The Equilibrium Thermal Physics of Supercritical Fluids

doi:10.4236/ijamsc.2013.12013

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International Journal of Analytical Mass Spectrometry and Chromatography, 2013, 1, 103-108

Published Online December 2013 (http://www.scirp.org/journal/ijamsc)

http://dx.doi.org/10.4236/ijamsc.2013.12013

Open Access IJAMSC

The Equilibrium Thermal Physics of Supercritical Fluids

Boris Sedunov

Computer and Information Systems Department, Russian New University (ROSNOU), Moscow, Russia

Email: [email protected]

Received October 23, 2013; revised November 20, 2013; accepted December 20, 2013

permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

ABSTRACT

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-

erties;

 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

B. SEDUNOV

104

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

fluid.

2.1. The Structure of the Supercritical Ridge

Zone

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

B. SEDUNOV 105

(a)

(b)

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

J/molK.

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

B. SEDUNOV

106

Fi gure 3. A flat, even at a small gravity, vapor-liquid bound-

ary at a saturation pressure formed due to the surface ten-

sion.

150

200

250

300

350

400

450

50 60 70 80 90100

Sound Spe ed (m/s )

Pressure

(

bar

)

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 temperatures—the

eliminat ion of pore s at h i gh pressures .

3. The Monomer Fraction Density Based

Models for the SF Solubility and

Diffusivity

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-

ity.

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:

 

Dif KDD



 (1)

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

B. SEDUNOV 107

0.0005

0.001

0.0015

0.002

05010

Dif (cm

/s)

Figure 5. The diffusivity values for supercritical CO2 at

pressures from up to down: 70, 80, 100, 150, 200 bar.

0.0005

0.001

0.0015

0.002

050 100

Dif (cm

/s)

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:





sol

SolKD D



.

(2)

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:



––2T

DDDCD

 (3)

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.

0.05

0.1

0.15

0.2

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].

0.05

0.1

0.15

0.2

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

B. SEDUNOV

Open Access IJAMSC

108

 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

process.

REFERENCES

[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.

44.

[7] B. Sedunov, “Structural Transition in Supercritical Flu-

ids,” Journal of Thermodynamics, Vol. 2011, Article ID:

194353.

[8] NIST, “Thermophysical Properties of Fluid Systems,”

2013. http://webbook.nist.gov/chemistry/fluid

[9] NIST, “Thermodynamics Research Center,” 2013.

http://trc.nist.gov/

[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.

http://dx.doi.org/10.1098/rspl.1879.0054

[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.

http://dx.doi.org/10.4236/ajac.2012.312A119

[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.

http://www.phasex4scf.com/

[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.

http://dx.doi.org/10.2113/gsecongeo.45.7.629