i x t u r e . A d i s c o n t i n u o u s change i n the slope of an e , p or e,T curve i s found a t a phase t r a n s i t i o n , which i n most cases can be determined very p r e c i s e l y . However, S t e i n e r [ r e f . 17] using t h i s method determined, f o r example a p , x - c r o s s - s e c t i o n w i t h two UCSP's at the same pressure i n the system [ m e t h y l i s o p r o p y l k e t o n + water] which i s very u n l i k e l y and not found using other methods [ r e f . 19]. U n t i l now these i n d i r e c t methods seem not to give r e l i a b l e r e s u l t s .
4 Literature
1. Sorensen, J . M . , W. A r l t , L i q u i d - l i q u i d e q u i l i b r i u m data c o l l e c t i o n DECHEMA chemistry data s e r i e s , F r a n k f u r t , (1979).
2. F r a n c i s , A . W . , L i q u i d - l i q u i d e q u i l i b r i a , (1963).
3. S c h n e i d e r , G . M . , B e r . Bunsenges. P h y s i k . Chem., 7 0 , (1966), 497.
4. S c h n e i d e r , G . M . , Chemical S o c i e t y S p e c i a l i s t P e r i o d i c a l Reports Chemical Thermodynamics, 2 , Londen, (1978).
5. G l a d e l , Y . L . , J . Durandet, R. Roux, Rev. I n s t . Franc. P e t r o l e , £ , (1954), 221.
6. Landauer, A . , R.N. L i c h t e n t h a l e r , J . M . P r a u s n i t z , J . Chem. Eng. D a t a , 2 5 ( 2 ) , (1980), 152. 7. P r a u s n i t z , J . M . , M o l e c u l a r Thermodynamics of F l u i d Phase E q u i l i b r i a , P r e n t i c e H a l l , (1969). 8. K o n i n g s v e l d , R . , T h e s i s , L e i d e n , (1967). 9. Chermin, H . A . G . , T h e s i s , Essex, (1971). 10. J e s c h k e , P . , T h e s i s , Bochum, (1980).
11. A l w a n i , Z . , G.M. S c h n e i d e r , B e r . Bunsenges. Phys. Chem., 7 3 , (1969), 294. 12. Young, C . L . , Chemical S o c i e t y S p e c i a l i s t P e r i o d i c a l Reports Chemical
Thermodynamics, 2 , Londen, (1978).
13. H e s t , van J . A . M . , G.A.M. Diepen, Phys. Chem. High Pressures Pap. Symp., (1962).
14. M o r i y o s h i , T . , S. Kaneshina, K. Aihara and K. Yobumoto, J . Chem. Thermo-dynamics, 2 » (1975), 537.
15. I d . , J . Chem. Thermodynamics, 9 , (1977), 495. 16. I d . , J . Chem. Eng. of J a p a n , 11, ( 1 9 7 8 ) , 341.
17. S t e i n e r , R . , E. Schadow, Z. Ph. Chem., 6 3 , (1969), 297. 18. S c h n e i d e r , G . M . , Pure & A p p l . Chem., 5 5 , (1983), 479.
19. Hunt, A . F . , J . A . Lamb, F l u i d Phase E q u i l i b r i a , 3 , (1979), 177. 20. Bozdag, 0 . , J . A . Lamb, F l u i d Phase E q u i l i b r i a , 6 , (1981), 191.
Chapter 3 THERMODYNAMICAL MODELS FOR THE DESCRIPTION OF LIQUID-LIQUID EQUILIBRIA
1 Introduction
I f two l i q u i d phases (denoted by ' and ") are i n thermodynamical
e q u i l i b r i u m , the pressure p (except f o r osmotic phenomena), the temperature T
i<\ (")
and the chemical p o t e n t i a l s y : ' and y : ' of each component i i n both phases have to be e q u a l . The chemical p o t e n t i a l o f component i i n a multicomponent mixture i s d e f i n e d a s :
where Gm i s the molar Gibbs energy of the m i x t u r e , Am i s the molar Helmholtz energy of the m i x t u r e , n. i s the number of moles of component i and k the number of components.
Quite a number of e m p i r i c a l and t h e o r e t i c a l models f o r the Gibbs and Helmholtz energy have been proposed. The models developed f o r the Gibbs energy u s u a l l y assume Gm to be independent of p r e s s u r e , i . e . Gm i s given as a
f u n c t i o n of temperature and composition o n l y . The equations f o r the Helmholtz energy, however, are o f t e n d e r i v e d from an equation of s t a t e f o r mixtures a n d , t h e r e f o r e , the dependence on pressure ( o r volume), temperature and composition i s taken i n t o account. Mostly models f o r the Gibbs energy a r e used to d e s c r i b e l i q u i d - l i q u i d e q u i l i b r i a a t ambient p r e s s u r e . In t h i s work the a p p l i c a b i l i t y
( 3 . 1 ) Or a s : k
of the van Laar equation (1910) [ r e f . 1 ] , the Non-Random-Two L i q u i d s (NRTL) equation (Renon & P r a u s n i t z , 1968 [ r e f . 2 ] ) , the U n i v e r s a l Quasi-chemical (Uniquac) equation (Abrams & P r a u s n i t z , 1975 [ r e f . 3]) and a new m o d i f i c a t i o n of the Uniquac equation w i l l be d i s c u s s e d . In order to apply those models t o high pressure e q u i l i b r i a , m o d i f i c a t i o n s are necessary t o account f o r the pressure dependence of Gm. In a d d i t i o n the Helmholtz energy equation obtained from the l a t t i c e gas treatment proposed by Trappeniers [ r e f . 9 , 1 0 ] and
extended by K l e i n t j e n s (1979) [ r e f . 4] i s a p p l i e d .
To get a b e t t e r understanding of the background o f the s e v e r a l models they are now d i s c u s s e d i n some d e t a i l .
2 Excess Gibbs energy models
In order t o be able to c a l c u l a t e l i q u i d - l i q u i d e q u i l i b r i a an e x p r e s s i o n f o r the chemical p o t e n t i a l i s needed as a f u n c t i o n of p , T and composition x . The chemical p o t e n t i a l i s defined as the p a r t i a l molar q u a n t i t y of the Gibbs energy (equation 3 . 1 ) . To c a l c u l a t e the chemical p o t e n t i a l a model f o r the Gibbs energy i s needed. The models i n v e s t i g a t e d give expressions f o r the molar excess Gibbs energy G^ r a t h e r than f o r Gm (see chapter 2 . 2 ) .
2.1 The van Laar equation
Van Laar assumed t h a t upon mixing two l i q u i d s the excess volume (V ) and the excess entropy ( SE) are zero [ r e f . 1 ] . Using the van der Waals equation of s t a t e , w i t h simple mixing r u l e s f o r the pure component constants a
and b?, the f o l l o w i n g equation f o r G^ of a b i n a r y mixture i s obtained A x , x
( 3 . 3 )
x
l !
+ x 2where
In p r a c t i c e Gm c a l c u l a t e d using the van der Waals constants determined from pure component p , V , T data o r c r i t i c a l p r o p e r t i e s do not agree w i t h
experimental v a l u e s . Much b e t t e r r e s u l t s are obtained using A and B as a d j u s t a b l e parameters.
The r e l a t i v e l y simple van Laar equation i s o f t e n used t o c a l c u l a t e l i q u i d - l i q u i d e q u i l i b r i a . A disadvantage of t h i s equation i s t h a t i t cannot e a s i l y be extended t o rnulticomponent m i x t u r e s .
2.2 The NRTL equation
Except f o r the van Laar equation most other equations f o r G^ a r e based on a l a t t i c e model. The NRTL model C r e f . 2] takes i n t o account the d i f f e r e n c e s i n the i n t e r m o l e c u l a r i n t e r a c t i o n s between molecules o f d i f f e r e n t components. A s p e c i f i c molecule i n a mixture might p r e f e r to be surrounded by molecules o f the same s p e c i e s o r p r e f e r to have s p e c i f i c other molecules as next
neighbours. This s o - c a l l e d non random mixing i s d i f f e r e n t a t d i f f e r e n t temperatures. Wilson [ r e f . 5] proposed to i n t r o d u c e Boltzmann-factors i n the G^-equation i n order to account f o r the temperature dependence. The NRTL equation i s based on such a l a t t i c e model and the ideas of W i l s o n . For a binary mixture the r e s u l t i s :
<Ll _ v ( °2 1 6 2 1 a1 2 G12 RT xlx2 l %x + x2 G2 1 x2 + xl G where (3.4) ° 2 1 = ( 92 1 " 91 1) / R T a12 = (g12 " 9 2 2 ^/ R T G2 1 = e x p ( - a1 2 a2 1) G1 2 = e x p ( - a1 2 a1 2) ( 3 . 5 ) (3.6) (3.7) (3.8) From the d e r i v a t i o n f o l l o w s t h a t the parameter a ^2 = 2/1, where the
c o o r d i n a t i o n number 2 o f the l a t t i c e u s u a l l y i s taken to be 10. However a can a l s o be used as an a d j u s t a b l e parameter. A g ^2 (= g ^2 - g2 2) i s the d i f f e r e n c e between the i n t e r a c t i o n energy of a molecule 1 surrounded by
molecules 2 ( 9 ^2) a n d a molecule 2 surrounded by molecules of the same kind ( g2 2) . For A g2j (= g2^ - g ^ ) t h i s i s analogous. U s u a l l y both A g1 2 and A g2 1 are used as a d j u s t a b l e parameters.
The exponential terms ( e q . ' s ( 3 . 7 ) and ( 3 . 8 ) ) were introduced by Renon [ r e f . 2] to d e s c r i b e the non random m i x i n g , i . e . the chance of f i n d i n g next t o a molecule A a l a t t i c e hole occupied by a molecule A o r B r e s p e c t i v e l y . For T •* ^ those f a c t o r s are equal t o o n e , which means t h a t the mixing process i s completely random.
The NRTL equation very o f t e n gives an e x c e l l e n t d e s c r i p t i o n o f the phase behaviour o f l i q u i d mixtures c o n t a i n i n g non-polar molecules only o r l i q u i d mixtures w i t h e x c l u s i v e l y p o l a r m o l e c u l e s . Problems u s u a l l y a r i s e approaching the c r i t i c a l r e g i o n .
2.3 The Uniquac equation
The Uniquac equation i s d e r i v e d by phenomenological arguments based on a two l i q u i d t h e o r y . In a d d i t i o n t o the assumption made f o r the NRTL equation the molecules are now d i v i d e d up i n t o molecular segments each occupying a l a t t i c e s i t e . The GE equation d e r i v e d i s the sum of two c o n t r i b u t i o n s one due
m E
to the d i f f e r e n t arrangements of the molecular segments on the l a t t i c e (Gm ( c o m b i n a t o r i a l ) ) and one due to the d i r e c t i n t e r a c t i o n s between the molecular segments of d i f f e r e n t molecules (G^ ( r e s i d u a l ) ) .
The Uniquac equation contains three c o n c e n t r a t i o n measures: the mole f r a c t i o n x , the segment f r a c t i o n $ and the s u r f a c e f r a c t i o n 8 . The segment f r a c t i o n appears i n the c o m b i n a t o r i a l p a r t of the Uniquac e q u a t i o n , as the number o f segments per molecule e f f e c t the arrangement o f the molecules i n the h y p o t h e t i c a l l a t t i c e . The s u r f a c e area of a segment r e f l e c t s t h e number of i n t e r m o l e c u l a r i n t e r a c t i o n s p o s s i b l e f o r a segment and t h e r e f o r e shows up i n the r e s i d u a l p a r t . The average segment f r a c t i o n <$> and the average s u r f a c e f r a c t i o n e are d e f i n e d a s : x. r . - - 1 1 ( 3 . 9 ) z x. r . x . q . 6. = 1 1 (3.10) i Z X. q .
where r^ and q^ are pure component s t r u c t u r a l parameters r e p r e s e n t i n g the van
der Waals volume and area of a molecule of type i r e l a t i v e t o those o f a standard segment which i s chosen to be a {CH2) group. These parameters can be
c a l c u l a t e d from
(3.11) (3.12) where i s the number of segments o f type k i n molecule i and and are the r e l a t i v e r a t i o s f o r the van der Waals volume and area of such a segment. R^ and can be determined by a method d i s c u s s e d by Abrams and P r a u s n i t z [ r e f . 3 ] .
The d e r i v a t i o n o f the Uniquac equation a c c o r d i n g t o Maurer and P r a u s n i t z [ r e f . 6] i s presented here i n order to understand c l e a r l y a m o d i f i c a t i o n proposed i n t h i s work and d i s c u s s e d i n chapter 3 . 2 . 4 .
For b i n a r y mixtures of molecules of a r b i t r a r y s i z e and shape, the mole-c u l e s of mole-component 1 are mole-c o n s i d e r e d to mole-c o n s i s t o f r^ segments omole-cmole-cupying Ev^.R^ s i t e s of a l a t t i c e . Each molecule has an e x t e r n a l s u r f a c e area p r o p o r t i o n a l to q p S i m i l a r parameters are d e f i n e d f o r molecules 2. The c o o r d i n a t i o n number Zof the l a t t i c e i s assumed to be the same f o r both components. For p o l y -segmented molecule 1 the number o f neighbouring segments belonging to other molecules i s Zq^. The energy r e q u i r e d to v a p o r i z e i s o t h e r m a l l y such a molecule from a pure l i q u i d to i t s i d e a l gas s t a t e i s J Z q ^ U ^ , where i s the
p o t e n t i a l energy between two nearest neighbour segments of two d i f f e r e n t molecules of the same k i n d . The v a p o r i z e d molecule i s now condensed i s o
-t h e r m a l l y i n -t o a h y p o -t h e -t i c a l f l u i d where i -t i s surrounded by Iq^e^^ segmen-ts belonging t o molecule 1 and 2q^e2^ s e9m e n t s belonging to molecule 2 . i s
the l o c a l composition o f segments 1 about a c e n t r a l segment 1 and ^s the
l o c a l composition of segments 2 about a c e n t r a l segment 1 , i . e . ft«, + e2j = 1.
The energy r e l e a s e d by t h i s condensation process i s ^ ( e ^ q ^ U ^ + 62 ic' iU2 1 ^
where Up^ i s the p o t e n t i a l energy between two nearest neighbour segments of the two d i f f e r e n t molecules 1 and 2. For a molecule 2 the same t r a n s f e r i s made from the pure l i q u i d to the h y p o t h e t i c a l f l u i d . The t o t a l change i n energy i n t r a n s f e r r i n g x^ molecules of component 1 from a pure l i q u i d and x2
moles of component 2 from a pure l i q u i d i n t o a l i q u i d m i x t u r e , i . e . the excess energy UE, i s given by:
where N. i s Avogadro number. The l o c a l s u r f a c e f r a c t i o n s e . . ( i , j = 1,2) are assumed t o be r e l a t e d t o the o v e r a l l s u r f a c e f r a c t i o n s and e2 using Boltzmann f a c t o r s , as already d i s c u s s e d i n chapter 3 . 2 . 2 ,
62 1/ 91 1 = 62/ el e x p C" ^Z(U2 1 " Ul l ) /R T ] (3.14) 61 2/ 92 2 = 6l/ 92 e x p C" ^Z(U1 2 " U2 2 ^/ R T ] (3.15) where 61 = Xlq l/ ( xlql + X2q25 <3-1 6) 62 = x2q2/ ( x1q1 + x2q2) (3.17) e2 1 + en = I (3.18) 912 + 922 = 1 (3-19)
With the equations ( 3 . 1 4 ) , ( 3 . 1 5 ) , (3.18) and (3.19) equation (3.13) can be r e w r i t t e n a s : UE = x1e2 1q1A U2 1 + x2e1 2q2A U1 2 (3.20) where A U2 1 = \ll\in - Un] NA A U1 2 = J Z [ U1 2 - U2 2] NA and 921 = 82T21/ /^81 + 92T21^ e12 = elT1 2/ ( e2 + 91T1 2) where 20
T2 1 = e x p ( - AU2 1/RT) t1 2 = e x p ( - AU1 2/RT)
To o b t a i n an expression f o r the molar excess Helmholtz energy A^, the f o l l o w i n g equation i s used
d(A^/T)/d(l/T) = \>l (3.21) Equation (3.21) i s used t o o b t a i n AE. I n t e g r a t i n g from 1/T t o 1/T g i v e s
^ = ƒ U d ( l / T ) + constant o f i n t e g r a t i o n (3.22) 1 / T q m
The constant o f i n t e g r a t i o n i s determined with 1/TQ •* 0 , the boundery
c o n d i t i o n the equation o f Guggenheim (1952) [ r e f . 7] f o r athermal mixtures of molecules o f a r b i t r a r y s i z e and shape i s used
(Am/ R TU e r m a l = " (Sm/ R>athermal =x l 1 n( * l/ xl > + x2 l n( * 2/ x2 >
+ ! Z C q1x1 In ( e ^ ) (3.23)
+ q2x2 In (e2/<(>2)]
At low pressures the f o l l o w i n g assumption i s made (Hildebrand e t a l . , 1950) [ r e f . 8] (Am>T,V - (Gm>T,p <3-2 4> Equation (3.21) then y i e l d s (Am/R T>T,V * (Gm/ R T)T, p " ^ ' ^ c o m b i n a t o r i a l + (Gm/ R T> r e s i d u a l <3-2 5> where ( ^ c o m b i n a t o r i a l =x l l n( Vxl > + x2 1 n( *2/ x2) + *Z C t» lxl (3.26) 1 n(e1/c()1) + q2x2 ln(e2/<f>2)]
(Gm/ R T> r e s i d u a l = " xl * l 1 n [ 6l + 62 e xP a U2 1/ R T)]
(3.27) - x2q2 l n [ e2 + 8, exp (- AU1 2/RT)]
which i s the Uniquac e q u a t i o n . 2.4 The Muniquac equation
In t h e i r d e r i v a t i o n of the Uniquac equation Maurer and P r a u s n i t z used the assumption that l o c a l compositions can be r e l a t e d to o v e r a l l compositions through Boltzmann f a c t o r s (3.14) and (3.15)
- 6 1 e « p (- ^ ) , 3 . 1 5 ,
'22 °2
For an i d e a l s o l u t i o n ^ l ^ l l = 62/'er A r e a l Din a ry s o l u t i o n should show ideal behaviour a t i n f i n i t e d i l u t i o n of one component i n the other o n e , i . e . the l o c a l and the o v e r a l l s u r f a c e f r a c t i o n s should approach each other a s y m p t o t i c a l l y a t i n f i n i t e d i l u t i o n . In other words the e x p [ - AU^/RT] and exp[- AU1 2/RT] i n (3.14) and (3.15) should approach a s y m p t o t i c a l l y the value of 1. This behaviour i s not obtained by equation (3.14) and (3.15) and t h e r e f o r e i t i s proposed to change equations (3.14) and (3.15) i n t o :
Jl l u l
The d i f f e r e n c e i n the behaviour of equation (3.14) and equation (3.28) i s shown s c h e m a t i c a l l y i n F i g . 3 . 1 . The s o l i d l i n e i n t h i s f i g u r e represents i d e a l behaviour of a l i q u i d m i x t u r e . The dashed s t r a i g h t l i n e s d e v i a t e from that immediately i n case e „ / e ^ > 0 . The dashed dotted l i n e s d e v i a t e from an i d e a l behaving mixture only i f e2/ 8 ^ i s l a r g e enough. That i s e x a c t l y the behaviour wanted. Two cases are shown one with the e x p [ . . . ] term > 1 and one with the e x p [ . . . ] term < 1.
H
exp(--)>1 exp(- • • )=1
exp(-0<1
Figure 3.1 Relation of local and overall surface fractions in liquid mixtures at infinite dilution.
Ideal behaviour,
Uniquac equation according to eq. (3.14). Muniquac equation according to eq. (3.28).
A s i m i l a r r e s u l t i s obtained comparing equations (3.15) and ( 3 . 2 9 ) . F o l l o w i n g the d e r i v a t i o n of Maurer e t a l . [ r e f . 4] the r e s i d u a l p a r t of the Muniquac equation i s obtained a s :
GE 6, e„ U „ . - I L ,
FT (
Res-) " "
xl ^ l 4
1 n C 6l
+ 92 6 X P " e l
" T T
]"
e2 91 U12 " U2 2
x2q2 — l n C B2 + ex exp - — ]
(3.30)
Comparing the r e s i d u a l terms of the Uniquac and Muniquac equation (3.27) and (3.30) the d i f f e r e n c e between both equations i s t h a t the r a t i o s eu/fij and en/ e0 which are introduced i n the exponential terms now a l s o appear i n
1 • E
E G ( R e s ) R T A 0.4 0 3 0.2 0.1 .10 .30 .50 .70 .90 A ^ x B
Figure 3. 2 Effect of the modification in the residual part of the Uniquac equation for the simple case that tJJ21/RT = AU /RT - 1 and q1 = q£ = 1.
0 Muniquac residual equation. 0 Uniquac residual equation.
r e s i d u a l f o r both equations which i s shown i n F i g . 3 . 2 , f o r the simple case that AU2 1/RT = AU1 2/RT = 1 and = q2 = 1. Although the m o d i f i c a t i o n was introduced to o b t a i n a p h y s i c a l l y c o r r e c t behaviour a t i n f i n i t e d i l u t i o n the G^(Res,) equation i s s i g n i f i c a n t l y e f f e c t e d over the whole composition range.
3 Helmholtz energy model
The mean f i e l d l a t t i c e gas model proposed by Trappeniers [ r e f . 9 , 1 0 ] was extended by K l e i n t j e n s [ r e f . 1 1 , 1 2 , 1 3 ] to d e s c r i b e q u a n t i t a t i v e l y L + G and
L + L phase behaviour of m i x t u r e s . In t h i s model, a pure component i s represented by a mixture of occupied and vacant s i t e s of a r e g u l a r l a t t i c e . Pressure and temperature e f f e c t the number of vacant s i t e s , but the volume per l a t t i c e s i t e V Q remains c o n s t a n t . For a substance with molar mass M, the d e n s i t y d i s r e l a t e d to the f r a c t i o n <|>. of occupied s i t e s by
where m^ i s the number of s i t e s occupied by a molecule which i s equal to the number of segments of a molecule. The change i n Helmholtz f r e e energy AA f o r a pure component ( i . e . i s a pseudo-binary mixture of holes and occupied s i t e s ) i s given by:
AA
RT = "O 1 n * 0 +
" l
1 n *1 + n0 *1 where = an entropy c o r r e c t i o n termyl = 1 - C T J / O Q
°-\ + ~1 :—
1 1 " Y| *2
(3.32)
°1^C T0 = r a t l° ° ^ c o n t a c t s u r f a c e area of a pure component and holes 9 j . = i n t e r a c t i o n energy parameter.
The equation of s t a t e f o l l o w s using
(3.33)
3AA\
where V = (n^ + n^ m)Vp and consequently
dV = vQ d nQ (3.34)
S u b s t i t u t i o n from (3.34) i n (3.33) gives
p = - i - M (3.35)
p vQ 3nQ I >
which leads with (3.32) to
P V0 1 2 ? - ?
~ RT~
= l n* 0
+ ( 1~ m
^*l
+*1
{ a + 91 1
( 1" '
Yl * l
) } ( 3 , 3 6 )This equation contains at l e a s t four pure component parameters V Q , a, and 3 i i ( T ) which can be obtained from PVT d a t a . For n-alkanes K l e i n t j e n s [ r e f . 12]
used Tompa's model to determine m^ [ r e f . 1 4 ] and d i v i d e d the molecules i n t o fCH3) and ( C ^ C H , , ) groups, = J nc + 1 where nc = the number of carbon atoms
i n the p a r a f i n . In t h i s work the value of m^ i s obtained from the van der Waals volume of the molecule d i v i d e d by the value assigned to a {CHpCHp) group.
For the c a l c u l a t i o n of b i n a r y l i q u i d - l i q u i d e q u i l i b r i a an e x p r e s s i o n f o r the change of the Helmholtz f r e e energy f o r a pseudo-ternary mixture of component 1 , component 2 and holes i s used.
^ = nQ In <j>0 + n j In ^ + n2 In $2 + nQ ^ {a + gn( l - y ^ Q "1} +
N0 * 2 { A + 92 2 ^ " ^ 2 ^ "1 } + ml " l <T'2 { A1 2 + 9m ^ " "
G0 1 * 0 1 h " G0 2 * 0 2 * 2 ( 3 > 3 7 )
where Q e 1 - <t>j - y2 ^
In e q . ( 3 . 3 7 ) the l a s t two terms do not have to be d i s c u s s e d f u r t h e r as they do not show up a t the spinodal and c r i t i c a l c o n d i t i o n s or cancel i n
e x p r e s s i o n s f o r the chemical p o t e n t i a l s and p r e s s u r e . Only two b i n a r y parameters and gm are i n t r o d u c e d which have to be determined from
experimental data of the m i x t u r e . With e q . ( 3 . 3 5 ) the f o l l o w i n g equation of s t a t e i s o b t a i n e d .
pv , ,
p j - = In +0 + ( 1 - — ) ^ + ( 1 - - ) *2 + ( V l + a2*2) ( *1 + *2)
{ A1 2 + 9m ^ " Y2 ^Q _ 2 } * 1 * 2 + { 9l l (! " V h + 92 2 ^ " yZ^ * 2}
(Q " *0) Q "2 ( 3 - 3 8 )
According to Gibbs two equations can be d e r i v e d f o r the c r i t i c a l p o i n t s . However, as the mixture c o n s i d e r e d here i s a pseudo-ternary one, the s p i n o d a l and c r i t i c a l equations are more c o m p l i c a t e d than f o r a b i n a r y m i x t u r e . The spinodal equation i s d e f i n e d by:
J _ = A -A - A2 = 0 ( 3 . 3 9 )
sp <p-[<?^ <P2<tL2 <t>^92
2 2 where A
h h ~ «2( A A / N+) / «+ 1a *2
and the c r i t i c a l equation Jc r by:
J „ . . = « J,. „ / « * • , A, . - 8 J A = 0 (3.40) c r sp y l
^2^2
P ^ l ^ A d e t a i l e d d e r i v a t i o n of these equations i s a v a i l a b l e [ r e f . 11]. 4 Literature 1. P r a u s n i t z , J . M . , M o l e c u l a r thermodynamics of f l u i d phase e q u i l i b r i a , P r e n t i c e - H a l l , (1969). 2. Renon, H . , J . M . P r a u s n i t z , A . I . C h . E . J . , 14, (1968), 135. 3. Abrams, D . S . , J . M . P r a u s n i t z , A . I . C h . E . J . , 21, (1975), 116. 4. K l e i n t j e n s , L . A . , T h e s i s , Essex, (1979). 5. W i l s o n , G . M . , J . Am. Chem. S o c , 8 6 , (1964), 127. 6. Maurer, G . , J . M . P r a u s n i t z , F l u i d Phase E q u i l i b r i a , 2 , (1978), 91. 7. Guggenheim, E . A . , M i x t u r e s , Clarendon Press O x f o r d , (1952).8. H i l d e b r a n d , J . H . , R.L. S c o t t , The S o l u b i l i t y of N o n e l e c t r o l y t e s , R e i n h o l d , New Y o r k , (1949).
9. Schouten, J . A . , C.A. ten Seldam, N . J . T r a p p e n i e r s , P h y s i c s , 73, (1974), 556. 10. I d . , Chem. Phys. L e t t e r s , 5., (1970), 541. 11. K l e i n t j e n s , L . A . , T h e s i s , Essex, ( 1 9 7 9 ) , 96 12. K l e i n t j e n s , L . A . , R. K o n i n g s v e l d , C o l l o i d & Polymer S e i . , 258, ( 1 9 8 0 ) , 711. 13. K l e i n t j e n s , L . A . , R. K o n i n g s v e l d , J . Electrochem. S o c , 127, (1980), 2352. 14. Tompa, H . , Polymer S o l u t i o n s ( B u t t e r w o r t h , (1956)).
Chapter 4 E X P E R I M E N T A L E Q U I P M E N T
1 The autoclave
The autoclave used i n t h i s work i s s u i t a b l e f o r pressures up t o 400 MPa and has been d e s c r i b e d i n d e t a i l p r e v i o u s l y by van Hest & Diepen [ r e f . 1 ] and others [ r e f . 2 , 3 , 4 , 5 ] . I t i s provided w i t h two f l a t sapphire windows and magnetic s t i r r i n g . The autoclave i s shown s c h e m a t i c a l l y i n F i g . 4 . 1 . With t h i s type of autoclave phase changes can be observed v i s u a l l y by p l a c i n g a g l a s s -vessel c o n t a i n i n g a mixture to be i n v e s t i g a t e d between the windows o f the a u t o c l a v e . The autoclave c o n s i s t s of three major p a r t s , a s t e e l c y l i n d e r K (the autoclave h o u s i n g ) , a plug E and a c l o s i n g nut F. For s e a l i n g the autoclave windows and the autoclave plug v i t o n 0 - r i n g s a r e used (maximum temperature 445 K ) . The temperature i s measured w i t h a platinum r e s i s t a n c e thermometer placed i n s h a f t D, which reaches to the top of the g l a s s v e s s e l .
2 The autoclave set up
For a l l the measurements an autoclave s e t up was used which i s s c h e m a t i -c a l l y given i n F i g . 4 . 2 [ r e f . 6 ] . One part of the s e t up i s f o r -c o n t r o l l i n g and measuring the temperature and the other part f o r c o n t r o l l i n g and measuring the pressure i n the a u t o c l a v e .
2.1 Temperature c o n t r o l and measurement
In order to achieve a constant temperature the autoclave i s placed i n a 3
Figure 4.1 The autoclave.
A - degass valve; B - electromagnet for activating the magnetic stirring mechanism C; D shaft for resistance thermometer; E plug; F closing nut; G, J glass measuring vessel; H
-sapphire window; K - autoclave housing; L - mercury container.
Table 4.1 Temperature control of the autoclave
temperature content of the thermostat constancy of the autoclave
[K] temperature < 290 a l c o h o l / w a t e r 90:10 < .01 K
275 - 368 water < .01 K 335 - 400 g l y c e r o l 98% < .01 K 400 - 445* a i r * * < . I K * Maximum temperature f o r the v i t o n 0 - r i n g s , which seal the a u t o c l a v e . * * The autoclave i s heated w i t h two heat j a c k e t s .
Table 4.1 contains the l i q u i d s usad i n d i f f e r e n t temperature ranges. This t a b l e a l s o shows the achieved constancy of the temperature (AT < + 0.01 K ) . Above 400 K i t i s used as an a i r b a t h . In the l a t t e r c a s e , however, the temperature i s only s t a b l e w i t h i n + 0.05 K. The temperature i n the autoclave was determined w i t h a platinum r e s i s t a n c e thermometer I. The e l e c t r i c
r e s i s t a n c e of t h i s thermometer was measured w i t h a r e s i s t a n c e measuring bridge ( B l e e k e r ) . The combination of r e s i s t a n c e thermometer and r e s i s t a n c e measuring bridge i s c a l i b r a t e d with the help of a standard thermometer [ r e f . 7]. 2.2 Pressure c o n t r o l and measurement
Pressures up to 50 MPa are generated with a handpump N, which pumps water from a r e s e v o i r i n t o the a u t o c l a v e . Higher pressures are obtained using a pneumatically d r i v e n o i l pump B ( H a s k e l ) . Fine adjustments are p o s s i b l e with the s p i n d l e pump 0 . The pressure i s t r a n s m i t t e d from o i l to water w i t h a s o -c a l l e d o i l / w a t e r s e p a r a t o r M. The pressure i s measured using a pressure
balance w i t h r o t a t i n g weights P ( ' t H a r t ) . The a b s o l u t e e r r o r i s .1 MPa, but pressure changes of .01 MPa can be d e t e c t e d . Because the weights are r o t a t i n g the system always l o s e s some o i l along the p i s t o n . To compensate f o r t h a t i t i s necessary to pump some more o i l i n t o the systen once i n a w h i l e i n order to avoid a decrease i n p r e s s u r e . The combination of the pressure balance and the pneumatic pressure pump i s used as a manostat. This i s described i n d e t a i l by de Loos [ r e f . 6 , 8 ] . The b a s i c p r i n c i p l e of the manostat i s t h a t the pressure pump, which i s seperated from the autoclave with a pneumatic v a l v e , holds a
pressure that i s always a b i t higher than the pressure i n the a u t o c l a v e . If the p i s t o n h o l d i n g the r o t a t i n g weights lowers to the l e v e l of an e l e c t r i c a l s e t p o i n t t h i s pneumatic valve opens. Now some o i l flows i n t o the system and r a i s e s the p i s t o n above the set p o i n t and the valve c l o s e s a g a i n . A l l t h a t i s achieved without any change of the t o t a l pressure i n s i d e the a u t o c l a v e .
Z Literature
1. H e s t , van J . A . M . , G.A.M. Diepen, Phys. Chem. High Pressures Pap. Symp., (1962).
2. De Loos, T h . W . , T h e s i s , D e l f t , (1981). 3. K o o i , van der H . J . , T h e s i s , D e l f t , (1981). 4. A a l d i j k , L . , T h e s i s , D e l f t , (1971).
5. Berkum, van J . G . , G.A.M. Diepen, J . Chem. Thermodynamics, (1979), 11. 6. De Loos, T h . W . , W. Poot, G.A.M. Diepen, Macromolecules, 16, (1983), 111. 7. O t t , P . L . , T h e s i s , D e l f t , (1983).
Chapter 5 EXPERIMENTAL BINARY [2-METH0XYETHAN0L + ALKANE] EQUILIBRIA AND CRITICAL CURVES
1 General remarks
To check whether the Gibbs energy models and the g a s - l a t t i c e model as described i n chapter 3 are capable to represent l i q u i d - l i q u i d e q u i l i b r i a a t elevated pressures r e l i a b l e experimental data are necessary. Therefore l i q u i d -l i q u i d e q u i -l i b r i a i n b i n a r y 2-methoxyethano-l [2-ME + a-lkane] systems were measured, i n c l u d i n g c r i t i c a l p o i n t s . The alkanes i n v e s t i g a t e d a r e : n-hexane
( C g ) , n-heptane ( C y ) , n-octane ( Cg) , 2-methylpentane ( 2 - C g ) , 2-methylhexane ( 2 - C6) , 2-methylheptane ( 2 - C7) , 2,4dimethylhexane ( 2 , 4 C g ) , 2 , 2 , 4 t r i m e t h y l -pentane ( 2 , 2 , 4 - C g ) and n-dodecane ( C ^ ) - This choice makes i t p o s s i b l e to reveal how c h a i n l e n g t h and degree of branching o f the alkane e f f e c t s the phase behaviour. The r e l e v a n t p h y s i c a l c o n s t a n t s , the d e l i v e r e r and the p u r i t y of the chemicals used are contained i n Table 5 . 1 .
2 Preparation of a binary \_2-met~hoxyethanol + aVka.ne~\ mixture
To prepare a mixture of a p a r t i c u l a r composition measured volumes of 2-ME and the alkane are i n t r o d u c e d with an "Agla" micrometer s y r i n g e i n t o the measuring v e s s e l r e s p e c t i v e l y . The t o t a l volume of both components was always 0 , 5 m l . The vessel a l s o contains one or two metal k e r n e l s , surrounded w i t h g l a s s , which w i l l s t i r r the mixture by means of the magnetic s t i r r i n g mechanism.
To prevent evaporation of the components o r d i f f u s i o n of water i n t o the 2-ME the vessel ends i n a narrow g l a s s tube of a length of 5 cm and an i n t e r n a l diameter of 20 urn. (See gasrack F i g . 5 . 1 ) . A f t e r f i l l i n g the v e s s e l
Table 5.1 The relevant physical constants, the deliverer and the purity of the chemicals used
component m o l . m e l t i n g b o i l i n g d e n s i t y r e f r a c t i v e producer p u r i t y
weight p o i n t p o i n t at T=20°C index weight
[ ° C ] .1 MPa Ekg/m3] n2 0 per cent
[ ° C ] n-hexane 86.18 - 9 5 . 0 68 95 660 3 1 37506 Fluka 99 7 n-heptane 100.21 - 90 61 98 42 683 76 1 38777 Merck 99 5 n-octane 114.23 - 56 79 125 66 702 5 1 3974 ii M n-dodecane 170.34 - 9 6 216 3 748.7 1 4216 Fluka II 2-methylpentane 86.18 -153 67 60 27 653 2 1 3715 Merck 99 7 2-methylhexane 100.21 -118 27 90 65 678 69 1 38485 II 2-methylheptane 114.23 -109 0 117 65 698 0 1 39494 II II 2 , 4 d i m e t h y l -hexane 114.23 ? 109 4 700 36 1 39534 II 11 2 , 2 , 4 t r i m e t h y l -pentane 114.23 -107 38 99 2 691 9 1 3915 II H 2-methoxyethanol 76.11 - 85 1 125 0 964 7 1 4024 II 99 5
w i t h both components i t i s attached to a s o - c a l l e d gasrack ( F i g . 5.1) [ r e f . 1] to be degassed and f i l l e d with mercury, which acts as the pressure i n t e r -mediate between the mixture and the p r e s s u r i s i n g f l u i d (water) i n the
a u t o c l a v e . The gasrack and the measuring v e s s e l can be evacuated using an o i l d i f f u s i o n pump. The pressure i n the gasrack i s measured w i t h an i o n i s a t i o n vacuum meter (Penning tube ( P h i l i p s ) ) and i s approximately 3 a 4.10 Pa.
The procedure i s as f o l l o w s . F i r s t the mixture i s f r o z e n and the gasrack i s evacuated with valve c l o s e d . Then i s opened and the measuring vessel i s evacuated. Now valve i s c l o s e d and the mixture i s melted again i n vacuum and consequently degasses. Then i t i s f r o z e n again and by opening the vessel i s again evacuated. This procedure i s repeated three t i m e s . Then the vessel i s turned upside down w h i l e the mixture i s kept f r o z e n . A f t e r c l o s i n g valve K. the pressure i n the r i g h t hand s i d e of the gasrack i s r a i s e d using
nitrogengas and mercury i s pressed from vessel N i n t o the measuring vessel to a pressure of .1 MPa. A f t e r t h a t the v e s s e l i s disconnected and assembled to a c o n t a i n e r which i s f i l l e d with mercury and t r a n s f e r r e d to the a u t o c l a v e .
The mercury i n vessel N i s degassed beforehand w i t h the procedure as f o l l o w s . A f t e r pouring the double d e s t i l l e d mercury i n t o g l a s s v e s s e l 0 the gasrack i s evacuated and valve i s c l o s e d . Now some n i t r o g e n pressure i s given on the r i g h t hand s i d e of the gasrack and consequently the mercury i s pressed s l o w l y from vessel 0 to vessel N. While the mercury drops i n t o vessel N the enclosed gas i s l i b e r a t e d and pumped away.
3 Isopleths
If the mixture of known composition i s kept at a constant temperature the v a r i a t i o n of the pressure a l l o w s to determine t h a t p r e s s u r e , at which a second l i q u i d phase appears or d i s a p p e a r s . R e p e t i t i o n of t h i s procedure at various temperatures a l l o w s to determine an i s o p l e t h (the b o r d e r l i n e between the one-phase and the two-one-phase region f o r a given c o m p o s i t i o n ) .
For mole f r a c t i o n s 0.2 < x < 0 . 8 i n the mixture [x 2-ME + ( l - x ) a l k a n e ] the accuracy of the measured pressures at which the second phase appeared or disappeared was + 0 . 0 5 MPa. This i n c r e a s e s to + 0 . 4 MPa i n the mole f r a c t i o n ranges 0.08 < x < 0 . 2 and 0 . 8 < x < . 9 2 . This l a r g e r u n c e r t a i n t y i s caused by the f a c t t h a t , i n these c o n c e n t r a t i o n r a n g e s , the volume of the second phase i s very small and i n c r e a s e s only s l i g h t l y with a r a i s e of the pressure of .1 MPa. Furthermore an i n c r e a s i n g delay i n the appearance of the second phase o c c u r s . The mole f r a c t i o n s x £ 0.08 and x % 0.92 are the l i m i t s of an accurate o b s e r v a t i o n of the appearance and disappearance of the second phase f o r the systems i n v e s t i g a t e d i n t h i s work. The e r r o r i n the mole f r a c t i o n s i s mainly due to l o s s e s i n t o the mercury and i s estimated from the r e p r o d u c i b i l i t y of the measurements to be l e s s than 0 . 0 0 1 .
A l l the data p o i n t s of the i s o p l e t h s of the b i n a r y [2-ME + alkane] systems i n v e s t i g a t e d are given i n Appendix B.
F i g . 5 . 2 presents some s e l e c t e d i s o p l e t h s of the b i n a r y system [x 2-ME + ? 2
( l - x ) o c t a n e ] . The s l o p e dp/dT of the i s o p l e t h s i s p o s i t i v e , d p/dT i s always p o s i t i v e , t o o . At low and high c o n c e n t r a t i o n s of 2-ME and with i n c r e a s i n g molecular weight t h i s behaviour i s even more pronounced. The i s o p l e t h s shown are c h a r a c t e r i s t i c f o r a system w i t h an upper c r i t i c a l s o l u t i o n temperature
(UCST) and a lower c r i t i c a l s o l u t i o n pressure (LCSP). For [2-ME + n-octane] a t low pressures a dp/dT of 50 bar/K i s found and a t high pressures the slope increases t o 100 bar/K.
At pressures of 40 MPa i t was not p o s s i b l e to perform accurate
experiments f o r a l l compositions of t h i s system. This i s caused by the f a c t t h a t the r e f r a c t i v e i n d i c e s o f both phases become more o r l e s s i d e n t i c a l . The pressure range of the s o - c a l l e d i s o o p t i c a l region [ r e f . 2] i s + 20 MPa f o r 2-ME c o n c e n t r a t i o n s c l o s e to the c r i t i c a l c o n c e n t r a t i o n . (The i s o p l e t h s a t the high temperatures i n F i g . 5 . 2 ) . At these c o n c e n t r a t i o n s the appearance o f the second l i q u i d phase i s accompanied by a turbulence which can be detected v i s u a l l y although the r e f r a c t i v e i n d i c e s of both phases are almost e q u a l . For 2-ME c o n c e n t r a t i o n s f u r t h e r away from the c r i t i c a l composition i . e . f o r the systems a t lower T the volume of the second phase which i s formed i s
r e l a t i v e l y small and h a r d l y to be seen. Consequently the pressure range of the i s o o p t i c a l region i n c r e a s e s f o r these c o n c e n t r a t i o n s to + 40 MPa.
Table 5.2 Refractive indices of the pure components at .1 MPa and the isooptical pressure for the L2-ME + alkanel mixtures
compound nn2 D2 D° (.1 MPa) ° (.1 MPa) i s o o p t i c a l pressure + 20 MPa n-hexane 1.37506 230 2-methylpentane 1.3715 240 n-heptane 1.38777 120 2-methylhexane 1.38485 130 n-octane 1.3974 40 2-methylheptane 1.39494 40 2,4-dimethylhexane 1.39534 40 2 , 2 , 4 - t r i m e t h y l p e n t a n e 1.3915 50 n-dodecane 1.4216
-2-methoxyethanol 1.4024 -4 0