Pure & Appi. Chem., VoL 47, pp. 277—291. Pergamon Press, 1976. Printed in Great Britain.
HIGH PRESSURE THERMODYNAMICS OF MIXTURES
G. M. SCHNEIDER
University of Bochum/GFR, Institute of Physical Chemistry, German Federal Republic
Abstract—In the present survey some important trends in the high pressure thermodynamics of fluid mixtures of
non-electrolytes are reviewed.
First the pressure dependence of excess functions such as the excess Gibbs energy GE, the excess enthalpy HE,
the excess entropy SE and the excess heat capacity CE is discussed. It can be obtained from the knowledge of the
excess volume yE as
a function of pressure, temperature and composition. Experimental results demonstrate that
the variations of yE
as
a function of pressure and temperature can be important and comparable to those of the
molar volumes themselves. A detailed discussion shows that further progress in this field depends on the
development of an accurate equation of state for mixtures.
Since furthermore direct calorimetric measurements are practically completely lacking at high pressures most
thermodynamic informations have to be deduced up to now from high pressure phase equilibria and critical
phenomena where our knowledge is much better. The pressure dependence and critical phenomena of liquid—gas,
liquid—liquid and gas—gas equilibria will be shortly reviewed. Mainly binary systems will be treated but phase
separation phenomena in some ternary and multicomponent systems will be equally considered. Recent results
concerning the rate of phase separation will be additionally presented.
New developments during recent years have shown that the limits between liquid—gas, liquid—liquid and gas—gas
equilibria are not well defined and that continuous transitions occur. This continuity will be demonstrated for binary
mixtures of hydrocarbons with carbon dioxide and methane and for some inert gas systems.
The significance of high pressure phase equilibria in fluid mixtures for practical applications is shortly discussed,
e.g. for high pressure extractions, supercritical fluid chromatography and for some other high pressure techniques
and processes.
Methods for the calculation of fluid phase equilibria in mixtures under high pressure are reviewed. They start
from equations of state or from theories of mixtures using sometimes sophisticated mixing rules for the parameters.
Some results are presented and compared with experimental data.
Finally a characteristic example for the pressure dependence of chemical equilibria in liquid solutions is
considered and the standard value of the reaction volume i V° is determined from measurements of the equilibrium
constant K as a function of pressure at constant temperature. It is shown how standard values of the reaction
enthalpy iH° can be obtained from temperature jump experiments in such solutions at high pressure.
INTRODUCTION
The thermodynamics of mixtures is one of the most
important fields of thermodynamics. It has been of funda-
mental interest to chemists for a long time especially
with respect to the thermodynamic and theoretical de-
scription of mixtures and solutions, to chemical equilibria
and to some important methods of separation. An enorm-
ous amount of experimental and theoretical work has
already been done in the thermodynamic investigation of
mixtures but up to now the pressure variable has been
widely neglected although in the first decades of our
century very promising efforts had been done especially by
the Dutch school.
Thus the high pressure thermodynamics of mixtures is
still in a developing state whereas much more research
activities have already been dedicated to the high pressure
properties of pure compounds, the most recent literature
being regularly reported in the Bibliography of High
Pressure Research' (for literature up to 1969, see Ref. 2).
In the present review some important trends in the high
pressure thermodynamics of fluid mixtures of non-
electrolytes will be discussed; the accent will be on the
discussion of some special high pressure effects that
might be of interest to thermodynamicists working in
other fields. Polar and ionic fluids at high pressures and
temperatures have been treated by Franck3 at the Third
International Conference on Chemical Thermodynamics.
Electrolyte solutions under pressure have been recently
reviewed by Hamann4 and critical phenomena in electro-
lyte solutions by Horvath."3 Throughout this paper the
pressure unit bar will be used (1 bar = 10
Pa =
0.1
MPa;
1 kbar = 10
bar; 1 Mbar =
106
bar).
In Fig. 1 some pressures that are important in nature
and chemical industry are summarized, making use of a
similar compilation by Pilz.5 On the left hand side some
pressures relevant in nature are given in logarithmic
scale: critical pressures are between 2.3 bar for helium
and approximately 1.5 kbar for mercury; the deepest
ocean corresponds to approximately 1 kbar and the center
of the earth to approximately 4 Mbar.
More interesting for chemists is the right hand column in
Fig. 1 where some high pressure processes and techniques
are compiled that are relevant to the European chemical
industry. Between 100 and 1000 bar we have high pressure
liquid chromatography (LC) and supercritical fluid
chromatography (SFC), crystal growth, some hydrogena-
tions; the syntheses of ammonia, of methanol, of some
organic compounds by oxo synthesis and of acetic acid;
and additionally some polymerisations. The fabrication of
polyethylene occurs between 1.5 and 3 kbar and the
synthesis of diamond at approximately 50 kbar. For all
these methods and processes the high pressure ther-
modynamics of mixtures is of fundamental importance.
Hydrostatic extrusion, high pressure densification and
sintering will become more and more interesting for
material research. Explosive welding and plating are
indispensable techniques for the fabrication of chemical
reactors and tubing.
EXCESS FUNCTIONS AND EQUATION
OF STATE OF MIXTURES
For the thermodynamic description of mixtures (espe-
cially of liquid mixtures) excess functions are often
277
278
G. M. SCHNEIDER
itself where
1
p(crit) helium
Fig. 1. Pressure scale in nature and chemistry (according to a
similar compilationby Pilz5).
used.' A molar excess function ZmE
is defined as the
difference of the molar function Zm between a real and an
ideal mixture at the same values of temperature T, total
pressure p and mole fraction x, of all constituents i, where
Z=G, H, S, V, C, etc.
XmE = Zm (real) — Zm
(ideal) for p, T, x1 = const. (1)
It can be easily shown that the following relations hold for
the pressure dependence of the excess functions
= VmE
\
t9p IT,;
= V
E -
TI'--'\
' t9J. ,/T,;
m
\, 3T ),
(3Sm'\ —
(8Vm'\
9p )T,x1
' aT IpX
—
132
VmE
-J --l---.
\
3p iT,;
\ 31
It follows from the right hand sides of the relations (2—5)
that the pressure derivatives of all molar excess functions
are completely described by the molar excess volume VmE
as a function of temperature, pressure and mole fractions.
From this the excess functions themselves can be ob-
tained by integration e.g. the molar excess Gibbs energy
by
GmE(p) —
GmE(pO)
= J
VmE
dp for T, x = const. (6)
where GmE (O)
is
the value of the molar excess Gibbs
energy at the reference pressure p°.
Thus the problem of thermodynamics of mixtures re-
duces to the knowledge of the molar excess volume VmE
as a function of pressure, temperature and composition or,
more generally, to the knowledge of an equation of state
for mixtures at high pressure.
Such an equation of state can be based on the excess
volume VmE or on the molar volume Vm of the mixture
Vm (real) = Vm (ideal) + VmE
Xi V'nj + VmE. (7)
According to eqn (7) Vm 5 made up of two terms: (1) of
the ideal term taking account of the molar volumes V of
the pure components i and (2) of the molar excess volume
Vm'
as
a correction term.
For pure liquids already many p VT measurements have
been undertaken up to quite high pressures. The data
have been correlated with the well-known equations of
state of Tait,9 Hudleston,'° Hayward,1' Chaudhuri,'2
Witt,13 and others. Several authors have shown recently
that among the two-parameter equations the old Tait
equation does astonishingly well.2'7'14'15 Up to now very
few measurements, however, exist on mixtures and here
the equations cited are not yet well tested especially with
respect to the mixing rules for the parameters. The
volumes of coexisting phases at high pressures have been
measured by Tsiklis et
a!.114
The
situation is not at all good for the molar excess
volumes VmE (for a review see16; for references
see16'110'111), Up to now only very few VmE data exist at
high pressures e.g. by Hamann and Smyth,'7 Engels,12°
Lamb and Hunt,21 Korpela,22 and Götze.23'24 VmE data can
be obtained from very accurate p VT data measured
separately on different mixtures and on all pure compo-
nents or more precisely from measuring VmE directly as a
function of pressure, temperature and composition. In
Figs. 2 and 3 some results of Götze24'25 are shown for the
system water-acetonitrile. In Fig. 2 the excess volume
Vm' in cm3 moV1 is plotted against the mole fraction of
acetonitrile for 100, 1000 and 2500 bar and for each
pressure at 25, 50 and 75°C. The pressure influence is very
remarkable: with increasing pressure the inflection point
becomes more and more accentuated and at the highest
pressures and temperatures the excess volume changes
sign for mixtures rich in acetonitrile. In Fig. 3 the pressure
derivatives of the excess functions that follow from the
eqns (2—4) with the Vm' data of Fig. 2 for x = 0.5 and
T = 50°C are plotted for the water-acetonitrile system.
All derivatives are negative in the pressure range 100—
2500 bar. In contrast to (3GmE/3p)T,x is monotonous,
(3HmE/3p)T,Xi and (3SE/ap)T,x run through minima be-
tween 500 and 1000 bar which correspond to inflection
points in the HE(p)
and
SE(p) curves respectively.
Figure 2 shows that the absolute values of the excess
volumes VmE are small in comparison to the molar
volumes Vm themselves whereas the variations of the
excess volumes with pressure, temperature and composi-
tion can be important and comparable to those of the
molar volumes. It follows that an equation of state based
on the excess volume as a function of pressure, tempera-
ture and composition should be much more promising
than one based on the molar volumes. Since nearly
nothing is known up to now the problem of an equation of
state for mixtures is still a wide field for future research
activities (for a compilation see Ref. 111).
Since calorimetric data such as heat capacities of
mixtures or enthalpies of mixing26 are practically com-
pletely lacking at high pressures, most thermodynamic
information has to be deduced up to now from high
pressure phase equilibria and critical phenomena where
our knowledge is much better (for books see Ref s. 7,8,27;
for review articles see Ref s. 28—35, for experimental
techniques see Refs. 36, 75).
Nature
center of earth
mantle of earth
(3000 1cm)
Chemistry
explos. welding
explos. plating
bar
106.
1000
100 -
10
-
diamond
sintering, densificat.
earth (30km)
hydrostatic extrusion
p(crit) mercury
deepest ocean
p(crit) water
p(crit) non-polar subst.
p(crit) hydrogen
polyethylene
polymerisations
acetic acid
methanol, oxo-synthes.
ammonia
hydrogenations
crystal growth
chromatography(lc, sfc),
reactions with
compressed gases
distillations
(2)
(3)
(4)
(5)
High pressure thermodynamics of mixtures
279
dependence given in Figs. 4i—v are represented schemati-
cally. Their shapes correspond remarkably well to the
temperature-composition diagrams for constant pressure
(Figs. 4a—d) demonstrating that pressure and temperature
are equivalent thermodynamic variables.
The second horizontal row shows that with increasing
pressure upper critical solution temperatures will either
decline (Fig. 4i) or rise (Fig. 4j) or run through a tempera-
ture minimum (Fig. 41).
The third horizontal row demonstrates that with in-
creasing pressure lower critical solution temperatures
may either rise (Fig. 4m) or decline (Fig. 4n) or run
through a temperature maximum (Fig. 4p).
The pressure dependence of closed loops is schemati-
cally shown in the fourth horizontal row: closed loops
may either shrink with increasing pressure and disappear
completely at a definite pressure in the three-dimensional
temperature—pressure—mole fraction space (Fig. 4q) or
only appear at higher pressures (Fig. 4r) or resemble
hyperboloids (Fig. 4t); examples of types 4r and 4t
have been found in binary mixtures of 2-, 3- and 4-
methylpyridine with water and heavy water.234
Immiscibility surfaces of a saddle-like type such as
shown in the fifth horizontal row have also been found,
e.g. in mixtures of methane with hydrocarbons (Fig. 4u) or
of sulfur with hydrocarbons (Fig. 4v).233
Examples for all types represented in this Figure have
been summarized elsewhere.234 Here only one example
for type 4j will be given.
In Fig. 5 some very recent results of Paas4° are shown
for the methane-tetrafluoromethane system. In Fig. 5a
isobaric temperature-mole fraction profiles are plotted
which exhibit upper critical solution temperatures that rise
with increasing pressure. In Fig. Sb the corresponding
isothermal pressure—mole fraction which exhibit lower
critical solution pressures, are given.
It has been shown by several authors235'37 that some
thermodynamic information can be obtained from the
pressure dependence of critical solution temperatures.
With the assumption that Gm is analytict at and near a
critical solution point eqn (8) and with the additional
assumption of a regular solution relation (9) hold for the
pressure dependence of the critical solution temperature
in a demixing binary system
Fig. 3. Pressure dependence of the molar excess functions in the system (1 —
x)
water +
x
acetonitrile: (3G,,/3p),
(8H,,,E/ap)Tx
asafunction of pressure at T = 323.15 Kandx =
0.5
(accordingto results of Götze24).
(1-.x) H20 + x CH3CN
Fig. 2. Molar excess volumes V,,.E in the system (1— x) water +
x
acetonitrile as a function of pressure, temperature and mole
fraction (according to results of Götze24).
LIQUID-LIQUID EQUILIBRIA
In Fig. 4 all types of pressure dependence of
liquid—liquid equilibria in binary systems that are presently
known are schematically represented;233 examples exist
for all types except those with a question mark (Figs. 4g,
4k, 4o, 4s).
In the first vertical column on the left the different types
of temperature-mole fraction isobars are represented, i.e.
those with upper critical solution temperatures (UCST's,
Fig. 4a), lower critical solution temperatures (LCST's, Fig.
4b), closed loops (Fig. 4c) and a hyperbolic-like type (Fig.
4d).
In the first horizontal row pressure-mole fraction
isotherms (Fig. 4e—h) that belong to the types of pressure
tlt has, however, been shown that the assumption of G,,, being
analytical at a critical solution point is doubtful; for an explicit
critical discussion see Refs. 7, 31, 35.
0
E
a
-U
-oo
(8)
dT —
T.
.
(t92VE/8x2)
(32H,,,EIôx2)
0
E
a
E
p/bar
280
G. M. SCHNEIDER
U
I
L1[Li:
Fig. 5. Phase behaviour of the system (1 —
x)
methane + x tetrafluoromethane at elevated pressures (according to
results of Paas4°). (a) T(x)isobars; (b)p(x) isotherms.
dT T .
dp HC
It
has to be kept in mind, however, that the conditions for
the validity of relation (9) in general are largely
oversimplified.t
For
the
system
methane—
tetrafluoromethane it follows from the existence of upper
critical solution temperatures (corresponding to HE > 0
829-3437)
and
from the positive sign of dT/dp obtained
(9\
experimentally40 that VmE should be positive, a result that
had already been known for this system from normal
pressure measurements by Croll and Scott.38 Also
Paas4° could show that regular solution theory39 describes
quite well the sign and the order of magnitude of dT/dp
in this system.
For systems of type t in Fig. 4 the molar excess
enthalpy HmE should have to change its sign from minus
(corresponding to LCST's) to plus (corresponding to
UCST's) with increasing temperature at constant pressure
and it follows from eqn (9) that the molar excess volume
VmE should have to change its sign from minus to plus
with increasing pressure at constant temperature.234 The
\
'\ T=const
\
p
const"
FI]
THP
p —
Fig. 4. Pressure effects onimmiscibilityphenomenainliquidbinary systems (according to Refs. 30,32; see text).
T
00
I
x CF4
tlt has, however, been shown that the assumption of G,,) being
analytical at a critical solution point is doubtful; for an explicit
critical discussion see Ref s. 7, 31, 35.
High pressure thermodynamics of mixtures
281
curious result is that for these systems that deviate very
much from ideality, the molar excess enthalpy HmE
and
the molar excess volume VmE should both nearly vanish
just between the two immiscibility surfaces in Fig. 4t. The
change of the sign of the excess volume has already been
affirmed experimentally by EngeIs'"2° for the system 3-
methylpyridine-H20.
Liquid—liquid equilibria in ternary systems have been
investigated most frequently at normal and low pressure,
but for high pressure the experimental data are very
scarce.234 Recently the liquid—liquid equilibria in the
ternary system tetrafluoromethane-trifluoromethane-
ethane have been studied as a function of pressure by
Peter42'43 and Paas.
For aqueous solutions of organic compounds the influ-
ence of added salts on the mutual miscibility has also been
studied under pressure in some cases.45' A rather simple
example is given in Fig. 6 where the upper and the lower
solution temperatures are plotted for the system 1-
propanol-water-potassium chloride according to measure-
ments of Russo;45 the mass ratio water/1-propanol being
kept constant at a value of 1.5. Figure 6a gives the solution
temperatures as a function of the amount of potassium
chloride added at normal pressure and Fig. 6b shows the
80
i— mole KCI/l000g H20
a
1.6
1.5
1.4
12.6 12.2
11.8
11.4
11.0
10.6
7a2
12.5
—g KCI/lOOg H20
solution temperature as a function of pressure for a
mixture of constant composition. There is a striking
analogy between Figs. 6a and 6b, increasing the pressure
having a similar effect as decreasing the salt content.
Systems such as shown in Fig. 6 are well suited to study
the rate of liquid—liquid phase separation. Starting from
concentrations, temperatures and pressures in the
homogeneous region just below the plotted curves
heterogeneous states can be reached by a relatively small
temperature increase and the rate of demixing is obtained
from measuring the turbidity as a function of time. Meas-
urements were made by Jost47 and Limbach' with a
temperature jump apparatus that has been developed for
the investigation of fast reactions in solution under high
pressure.49 Here the temperature jump was created within
microseconds by discharging a high voltage capacitor
through the solution. The measuring cell with optical
windows was mounted in a high pressure autoclave. The
intensity of the transmitted light and of the light scattered
through an angle of 90 degrees could be measured
simultaneously as a function of time, stored in a transient
recorder and registered afterwards with a normal recorder.
In Fig. 7 the recorder traces thus obtained are presented for
one experiment on the system 1-propanol-water-potassium
b
p=lbar
x
I
77Xr
I 2liquid phases
;J
iliquid phase
40
U
L)
1'
80
40
I)
r-x
12.5gKCI/lOOgH2O
r
2liquid phases
qui phase
0
200 400 600
.— p/bar
Fig.
6. SaJt and pressure effects on liquid—liquid immiscibility phenomena in the system 1-propanol-water-potassium
chloride (according to results of Schneider and Russo45; w% water/w% 1-propanol =
const.
=
1.5).
100%
mt.
I
0%
mt.
I
0
transmitted light
scattered light (9 900)
Fig. 7. Rate of liquid—liquid phase separation
system 1-propanol-water-potassium chloride (according to
results of Limbach48; isT =
13°K;
A =
497
nm; p ibar;
concentrations: 50.3 wt% 1-propanol, 44.5 wt% water,
5.2wt% potassium chloride).
