Scatchard’s equation

There are two basic approaches to the calculation of vapor compositions from boiling point-liquid composition data or vapor pressure-liquid composition data (a) the coexistence equation (i) which requires the smoothing of experimental T-x or H-x data first, or (b) a correlating equation which relates the excess free energy with liquid composition. Various equations have been proposed, but Barker (2), who pioneered this method, employed Scatchard s equation (3). Raw or smoothed data are used, but the smoothing process may introduce unwarranted errors. 

A more desirable form for graphical use is known as Scatchard s equation ... 

Formalism According to Pitzer. The most common method for the evaluation of the activity and osmotic coefficients of an electrolyte in a binary mixture of strong electrolytes with a common ion is by Scatchard s Equations (23), the McKay-Perring treatment (24), Mayers Equations... 

Scatchard s equation (1946) is derived to explain Ai for proteins. The change in dimensions (e.g., mean-square end-to-end distance) is not of any concern in Scatchard s derivation. No model was assumed and no statistical mechanics was used. Scatchard successfully correlated the osmotic pressure with the distribution of diffusible solutes across a semipermeable membrane by manipulating the terms of activities of the components (such as protein, salt, and water) with changing composition of the solutions. The mathematical detail is simple but messy. According to Scatchard, the interactions involved in protein solutions are not limited to the exclusion of volume between the segments of macromolecules but also includes the Donnan effect and the binding of small ions to macro ions in a given system. For simplicity, let us consider a three-component system, and let 1 represent the solvent (or buffer), 2 the macromolecule (such as protein), and 3 a salt (e.g., NaCl). Scatchard derived an equation of the second vitial coefficient... 

Figure 19.11. A plot of —(1 — g)/m2 against from the freezing point data for potassium nitrate (KNO3) solutions in water. Data from G. Scatchard, S. S. Prentice, and P. T. Jones, J. Am. Chem. Soc. 54, 2690 (1932). See Equation (19.63).

The Bronsted-Guggenheim-Scatchard approach (abbreviated B-G-S equation in this document), (see section 6.1). 

As early as 1926, Hildebrand showed a relationship between solubility and the internal pressure of the solvent, and in 1931 Scatchard incorporated the CED concept into Hildebrand s equation. This led to the concept of a solubility parameter, S, which is the square root of CED. Thus, as shown below, the solubility parameter S for nonpolar solvents is equal to the square root of the heat of vaporization per unit volume ... 

Later Hildebrand defined the solubility parameter, S, as the square root of the cohesive energy density, ced, after Scatchard derived Equation (395)... 

Binding parameters and thermodynamic parameters. 3 mL solution containing 1.5x1 O 6 mol/L HSA was titrated by successive additions of kaempferol solution and the fluorescence intensity was measured (excitation at 280 nm). The binding parameters have been calculated using the Scatchard s procedure.I0. Quenching data were also analyzed according to the Stern-Volmer equation. " The thermodynamic parameters were calculated from the Van t Hoff equation. 

For multicomponent systems in general, we shall follow Scatchard s (1946) formulation of the expressions for the activities of the components and their derivatives with respect to the masses of the components — that is, the coefficients which enter into equation (27). We shall use the subscript K to denote any component made up of small molecules or ionsx, and i to denote a small ion which is a constituent of one or more components, but is not itself a component. Again denoting the protein as component 2, we have, following Scatchard ... 

Linear transformations of the equation v = V , S/(K, -i S), where v = initial veloci, = maximal velocity when enzyme is saturated with substrate, K = Michaelis constant, S = starting substrate concentration. Also included is the Scatchard linear equation for ligand binding, where b = concentration of bound ligand, f = concentration of free ligand, K, = dissociation constant, b, = maximal concentration of b when ligand is saturating. 

These equations are identical with those given by Van Laar, but the assumptions are somewhat different. In the case of the Van Laar, Scatchard, and Hildebrand derivations, both A and B should be positive, while in Cooper s equation B could be either positive or negative. 

For the monomers in the polymerization under consideration the fugacity coefficients were estimated by Redlich-Kwong equation of state and were found to be close to unity. The activity coefficients (8) for the monomers were estimated by Scatchard-Hildebrand s method (5) for the most volatile monomer there was a temperature dependence but none for the other monomer. These were later confirmed by applying the UNIFAC method (6). The saturation vapor pressures were calculated by Antoine coefficients (5). 

The extended Van Arkel equation contains the same volume factor as that of Hildebrand-Scatchard. Van Arkel s extended equation is then as follows ... 

The symbols have the same physical significance as those in Pitzer s previous papers (1,2,3,4,5), which are based on a different theoretical framework from that of Scatchard and use the ions of the mixed electrolytes as components. The 0mn (the doublet cation-cation interaction) represents the interactions between H+ and (Et)4N+, whereas mnx (the triplet ion interaction) indicates the interactions between H+, Br", and (Et)4N+. Thus, the quantities 0, 0, and are properties characteristic of the mixture, whereas By, BCY, and C are the properties of the single electrolyte solution, and are functions of the ionic strength. Equation 10 can be further reduced after imposing the conditions that Mnx = 0, 0 Mn = 0, and y2 (at the limit) = 0 ... 

It has been shown, however, by Scatchard that such ion-ion interaction is to be expected if one uses further terms in the Debye-Hiickel equation. If Mayer s somewhat complex theory of electrolytes is applied to polyvalent ions, quantitative agreement is also possible. What is implicit in all of these effects and what is quite reasonable is that polyvalent ions of... 

An alternative approach to binding is based on the Scatchard equation [96]. If a protein has n independent and identical binding sites with intrinsic binding constants K and a fraction 0 of these are occupied at a given surfactant concentration [S], then a simple kinetic argument, in which the rate of binding is proportional to [S] times the fraction of vacant sites (1-0) and is equated to the rate of dissociation from the occupied sites proportional to 0, gives... 

Method of Worn. (Van Laar, Mar- GULES, SCATCHARD- Hamer) Effective Volumetric Ratio Polynomial Equation for the excess Gibb s energy, interaction parameters, relatively simple to use WoHL, K., TYans. Am. Chem. Eng. 42 (1946) 215 [1.13]. 

Sis a parameter of intermolecular interaction of an individual liquid. The aim of many studies was to find relationship between energy of mixing of Uquids and their S The first attempt was made by Hildebrand and Scatchard who proposed the following equation ... 

Linear transformations of the equation v=V S/Km + S. Error bars are also shown. 1. Lineweaver-Burk, or double reciprocal plot 2. Eadie-Hofstee plot 3. Hanes-Wilkinson plot 4. Eisenthal-Cornish Bowden, or direct linear plot 5. The Scatchard plot which is used for determination of ligand binding constants. 

They calculated y,-, by a slightly modified version of Equation 9.24, based on Scatchard and Hildebrand s regular solution theory, and found (< i)pure liquid t by a Pitzer-type equation... 

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