The partition isotherms

If a pure liquid is heated, a temperature-dependent vapour pressure is established above its surface. In a two-component system, consisting of two completely miscible liquids A and B, the vapour pressure P is equal to the sum of the partial pressures of the individual components (Dalton). In a gas-chromatographic system, A is equivalent to the substance to be separated, whereas B constitutes the stationary phase. Generally speaking, the working temperature must be selected such that the vapour pressure of the stationary phase is negligible. The vapour pressure above the stationary phase is then produced almost entirely by substance A, and apart from temperature is dependent on the concentration of A in the stationary phase. 

The relationship between the concentration of A in the stationary phase (Cs) and the concentration of A in the mobile phase (Cm) with regard to a particular component is measured at constant temperature. For this reason, the curves generated by plotting Cs against Cm are known as partition isotherms (see Fig. 45). 

The shape of the partition isotherms in the chromatographic process has a decisive influence on the shape of the peak. Let us now differentiate between two possible cases  

Ideal distribution of substance A between the mobile and stationary phases 

The function displays a linear relationship. The partition coefficient Ko remains constant over a large concentration range. 

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Essentially, extraction of an analyte from one phase into a second phase is dependent upon two main factors solubility and equilibrium. The principle by which solvent extraction is successful is that like dissolves like . To identify which solvent performs best in which system, a number of chemical properties must be considered to determine the efficiency and success of an extraction [77]. Separation of a solute from solid, liquid or gaseous sample by using a suitable solvent is reliant upon the relationship described by Nemst s distribution or partition law. The traditional distribution or partition coefficient is defined as Kn = Cs/C, where Cs is the concentration of the solute in the solid and Ci is the species concentration in the liquid. A small Kd value stands for a more powerful solvent which is more likely to accumulate the target analyte. The shape of the partition isotherm can be used to deduce the behaviour of the solute in the extracting solvent. In theory, partitioning of the analyte between polymer and solvent prevents complete extraction. However, as the quantity of extracting solvent is much larger than that of the polymeric material, and the partition coefficients usually favour the solvent, in practice at equilibrium very low levels in the polymer will result. 

Figure 1.9. The dependence of the boundary profile on the form of the partition isotherm. (Courtesy of John Wiley-Interscience. ) c=concentration (cm 5/mole) of solute in gas phase q=concentration in liquid or adsorbed phase t=time for band to emerge from column (1) se1f-sharpening profile (2) diffuse profile (3) gaussian profile.

When the partition isotherm is nonlinear (typically when surface adsorption is involved), the elution peaks exhibit long tails. Tailing is also caused by slow diffusion of the probe in the polymer and by technical artifacts mixing in the injection chamber, etc. 

Fig. 46 a - c. Relationship of the shape of the peak profile (change of concentration in the carrier gas over time) to the partition isotherm a = linear b = concave c = convex isotherms (Schomburg)... 

The effect of sample concentration on the chromatographic behaviour of ionic solutes on Sephadex G-10 can be qualitatively described by means of the partition isotherms derived on the basis of the ion partition model proposed by Shibukawa et al. [ref. 69] (see Sec. 2.2.1). The overall partition isotherms are anticipated to be represented as shown in Fig. 9 from the analogy with typical ion-exchange isotherms [ref. 70], if the partition isotherm of process (A) is linear. It is thus predicted that, in the system where the contribution of the steric exclusion effect can be neglected, the difference in the affinity for the internal gel phase between sample ion, s , and coion Y , that is, the equilibrium constant of ion-exahange process (B), K, determines the sample concentration dependence of the elution behaviour of sample ion in the following manner. When > 1, the elution volume or the distribution coefficient of sample ion decreases and the elution profile is more skewed with sharp leading... 

At equilibrium, the dependence of a component concentration in the stationary phase on its concentration in the mobile phase is given by an equation which is called the partition isotherm. For the low concentrations of injected substance employed in gas chromatography there are three types of isotherms, shown in Figure 2.3. 

In the present study we try to obtain the isotherm equation in the form of a sum of the three terms Langmuir s, Henry s and multilayer adsorption, because it is the most convenient and is easily physically interpreted but, using more a realistic assumption. Namely, we take the partition functions as in the case of the isotherm of d Arcy and Watt [20], but assume that the value of V for the multilayer adsorption appearing in the (5) is equal to the sum of the number of adsorbed water molecules on the Langmuir s and Henry s sites ... 

Isotherms. When a fibei is immersed, in a dyebath, dye moves fiom the external phase into the fibei. Initially the late is quick but with time this slows and eventually an equiUbrium is reached between the concentration of dye in the fiber and the concentration of dye in the dyebath. For a given initial dyebath concentration of a dye under given dyebath conditions, eg, temperature, pH, and conductivity, there is an equiUbrium concentration of dye in fiber, Dj and dye in the dyebath external solution, D. Three models describe this relationship simple partition isotherm, Freundhch isotherm, and Langmuir isotherm. 

With simple partition the situation is comparable to the partition of a solute between two solvents. The bonding forces involved between uncharged dye and uncharged fiber, and uncharged dye and uncharged solvent are considered to be the same. The dye is sometimes referred to as in soHd solution in the fiber. This type of isotherm is found in practice with disperse dyes on cellulose acetate and polyester. It represents the dyeing situation with the minimum restrictions for the dye to enter the fiber the only restriction is when the fiber solution becomes saturated. 

The working capacity of a sorbent depends on fluid concentrations and temperatures. Graphical depiction of soration equilibrium for single component adsorption or binary ion exchange (monovariance) is usually in the form of isotherms [n = /i,(cd or at constant T] or isosteres = pi(T) at constant /ij. Representative forms are shown in Fig. I6-I. An important dimensionless group dependent on adsorption equihbrium is the partition ratio (see Eq. 16-125), which is a measure of the relative affinities of the sorbea and fluid phases for solute. 

In the low concentration limit the adsorption isotherm is alinear law as was the partition coefficient,andjustastheisothermdeviatesfromlinearityoutsideofthelowconcentration limit,sotoodoesthepartitionrelationbetweenthetwoliquidphases. 

Current use of statistical thermodynamics implies that the adsorption system can be effectively separated into the gas phase and the adsorbed phase, which means that the partition function of motions normal to the surface can be represented with sufficient accuracy by that of oscillators confined to the surface. This becomes less valid, the shorter is the mean adsorption time of adatoms, i.e. the higher is the desorption temperature. Thus, near the end of the desorption experiment, especially with high heating rates, another treatment of equilibria should be used, dealing with the whole system as a single phase, the adsorbent being a boundary. This is the approach of the gas-surface virial expansion of adsorption isotherms (51, 53) or of some more general treatment of this kind. 

The simplest type of isotherm is the linear-distribution coefficient, A d.12 It is also called the partition coefficient, 7. 58 The equation for calculating adsorption at different concentrations is... 

There are three approaches that may be used in deriving mathematical expressions for an adsorption isotherm. The first utilizes kinetic expressions for the rates of adsorption and desorption. At equilibrium these two rates must be equal. A second approach involves the use of statistical thermodynamics to obtain a pseudo equilibrium constant for the process in terms of the partition functions of vacant sites, adsorbed molecules, and gas phase molecules. A third approach using classical thermodynamics is also possible. Because it provides a useful physical picture of the molecular processes involved, we will adopt the kinetic approach in our derivations. 

The retention of analyses in RP-HPLC markedly depends on the adsorption of the organic constituent of the mobile phase on the surface of the stationary phase. The excess adsorption isotherms of ACN, THF and methanol were measured on silica support modified with C, C6, C8, C10, C12 and C18 monomeric phase and a model was developed for the description of the retention of solutes from the binary mobile phase. The dependence of the retention factor on the partition coefficient can be described by... 

For the linear isotherm model, the parameter (Kd) that relates both sorbate and solute is called the partition coefficient. A number of studies have developed empirical relationships for partition coefficients in natural solid phases and several of these studies are summarized in Table 1.Various theoretical-based methods of partition coefficient estimations also exist (Table 1, Eqs. a- f). 

The values of q are plotted as a function of the equilibrium concentration. For constituents at low or moderate concentrations, the relationship between q and C can be generated. If n = 1, the (q-C) relationship will be linear (Eq. 9), and the slope of the line (i.e.,ITd) defines the adsorption distribution of the pollutant. Kd is generally identified as the distribution or partition coefficient, and is used to describe pollutant partitioning between liquid and solids only if the reactions that cause the partitioning are fast and reversible, and if the isotherm is linear. For cases where the partitioning of the pollutants can be adequately described by the distribution coefficient (i. e.,fast and reversible adsorption, with linear isotherm), the retardation factor (R) of the subsurface environment can be used as follows ... 

Suppose we have an ideal chromatographic column, no dispersion, and a linear partition isotherm. The result is only a... 

The partition function and the binding isotherm of a general two-site model were discussed in Section 4.2. Here, we examine a special case of direct correlation only. The model is essentially the same as in the previous section, except that now... 

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