International Journal of Analytical Mass Spectrometry and Chromatography, 2013, 1, 103-108
Published Online December 2013 (
Open Access IJAMSC
The Equilibrium Thermal Physics of Supercritical Fluids
Boris Sedunov
Computer and Information Systems Department, Russian New University (ROSNOU), Moscow, Russia
Received October 23, 2013; revised November 20, 2013; accepted December 20, 2013
Copyright © 2013 Boris Sedunov. This is an open access article distributed under the Creative Commons Attribution License, which
permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
To discover particular features of pure supercritical fluids, important for the supercritical fluid extraction and cleaning
technologies, the preprocessed and generalized experimental data from the US National Institute of Standards and
Technology (NIST) online database have been analyzed. Th e soft transition between gas-like and liquid-like structur es
in pure supercritical fluids has been considered in comparison with the abrupt vapor-liquid phase transition. A rough,
diffused and boiling boundary between these structures in conditions of extra high gravity is opposed to a flat
vapor-liquid boundary at a moderate gravity. The model for molecular diffusivity in carbon dioxide at temperatures near
the critical temperature discovers its proportionality to the monomer fraction density. The cluster fraction based model
for small molecular weight solids’ solubility in supercritical fluid s has been suggested and successfu lly compared with
the well-known experimental results for th e solubility of silica in water. The model shows that at growing pressure the
dissolution process has already started in a real gas and discovers the cluster fractions’ role in the solubility process.
Keywords: Supercritical Fluid; Real Gas; Cluster; Extraction; Cleaning; Diffusivity; Solubility
1. Introduction
Supercritical Fluids (SFs) present a growing interest for
modern chemical technologies, such as the Supercritical
Fluid Extraction (SFE) [1,2] and Supercritical Fluid Cle an -
ing (SFC) [3]. Staying between gases and liquids, they
provide a large solub ility of so lid subs tances, characteris-
tic to liquids, and a large diffusivity of extracted mole-
cules, specific for gases. In the analytical chemistry, the
products of the supercritical fluid extraction perfectly
match the requirements to candidate substances for fur-
ther analysis in chromatography columns and in mass
spectrometers. The joint utilization of SFE, chromatog-
raphy and mass spectrometry opens remarkable possibili-
ties for pharmacology and food and perfume industries.
A high resolution of the Supercritical Fluid Chromatog-
raphy and Supercritical Fluid Chromatography—Mass
Spectrometry in pharmacological analysis was demon-
strated by Ilia Brondz and Anton Brondz in [4]. High
diffusivity and low viscosity of SFs permit higher flows
of substances in chromatography columns, which provide
a quicker separation, better resolution and effectiveness
of the chromatography process and higher purity of se-
lected products. To improve further the SF based tech-
nologies, it is needed to penetrate deeper in the thermal
and molecular physics of SFs.
The thermal physics [5] of supercritical fluids is of a
great value for researchers and practitioners:
For research ers, it is important to build precise mod-
els of SF structures and their thermo physical prop-
For practitioners, it is desirable to use correlations
between optimal parameters of technological proc-
esses and thermo physical properties of SFs.
This investigation proceeds answer the questions [6,7]
arising in courses of Thermal and Molecular Physics of
Fluids, such as:
How the extension of the saturation curve can be
traced in the supercritical region?
What are the changes in the fluid structure, when we
come from the gaseous phase to the liquid phase
around the critical point?
In what conditions it could be possible to visualize
the boundary between the gas-like and liquid-like su-
percritical fluids?
What factors provide high diffusivity of molecules in
supercritical fluids?
What factors provide high solubility of solids in the
gas-like supercritical fluid?
The investigation is based on precise experimental
data contained in electronic databases for thermophysical
properties of pure fluids. The principal source of initial
data is the database for thermophysical pr operties of f luid
systems [8]. Due to advanced computer processing of
raw experimental data from different sources, this da-
tabase contains the generalized data with an improved
resulting precision. The data improvement process is de-
veloped by the Thermodynamics Research Center (TRC)
of the US National Institute of Standards and Techn olog y
[9] and is named as the critical evaluation of experimen-
tal data. Its main idea sounds as: “critical evaluation is a
process of analyzing all available experimental data for a
given property to arrive at recommended values together
with estimates of uncertainty, providing a highly useful
form of thermodynamic data for our customers. The
analysis is based on intercomparisons, interpolation, ex-
trapolation, and correlation of the original experimental
data collected at TRC. Data are evaluated for thermody-
namic consistency using fundamental thermodynamic
principles, including consistency checks between data
and correlations for related properties. While automated
as much as possible, this process is overseen by experts
with a great deal of experience in the field of thermody-
namic data” [9].
The principal goal of this paper is to analyze deeply
the structural changes [7] of the supercritical fluid near
the extension of the saturation curve to the supercritical
region. More practical goals are to correlate the pressure
driven steep solubility growth in SFs [1,10] with the
structural transition from the gas-like to the liquid-like
supercritical fluid and to correlate the high diffusivity [1]
of molecules in supercritical fluids with the monomer
fraction density [11].
2. The Nature of the Gas-To-Liquid
Transition in the Supercritical Zone
In our approach the transition from the gas-like to the
liquid-like supercritical fluid is named as the Soft Struc-
tural Transition [7,12,13 ]. It differs from the well known
Structural Transitions in solids [14]. The solids have long
range correlations of their elementary cells’ structures.
But the structure of a supercritical fluid is not homoge-
neous in space and dynamically changes. It can have
only short r ange structural correlations with a correlation
radius smaller than the dimensions of clusters and pores.
And it differs also from the first order liquid-vapor phase
transition that is based on the collective behavior of
molecules in a liquid state with a radius for correlations
of potential energy much larger than the dimensions of
pores. But the correlation radius for potential energy in
supercritical fluids is smaller than the dimensions of
clusters in the gas-like fluid or pores in the liquid-like
2.1. The Structure of the Supercritical Ridge
The lack of the long range structural and energy correla-
tions in supercritical fluids may be explained by large
density fluctuations discovered by the group of research-
ers headed by Dr. Keiko Nishikawa [15]. This group has
investigated by the X-ray diffraction and Raman spec-
troscopy the density fluctuations in supercritical fluids.
They discovered the peak line Pr (T) of density fluctua-
tions on the (T, P) diagram and named it as the ridge. A
similar sort of the ridge line Pr (T) may be seen on the
diagrams of many thermophysical properties, such as
heat capacities, compressibility factors, etc. [7,16].
The saturation curve Ps (T), marking the first order
phase transition between liquid and vapor phases on the
PT diagram, terminates at the critical point (Pc, Tc). The
ridge line Pr (T) is the extension of the saturation line Ps
(T) to the supercritical zone, but depends on the nature of
the property selected to build this lin e [7,16]. So, there is
a plurality of the extension lines, forming together the
zone of the soft structural transition. The zone widens
with a growth of the T – Tc difference, Figure 1, [17].
The data are from [8].
The ridge zone is the zone of large but controllable
changes in thermophysical properties and can be recom-
mended as the zone favorable for technological processes.
If the pressure P grows at a constant supercritical tem-
perature T, the fluid passes through the structural transi-
tion from the gas-like structure to the liquid-like one. But
this transition is not accompanied by abrupt changes of
thermophysical properties. In the ridge zone abruptly
changes only topology of the medium: it converts from a
scattered form, filled with clusters, to a condensed one,
filled with pores, [7,6,17]. The scattered clusters of the
gas-like structure over the ridge pressure collect together
into an infinite cluster, filled with pores of the same total
volume, as the total volume of clusters.
The supercritical fluid is a heterogeneous system [18]
that can change only its structure at changing pressure or
temperature. The structure of the supercritical fluid at
pressures lower the ridge pressure Pr (T) is fog-like and
consists of different size clusters flying in an ideal gas of
monomers, but over the ridge pressure the picture is quite
contrary: the structure is foam-like and contains pores of
different sizes spread in the continuous liquid media,
Figure 2. At the ridge pressure the average densities of
both structures are equal and close to the critical density
Dc. At the ridge pressure coexist large regions with the
gas-like structure and large regions with the foam-like
structure, mutually penetrating in each other. It gives rise
to the giant fluctuations of density, directly measured in
experiments of Dr. K. Nishikawa and her group.
In the earlier author’s work [16] the method o f estima-
tion the populations and bond parameters of clusters in
Open Access IJAMSC
Figure 1. The ridge zone in supercritical water in coor-
dinates T-P (a) and T-D (b), built by lines of constant Cp
equal to: 10000, 3000, 2000, 1500, 1000, 800, 500, and 300
Figure 2. A rough, diffused and boiling boundary between
gas-like and liquid-like structures, formed due to the lack of
surface tension at supercritical temperatures near the ridge
pressure in conditions of extra high gravity.
the gas-like SF was developed. It was shown that the
maximal number of particles in the cluster at the ridge
pressure falls with temperature shift from the critical
temperature. Clusters at the critical point are visible: the
critical opalescence becomes possible because the aver-
age dimension of clusters and pores at this point is near
the wavelength of the visible light. But over the critical
temperature the dimensions of the largest clusters and
pores are much smaller than one micrometer and even
tend to the nanometer size.
The small value for correlation radius may be respon-
sible for zero value of the surface tension at the boundary
between the liquid-like and gas-like structures, if such a
boundary may exist. To visualize this boundary it is
needed to place an equ ilibrium supercritical fluid in extra
high gravitational field, exceeding the Earth’s gravity by
many orders of magnitude. Even in these severe condi-
tions the boundary will be rough, diffused and boiling,
Figure 2.
If we compare this picture with the corresponding pic-
ture for the liquid-vapor equilibrium at subcritical tem-
peratures, it is clearly seen that the gas-like structure,
much above this boundary zone does not differ from a
real gas with clusters, and the liquid-like structure, much
lower the boundary zone, does not differ from a liquid
with pores, shown at the Figure 3.
For further perfection of the SF based technologies it
is important to understand the complex mechanisms of
clusters’ and pores’ formation in these media and their
influence on the diffusivity and so lubility.
2.2. The Parallel between the Liquid-Vapor
Phase Transition and the Soft Structural
Transition at the Supercritical Ridge Zone
The liquid and vapor phases have quite different struc-
tures. The liquid phase includes a continuous net of
molecules with varying numbers of bond s, interrupted by
pores of varying dimensions. The vapor phase includes
an ideal gas of monomers filled with molecular clusters
of varying dimensions and isomer configurations, Figure
3. This structural difference cannot disappear totally,
when we move from one phase to another around the
critical point. The properties of both structures in the
Supercritical fluid exhibit a large difference. Though the
transition is not accompanied by abrupt changes of ther-
mophysical properties, they at the ridge pressure exhibit
quick changes of values or their derivatives, seen at the
Figure 4.
The pressure dependences of the supercritical fluids’
thermophysical properties in some way repeat the similar
dependences for the system liquid-vapor. This statement
may be illustrated by the Speed of Sound (SS) pressure
dependences at near critical temperatures, Figure 4.
This figure clearly shows the parallel between the de-
scending parts of curves corresponding to the vapor
phase at subcritical temperatures and to the gas-like
structure at supercritical temperatures. And the ascending
parts corresponding to the liquid phase at subcritical
temperatures and to the liquid-like structure at super-
critical temperatures are also very similar. It is seen that
the gas-like supercritical structure inherits properties of
the vapor phase and the liquid-like supercritical structure
inherits properties of the liquid phase. An abrupt jump of
the Speed of Sound at the saturation pressure at subcriti-
cal temperatures becomes substituted by a smooth local
minimum at the ridge pressure and supercritical tem-
Open Access IJAMSC
Fi gure 3. A flat, even at a small gravity, vapor-liquid bound-
ary at a saturation pressure formed due to the surface ten-
50 60 70 80 90100
Sound Spe ed (m/s )
Figure 4. The speed of sound pressure dependence in car-
bon dioxide at subcritical temperatures: 27˚C—blue, 28˚C—
red; and supercritical temperatures: 32˚C—green, 35˚C—
violet. The critical parameters of CO2 are: Tc = 31.1˚C, Pc =
73.8 bar.
peratures. This minimum can give one more line Pr (T)
for the ridge zone. The curve for SS (P) at T = 35˚C
clearly shows that in the supercritical region there are
two branches with quite different character. The gaseous
branch with a descending speed of sound results from the
growth of large clusters’ population. And the ascending
branch of the SS (P) in the liquid-like structure has the
same mechanism as at subcritical temperaturesthe
eliminat ion of pore s at h i gh pressures .
3. The Monomer Fraction Density Based
Models for the SF Solubility and
The most widely used SF in SFE and SFC is Carbon Di-
oxide. Its critical temperature Tc = 304.1282 K [8] is near
the room temperature and critical pressure Pc = 73.773
bar [8] is low enough and does not put extra heavy re-
quirements to technological equipment. Moreover, the
carbon dioxide is not expensive and non-toxical. After
the SFE process is finished, it totally evaporates leaving
no harmful traces in the extracted products. Now, due to
a large scale applications of the CO2 based SFE and SFC
technology, the technological equipment becomes less
and less expensive, thus making this technology more
and more competitive [19]. For further optimization of
technologies based on CO 2 SF it is important to find cor-
relations between its thermophysical properties and
technological parameters, such as diffusivity and solubil-
3.1. The Model for Diffusivity of Molecules
in SCF
The well known high diffusivity of molecules in super-
critical fluids results from the high total volume of the
gaseous regions, filled with monomers and small clusters.
The Figure 5 was built basing on the data from [1] to
illustrate this fact.
It is reasonable to suppose that the diffusivity should
be proportional to the share of the monomer fraction
density Dm in the total density D of basic particles of the
fluid and inversely proportional to the viscosity µ of the
fluid. So, we come to a simple formula for diffusivity
(Dif) in SF:
 
Here Kd (T) is the coefficient that should be fit to ex-
perimental data at some fixed pressure. To compare the
model (1) values for Dif (T, P), Figure 6, with the ex-
perimental data from the Figure 5 we have selected a
constant value for K (T) equal to 0.06. The values for Dm
(T, P) have been computed by the method described in
the authors’ earlier publications [11,20].
It is seen that there is a general agreement between the
model and experiment, in spite of a very simple form of
the model formula (1). It confirms the hypothesis that th e
diffusivity in SF is proportional to the monomer fraction
density share Dm/D in the total density D. The curves for
diffusivity shift along the T-axis together with the shift of
the ridge line T-coordinate.
3.2. The Model for Solubility of Solids in SF
Another nontoxic and not expensive SF is water. Its
critical parameters are much higher, than for CO2: Tc =
647.096 K and Pc = 220.640 bar [8]. It makes more dif-
ficult the utilization of sup ercritical water in the selective
extraction technologies. But a great effectiveness of the
water based SFE in the chemical weapons and nuclear
wastes treatment and a wide utilization of the supercriti-
cal water in modern energy production technologies re-
quire finding more precise correlations of its solution
power with the thermophysical properties. They may be
useful also for a proper selection of materials for the
equipment details contacting with supercritical water.
The supercritical water plays a significant role in
Open Access IJAMSC
Dif (cm
Figure 5. The diffusivity values for supercritical CO2 at
pressures from up to down: 70, 80, 100, 150, 200 bar.
050 100
Dif (cm
Figure 6. The computed diffusivity values for supercritical
CO2 at pressures from up to down: 70, 80, 100, 150, 200 bar,
corresponding to the model (1) with K (T) = 0.06.
geological processes selectively dissolving solids and
forming mineral deposits [21]. The solubility (Sol) of
silica in SF water, Figure 7, is important both for the
mineralogy and for the energetic stations, where Silica
from SF water covers the blades of turbines thus pre-
venting from the normal energy production, [1]. At the
Figure 7 the solubility is given in weight percents.
It is reasonable to suppose that the monomer and
dimer fractions do not participate in the dissolution
process, but higher clusters play an active role in the
dissolution of solids. And this role is supported by the
total density D of the SF. Thus we come to a model:
Here D3+ is the molar density of basic particles con-
tained in all cluster fractions, starting from trimers. Ksol
(T)—the empirical coefficient adjusted to experimental
results at P = 750 bar.
The computation of the isothermal values for D3+ (P)
is performed via the equation:
Here C2 (T) is the apparent equilibriu m constant for the
Pressure-Density-Temperature (PDT) relations [11,20],
equal to minus second virial coefficient—B (T). So, in-
stead of the computations of populations for all cluster
fractions we calculate only the monomer fraction density
Dm and the molar density of basic particles contained in
the dimer fraction. The results of the solub ility computa-
tion according to the model (2), Figure 8, are in a good
agreement with the experimental results, Figure 7.
A comparison of model results with experimental data
points on a general correspondence of theory to practice.
A good correspondence of the theoretical solubility to
experimental values in a wide range of pressure and
temperature tells about co rrectness of the model (2).
It is important to notice that the curves for solubility
shift along the T-axis in a correspondence with the shift
of the ridge pressure and the width of the transition re-
gion from high to low values of solubility corresponds at
every pressure to the width of the ridge zone along the
T-axis. So, the soft structural transition in SF demon-
strates its influence on important technological parame-
ters: solubility and diffusivity.
4. Conclusions
The computer aided analysis of the preprocessed and
generalized experimental data for pure fluids from the
NIST online database discovers the properties of the
soft structural transition between gas-like and liquid-
like structures in pure supercritical fluids.
The analysis helps to find the monomer fraction den-
sity based features of pure SFs, such as the diffusivity
and solubility, impor tant for the SF technologies.
300 350 400 450 500
Solubili ty, W -t %
Temp eratu re,
Figure 7. The silica solubility in SF water for pressures
from down to up: 250, 300, 350, 400, 500, 600, 750, 1000,
1500 bar according to data from [1].
300 350 400 450 500
Sol ubil ity, W- t %
Temperatu re,
Figure 8. The results of model (2) for silica solubility in SF
water at pressures: 250, 300, 350, 400, 500, 600, 750, 1000,
1500 bar, adjusted to experimental data at P = 750 bar.
Open Access IJAMSC
Open Access IJAMSC
The model for molecular diffusivity in carbon diox-
ide in the near critical zone of temperatures and pres-
sures discovers its proportionality to the monomer
fraction density.
The cluster fractions based model for the solids’
solubility in supercritical fluids shows that the cluster
fractions play an important role in the solubility
[1] M. A. McHugh and V. J. Krukonis, “Supercritical Fluid
Extraction: Principles and Practice,” Butterworth Pub-
lishers, Stoneham, 1986.
[2] L. T. Taylor, “Supercritical Fluid Extraction,” John Wiley
& Sons Ltd., New York, 1996.
[3] J. McHardy and S. P. Sawan, Eds., “Supercritical Fluid
Cleaning. Fundamentals, Technology and Applications,”
Noyes Publications, Westwood, 1998.
[4] I. Brondz and A. Brondz, “Supercritical Fluid Chroma-
tography—Mass Spectrometry (SFC-MS) and MALDI-
TOF-MS of Heterocyclic Compounds with Trivalent and
Pentavalent Nitrogen in Cough Relief Medical Forms
Tuxi and Cosylan,” American Journal of Analytical C he m-
istry, Vol. 3, No. 12A, 2012, pp. 870-876.
[5] Ch. Kittel, “Thermal Physics,” John Wiley and Sons, Inc.,
New York, 1969.
[6] B. Sedunov, “Gas-Like and Liquid-Like Structures in
Supercritical Fluids,” The 2nd International Symposium on
Structural Thermodynamics (ISST-2010), Osaka, 2010, p.
[7] B. Sedunov, “Structural Transition in Supercritical Flu-
ids,” Journal of Thermodynamics, Vol. 2011, Article ID:
[8] NIST, “Thermophysical Properties of Fluid Systems,”
[9] NIST, “Thermodynamics Research Center,” 2013.
[10] J. B. Hannay and J. Hogarth, “On the Solubility of Solids
in Gases,” Proceedings of the Royal Society of London,
Vol. 29, 1879, pp. 324-326.
[11] B. Sedunov, “Monomer fraction Density,” International
Journal of Thermodynamics, Vol. 11, No. 1, 2008, pp. 1-
[12] B. Sedunov, “Soft Structural Transitions in Fluids,” The
22nd International Conference on Chemical Thermody-
namics (ICCT) and the 67th Calorimetry Conference
(CALCON), Buzios, 2012.
[13] B. Sedunov, “The Supercritical Zone of Extraordinary
Properties,” Industrial Use of Molecular Thermodynam-
ics (InMoTher 2012) Workshop, Lyon, 2012, Book of
Abstracts, p. 90.
[14] IUPAC, “Compendium of Chemical Terminology,” A. D.
McNaught and A. Wilkinson, Compiled, 2nd Edition,
Blackwell Scientific Publications, Oxford, 1997.
[15] K. Nishikawa, et al., “Local Density Enhancement in
Neat Supercritical Fluid due to Attractive Intermolecular
Interactions,” Chemical Physics Letters, Vol. 368, No. 1,
2003, pp. 209-214.
[16] B. Sedunov, “The Analysis of the Equilibrium Cluster
Structure in Supercritical Carbon Dioxide,” American
Journal of Analytical Chemistry, Vol. 3, No. 12A, 2012,
pp. 899-904.
[17] B. Sedunov, “Nanosized Objects in Equilibrium Super-
critical Fluids,” MATEC Web of Conferences, Nancy, Vol.
3, 2013, Article ID: 01062.
[18] M. A. Anisimov, “Thermodynamics at the Meso- and Nano-
scale,” In: J. A. Schwarz, C. Contescu and K. Puty e r a , Eds.,
Dekker Encyclopedia of Nanoscience and N anotechnology,
Marcel Dekker, New York, 2004, pp. 3893-3904.
[19] “PHASEX Corporation Web Site,” 2013.
[20] B. Sedunov, “Thermal Analysis of Thermophysical Data
for Equilibrium Pure Fluids,” Journal of Modern Physics,
Vol. 4, No. 7A2, 2013, pp. 8-15.
[21] G. C. Kennedy, “A Portion of the System Silica-Water,”
Economic Geology, Vol. 45, No. 7, 1950, pp. 629-653.