Figure 1 presents the phase diagram for pressure-temperature relations (bar/ Kelvin, K) for neat CO₂, with its triple and critical points. Built on thermophysical data from the NIST databases [13] [14] , Figure 1 develops the idea of ex-

Many other substances also can be used http://www.supercriticalfluid.org/Supercritical-fluids.146.0.html.

tending the saturation line to the SF zone, where a soft structural transition from the gas-like SF to the liquid-like SF takes place at a fluid density close to the critical density D_c [15] [16] . This extension is known as the Widom line. The equation for sublimation vapor pressure is taken from the NIST database [14] .

The phase diagrams indicate the conditions for triple and critical points; the regions for the existence of vapor, liquid, condensing, melting, freezing, existence of solid, deposit, and sublimation; and the lines of sublimation, melting, and saturation. The important conditions for the critical point and the extension of the saturation line to the SF zone are also evident.

The influence of modifiers on phase diagrams for pressure-temperature relations and the triple and boiling points, and lines of solid, liquid, and vapor separation will be demonstrated later.

Neat CO₂ in a supercritical state is primarily employed for the extraction of hydrophobic molecules and, at best, it can dissolve some moderately polar compounds. CO₂ alone is very seldom used. This is because neat CO₂ is rather lipophilic and polar/strong polar substances do not dissolve well in it. This issue can be resolved to some extent by elevation of pressure or modification of temperature, or both [3] . In addition to altering the pressure and temperature of neat CO₂, modifiers or cosolvents―which are polar in nature, such as water, methanol, ethanol, diethylamine, formic acid, acetic acid, and many other polar and even nonpolar solvents, and complexation agents―are added to SCCO₂ to enhance bond separation and the extraction of strongly polar compounds and ions. By using one component as modifier, or a mixture of different substances as modifiers, it is possible to achieve a wide range of analyte solubilities in the mobile phase of modified SCCO₂. Methanol and ethanol are often used as modifiers in SFC to enhance mobile phase polarity. Small m/z ions from these modifiers often do not disturb the fragmentation picture of large molecules under SFC- mass spectrometry (MS).

Methanol is an important modifier for SCCO₂ to achieve some polarity in this mobile phase [17] . However, ethanol is less toxic for the operator and the environment; it has been used in up to 30% molar volume in isocratic mode [18] . The practical experiments and data described in [19] are considered very important because of the large volumes of ethanol used as modifier. Multicomponent mixtures of CO₂, ethanol, and diethylamine, described in [18] , are not unique. Many complicated multicomponent mixtures of CO₂ with modifiers [17] , cosolvents, and even complexation agents are in use [11] .

Calculation of phase equilibria for multicomponent mixtures as described by Gernert et al. [20] can be very helpful for understanding the processes in SFC columns during bond separation in complex mixtures or the resolution of isomers. In section of columns under separation together with complex mobile phase also present analytes of different nature (which are present in initial analyzing mixture). Chueh and Prausnitz [21] [22] have modeled CO₂ with mixtures of some of these compounds (including methanol and ethanol) for the estimation of the critical parameters T_c and P_c. The most complete collection of SCCO₂ solubility data with multiple solutes is presented in “Solubility in Supercritical Carbon Dioxide” by Ram B. Gupta and Jae-Jin Shim [23] .

In some cases, polar supercritical phases have been used for chromatography and extraction and have advantages over CO₂ in the supercritical state [24] . This provides quantitative recovery and the possibility to overcome the matrix-ana- lyte interaction, which sometimes may be more important than a high solubility of the analyte in the solvent [24] . Polar supercritical principal phases, such as ammonia, can be used for the extraction of metal ions or polar substances from matrices by SFE or SFC [25] . Many salts and substances are readily soluble in ammonia in the supercritical state and this principal phase can also be modified to lend the principal mobile phase some lipophilic characteristics [25] . The pressure-temperature diagram for ammonia is presented in Figure 2, built on thermophysical data from the NIST databases [13] and [14] .

In several cases, ammonia in the supercritical state has advantages because it is polar, in contrast to CO₂ containing polar modifiers for the extraction of strongly polar compounds. This is because the addition of polar modifiers to CO₂ results in alteration of the critical temperature and critical pressure. Hence, the integrity of the supercritical phase should be carefully preserved by choosing

appropriate experimental pressures and temperatures. However, ammonia is extremely corrosive and toxic, which makes it the least preferred mobile phase.

The mobile phase in a chromatography column under SFC is not homogeneous, even when using neat CO₂, because of a pressure gradient fall from the head of the column to the bottom (outlet). However, modern SFC instrumentation is equipped with a backpressure regulator (BPR) that can automatically hold a constant pressure at the outlet of the column in programmed mode, and hence define the pressure at the head of the column. The main purpose of a BPR is to prevent the conversion of SCCO₂ to CO₂ gas at the outlet of the column as well as to prevent freezing at the outlet as a result of this, i.e., the analyte is kept in the mobile phase in the supercritical state until it arrives in the detector.

In real situations, the following are often present in a column: in the head section of the column, the CO₂ mobile phase in a liquid state (depending on conditions); in the middle of the column, a section of SCCO₂; while at the bottom of the column, there is the section where CO₂ often exists as condensed gas under a high pressure (as a fluid under subcritical conditions). Of course, the operator can achieve better conditions by using an isocratic mode of operation with only two different phases: liquid in the head section of the column and SF in the main part of the column. However, this is not achievable with a programmed mode of operation of conditions and parameters, such as programmed pressure, programmed temperature, and programmed concentration of modifier in the mobile phase, simultaneously.

The solvating power of a SF at a given temperature depends on its density. The solubility of a solute in SCCO₂ depends on the intermolecular interactions between solvent and solute. The solvating power of SCCO₂ can be defined in terms of the solubility parameter, δ. It is defined as the square root of the cohesive energy density and is used extensively in interpreting the thermodynamic properties of a solution. It is given as follows:

where ΔH is the heat of vaporization, V is the molar volume of the liquid, and С is the cohesive energy density.

For a SF, the heat of vaporization is zero at the critical temperature; therefore, the above equation leads to a negative value of С. The equation was later modified and Giddings proposed an equation based on studies in liquid chromatography using the van der Waals equation of state. The δ for a SFC is given as follows:

where ρ_r is the reduced density of the substance in supercritical state and ρ_r,liq is the reduced density of the liquid at its normal boiling point.

Thus, the solubility parameter for a given substance can be obtained if the fluid density and critical pressure are known, and it is unique for a given substance. When the solubility parameters of both the solute and solvent are similar, appreciable solubility of the solute will generally be observed in the given solvent. The solubility parameters of some SFs are given.

CO₂ in supercritical state is a nonpolar medium and it finds wide application in the extraction of organic compounds, such as natural products, which are generally hydrophobic in nature. SCCO₂ with the addition of modifiers/cosol- vents, such as methanol, is used for extraction of polar compounds. SFE of metal ions can be performed by adding suitable chelating agents to the SCCO₂ medium.

Modifiers or cosolvents are the substances that are added to SFs to alter solvent properties, i.e., to increase the polarity, aromaticity, chirality and the ability to complex various metal ions. For example, methanol or ethanol is added to CO₂ to increase polarity; toluene to impart aromaticity and neutral extractant; tri-n-butyl phosphate, dialkylalkyl phosphonates, dibutylbutyl phosphonate, and

diamylamyl phosphonate for solvation of metal complexes [11] . Addition of polar cosolvents may be effective in solubilizing the polar solute molecules. The modified CO₂ solvent may also absorb on the active sites of the matrix and prevent the readsorption of the analyte, and thereby its transfer to the supercritical phase. When modifiers are added to SFs, it is essential to evaluate the phase equilibria of the mixture and chemical interactions between them. Issues relating to estimation of the critical parameters for the mixtures should be carefully evaluated. At the critical point, the composition and density of the two phases become identical, and only one SF phase exists. For a given mixture at a given temperature, there will be a critical pressure and critical composition value.

Knowledge about the modifier-fluid phase diagram is essential to ensure that the solvent is in a single phase. For binary mixtures with a low interaction between components, the critical constants are approximately the arithmetic means of the T_C and P_C.

Here x_A and x_B are the molar shares of components A and B in the mixture. T_A and T_B are the critical temperatures and P_A and P_B are the critical pressures of components A and B, respectively.

In some mixtures, the cosolvent can interact with CO₂ and the resultant T_C and P_C may differ from the values based on a linear combination of the components T_C and P_C. Brunner et al. [26] estimated the critical parameters for the methanol-CO₂ system. CO₂ containing methanol can be taken to critical point only above a certain temperature, depending upon the composition of methanol in the CO₂ phase. For the methanol-CO₂ system at the CO₂ molar share equal to 0.84, the critical temperature is 50˚C and the critical pressure is 95.5 bar. However, linear estimation for the critical temperature of the above mixture is around 64˚C and linear estimation for the critical pressure is around 75 bar.

Because direct extraction of ionic species is inefficient in a neat SCCO₂ medium, chelating agents are generally added to a SCCO₂ medium to form neutral complexes. The latter are quite soluble in supercritical medium. A variety of chelating agents such as organophosphorous compounds, dithiocarbamates, and macrocyclic ligands have been used for the extraction of metal ions such as lanthanides, actinides, and transition metal ions [27] - [34] . Tributylphosphine (TBP) and trioctylphosphine oxide (TOPO) are some of the solvating extractants used in the SFE of actinides. Crown ethers are used for selective extraction of alkali/alkaline earth metal ions. The solubility of metal complexes in SFC medium is an important criterion for designing an efficient metal extraction process. In general, many chelating agents show relatively high solubility in SCCO₂. The β-diketones and fluorinated diketones have been used in SCCO₂ medium for the SFE of lanthanides and actinides. The extraction efficiency depends on the solubility of ligands as well as that of metal complexes in the supercritical medium.

In some cases, SCCO₂ will be delivered to an extraction vessel containing the ligand. The SCCO₂ stream containing the ligand is subsequently passed through another vessel containing the analyte (metal ion of interest). On some occasions, ligand is dissolved in methanol medium and mixed into SCCO₂ and the modified stream will be employed for metal ion extraction.

At a solution pressure P lower than P_c/2, the following dominates: distribution of solute molecules in a free space of a gaseous solution V_fs(T, P) approximately equal to the solvent’s molar volume V_s(T, P). The larger the V_s(T, P), the more moles of solute vapor may be distributed in this volume. The resulting contribution of this dominating mechanism in the solubility coefficient y_fs (Figure 6) is equal to:

where D_wsat(T) is the saturated water vapor molar density at temperature T. It can be taken from the NIST database [13] .

The solvent’s molar volume V_s(T, P) increases at low pressures as V_s(T, P) =

RT/P. In real gaseous solutions, V_fs(T, P) deviates from the ideal gas value. In a real gas, the free space for the distribution of solute molecules is created by freely moving particles of a different nature: monomers and clusters of the solvent, solute molecules themselves, and the solute?solvent clusters. In real gaseous solutions, for V_fs, instead of using the solvent’s molar volume V_s, it is more correct to use the free moving particles’ molar volume V_p = 1/D_p, where D_p = P/RT is the free moving particles’ density, responsible for pressure in a real gas [40] . Figure 7 shows the values of the free moving particles’ molar density D_p in gaseous CO₂ at 25˚C, in comparison with the total molar density D and the monomer fraction density D_m. The difference between D and D_p is due to the clusters’ contribution to the total density.

Figure 6 shows the comparison of the free space solubility y_fs of water in CO₂ with experimental solubility data at subcritical pressures of the solvent. This figure is typical for many solute-solvent systems at pressures lower than the critical

pressure. It shows that the experimental solubility data at a constant temperature differ from the free space solubility on the close to constant value Δy(P). In the pressure range of almost constant Δy(P) and at a fixed temperature T, we can estimate the average value Δy_av(T). This value is the number of solute moles bound with one mole of solvent molecules by molecular cross-interaction forces (central type in Figure 5). This increment over the free space solubility may be essential, especially close to critical pressures.

The Δy_av(T) to D_wsat(T) ratio is the equilibrium constant for the solute-solvent cross-interaction C_w?cd, reflected in Table 3 for the H₂O-CO₂ system. For the total water solubility y (T, P) in CO₂ at pressures P < P_c, we have the following:

The corresponding data for two other systems that were analyzed are presented in Table 4.

The expression (7) reflects both the solution’s colligative properties, reflected in RT/P, and individual properties of the solution system, reflected in the solute-solvent cross-interaction equilibrium constant C_w?cd.

Table 3 and Table 4 show that the equilibrium constants for the cross-inte- raction of water molecules in gaseous solutions with CO₂ and N₂O molecules are of the same order of magnitude, but with C₂H₆ molecules they are much smaller. The temperature dependence of equilibrium constants permits us to estimate the cross-interaction parameters: the bond energy E_cross and the corresponding entropic factor, and the effective volume of molecular cross-interaction

these are reflected in Table 5. To find these parameters, we plot ln(C_cross) versus 1000/T (see Figure 8). The linear trend for the experimental data provides the required parameters.

We see that despite a zero dipole moment value for the CO₂ molecule, its cross-interaction bond energy E_cross with polar water molecules is rather high due to the existence of the quadrupole moment [38] in the CO₂ molecule. Hence, the atoms in this molecule are charged.

A high cross-interaction bond energy E_cross in the H₂O-N₂O system results from a combination of dipole and quadrupole potentials of the N₂O molecule. As Table 1 shows, the dipole moment in the N₂O molecule is not very large.

A symmetric ethane molecule possesses no dipole or quadrupole moments. But its cross-interaction bond energy with the water molecule is high enough, although not as high as in H₂O-CO₂ and H₂O-N₂O systems. The cross-bond energy in the H₂O-C₂H₆ system results from the multipole moment [38] of the C₂H₆ molecule, caused by no zero atoms’ charges in this molecule, as discussed in [39] .

In the third type of solution, in three-dimensional (3D) clusters, the number of water molecular bonds with surrounding CO₂ molecules increases (see Figure 5). This results in an increase in water solubility with pressure in high-density gaseous CO₂ (see Figure 9).

Figure 9 demonstrates that at 75˚C and P < 40 bar, the solubility increment Δy(P) is almost constant according to the second type of solubility; however, at P > 40 bar, the increment essentially increases with pressure. Similar behavior of Δy(P) is seen at 50˚C (see Figure 4).

We now consider the solubility increment Δ_3D y caused only by the 3D clusters, by subtracting from the total solubility y the estimated values for the second type of solubility mechanism. According to [16] , at supercritical temperatures the number of 3D clusters rapidly increases as the total density D approaches the critical density.

It is possible to estimate approximately the number of particles in 3D clusters, providing maximum contribution to the solubility increment at supercritical pressures. For this we plot the ln(Δ_3Dy) versus ln(D_m) (see Figure 10) and estimate the tangent of the slope for the linear trend of the graph, K_3D:

The tangent of the slope for the graph is close to 7. Hence the 7-particle clusters provide the maximum contribution to the water solubility in CO₂ at 50˚C and pressures up to 100 bar. The proportionality factor k_3D between D_3D y and is 4.91 × 10⁻⁶ (l/mol)⁷ (see Figure 11).

The theory of three types of solubility developed for the water-CO₂ system may be applied to the ethanol-CO₂ system. The ethanol solubility y (T, P) in gaseous CO₂ has been calculated based on theoretical estimations from [19] for the CO₂ molar share in a gaseous ethanol-CO₂ system at temperatures of 25˚C, 40˚C, 45˚C, and 50˚C. In [19] , the authors used four theoretical models and achieved the best performance with the Peng-Robinson/Mathias-Klotz-Prausnitz approach [41] to estimate the molar share of CO₂ in the gaseous CO₂-ethanol system. Figure 12 demonstrates a good qualitative correspondence of the ethanol solubility pressure dependence with the theory of three types of solubility for polar substances in sub- and supercritical CO₂.

Figure 12 shows an almost constant value for the solubility increment Δy at P < 55 bar and the solubility minimum near P = 60 bar. Comparison of Figure 12 with Figure 4 shows that the solubility of ethanol in gaseous CO₂ at P > 30

bar is greater than the solubility of water at the same temperature and pressure. This is due to the ethanol vapor density being double that of the water vapor density under the same conditions. Furthermore, the solubility increment Δy for the ethanol-CO₂ system is larger than for the water-CO₂ system and larger than the free space solubility y_fs. This indicates that the equilibrium constant for the ethanol-CO₂ dimers is larger than for the H₂O-CO₂ dimers. For a detailed analysis of the cross-interactions in the ethanol-CO₂ dimers, more precise experimental solubility data are required.

In Section 1.3, the unwanted transformation in chromatography columns of a liquid-like SF to a condensed gas under high pressure was described. This transformation may occur when the pressure falls to the bubble pressure (discussed in [19] ). At the point of this transformation in the chromatography column with a CO₂-modifier system, the modifier’s molar share may be different for the liquid-like and gaseous sections. Under equilibrium conditions, at the bubble pressure, from the two-component liquid-like SF system in the gaseous bubble enters mainly CO₂, and the modifier’s molar share y_gas in the bubble becomes much lower than y_liq in the liquid-like section. This is illustrated in Figure 13, built on the basis of data from [19] , where y_gas is multiplied by a factor of 100 to compare it with the y_liq values at the same pressure.

Figure 13 shows that the 20% - 30% modifier content in the liquid CO₂- ethanol section of the chromatography column may be transformed to the gas phase, with only 1% of modifier in it. A similar process takes place at a subcritical temperature (see Figure 14).

Here again we see that the 20% - 30% CO₂-ethanol liquid section in the chromatography column may be transformed in the bubble point to a 0.4% CO₂- ethanol gaseous system. The content of the analyte may differ significantly from its content in the liquid section, which may distort the chromatography results.