Adsorption isotherms constant partitioning

Figures 1 and 2 show the effect of temperature on the extent of binding of DDT to both Pakim Pond Humic Acid and Boonton Humic Acid. The Y axis in these figures is the amount of DDT bound to the humic acid in nanograms of DDT per gram of humic acid. The X axis is the free, truly dissolved DDT in nanograms per liter. This is similar to the presentation of an adsorption isotherm. If the slope of the lines is multiplied by 1000 it becomes analogous to a weight-weight partition coefficient ([g DDT/g D0C]/[g DDT/g water]). This will be referred to as the binding constant.

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. 

Fig. 20.1. Correlation between the air/water partition coefficient, Kaw, determined from measurements of the surface pressure as a function of drug concentration (Gibbs adsorption isotherm) in buffer solution (50 mM Tris/HCI, containing 114 mM NaCI) at pH 8.0 and the inverse of the Michaelis Menten constant, Km obtained from phosphate release...

Statistical thermodynamics is used to obtain the partition function for species strongly bound to the surface (i.e., chemisorbed species). This approach can be used to derive the Langmuir adsorption isotherm, and to estimate the associated equilibrium constant, discussed in Section 11.5.3. The situation in which the adsorbed species is more weakly bound, and moves freely across the surace is considered in Section 11.5.4. 

The slope of this line is the distribution coefficient (Kd), which is the ratio of the arsenic concentration on the adsorbent (Cads) to the concentration of the associated remaining arsenic in the aqueous solution (Csdn). With each linear adsorption isotherm, Kd has only one value. That is, a linear distribution indicates that the partitioning of arsenic between the adsorbent and the solution is constant over the given range of arsenic concentrations (Eby, 2004), 221. If both concentrations (Cads and Csoin) are in the same units (such as molal), Kd is unitless. However, if the adsorbed concentration is given in molal and the dissolved concentration is molar, then Kd has the units of liter/kilogram. 

Adsorption isotherms are used to quantitatively describe adsorption at the solid/ liquid interface (Hinz, 2001). They represent the distribution of the solute species between the liquid solvent phase and solid sorbent phase at a constant temperature under equilibrium conditions. While adsorbed amounts as a function of equilibrium solute concentration quantify the process, the shape of the isotherm can provide qualitative information on the nature of solute-surface interactions. Giles et al. (1974) distinguished four types of isotherms high affinity (H), Langmuir (L), constant partition (C), and sigmoidal-shaped (S) they are represented schematically in Figure 3.3. 

Adsorption isotherms were obtained for four amino acids in an investigation of their interaction with calcium montmorillonite and sodium and calcium illite. Linear isotherms were obtained in the study of their adsorption by the calcium clay. These isotherms were described in terms of a constant partition of solute between the solution and the adsorbent Stem layer. Free-energy values were calculated using the van t Hoff relation [16,27]. 

Other ions in the solution. The self-energy of a dipole embedded in a dielectric sphere is the key to Onsager s theory of the dielectric constant of dipolar fluids. Equally, in any theory for, say, the surface energy of water, or adsorption of molecule, the self-energy of a molecule as a function of its distance from an interface is involved. In adsorption proper, the same selfenergy for a molecule appears in the partition function of statistical mechaiucs from which the adsorption isotherm is derived. 

The C-type (constant-partitioning) isotherm, which suggests a constant relative aflSnity of the adsorbate molecules for the adsorbent, is usually observed only at the low range of adsorption. Deviation from the linear isotherm is likely at high adsorption levels. Nevertheless, because many nonpolar organic compounds of interest in soils are adsorbed at quite low concentrations, the linear C-type isotherm is often a reasonable description of adsorption behavior. 

Equilibrium vapor pressure of bulk liquid Relative pressure, p/p 0 Statistical mechanical partition function Density in molecules/cubic centimeter Gas constant per mole Number of moles of adsorbed gas Number of moles of adsorbent Isosteric heat of adsorption Differential heat of adsorption Isothermal heat of adsorption Adiabatic heat of adsorption... 

In this paper, we present an exact calculation of the statistical mechanics of a lattice model of hydrocarbon adsorption in the quasi one-dimensional pores of zeolites, based on a matrix method that utilises the Constant Pressure partition. The model is tested on benzene adsorption, where it reproduces experimentally observed steps in isotherms. The model has been extended also to linear alkanes where it reproduces very accurately experimental adsorption isotherms as well as Monte-Carlo simulation results of ethane. 

Isotherms for the adsorption of phenylureas on ethylcellulose are of the C-type (71) (top of Figure 12). The constant partition or C-type isotherm is common when new sites become available as the solute is adsorbed from the solution. For every concentration of a particular chemical that is used, the same proportionate amount is partitioned between the adsorbent surface and the solution phase. The diflFerence in adsorption of one compound vs. another depends upon the structure and properties of the compounds and the preference by the adsorbent of the compounds over that of water molecules. The phenylureas were readily desorbed from ethylcellulose with water showing that the adsorption mechanism was physical in nature. 

The C-curve isotherm is characterized by an initial slope that remains independent of the concentration of a substance in the soil solution until the maximum possible adsorption. This kind of isotherm can be produced either by a constant partitioning of a substance between the interfacial region and an external solution or by a proportional increase in the amount of adsorbing surface as the surface excess of an absorbate increases. The example of parathion (diethyl p-nitrophenyl monothiophosphate) adsorption in Fig. 4.1 shows constant partitioning of this compound between hexane and the layers of water on a soil at 50 per cent relative humidity. The adsorption of amino acids by Ca-montmorillonite also exhibits a... 

This relationship resulted from the spontaneous tendency for the analyte, initially solubilized in an aqueous matrix, to partition into the surface monolayer of hydrophobic octadecyl- or octyl-bonded silica. Organic compounds as analytes will have a unique value for the thermodynamic distribution constant. One fundamental difference between LEE and SPE stands out Partition or adsorption of solute molecules onto a solid surface follows the principles of the Eangmuir adsorption isotherm, whereas EEE does not. We will briefly develop the principles below. This model assumes that an analyte A combines with a site of adsorption, S, in which there is a finite number of such sites according to... 

During the diffusion of solute molecules through the network of pores, some of the solute molecules are adsorbed onto the interior surface of the particle. This process of adsorption is normally very fast relative to the diffusion process, and so we can model it as local equilibrium between the solute in the pore fluid and the solute bound to the interior surface of the particle. This partitioning is often referred to as the adsorption isotherm, which implies constant temperature conditions. When the solute concentration in the pore is low enough, the relationship becomes linear (Henry s law) hence, mathematically, we can write... 

The final type of isotherm is the C class. Such systems exhibit an initial linear portion of the isotherm, indicating a constant partitioning of the adsorbate between the solution and the solid. Such isotherms are not found for homogeneous solid surfaces but occur in systems in which the solid is micropo-rous. This classification system has proved very useful in providing information about the mechanism of adsorption. 

Adsorption-partition constants were determined for benzene by use of the following log form of the Freundlich isotherm log x/m = 1/n log C + log K, where X is the weight of benzene sorbed (ng), m is the weight of adsorbent (g), C is the equilibrium concentration of benzene in solution, and K and 1/n are constants. K is a measure of the degree or strength of adsorption while 1/n is used as an indication of whether adsorption remains constant (as indicated by a 1/n value of unity) or decreases with increasing adsorbate concentrations. 

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. 

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