Thermodynamics of Fractions and Its Application to the Hydration Study of the Swelling Process in Functionalized Polymer Particles

Abstract

A fraction of a three-component system is defined as a thermodynamic entity that groups two components. In this work, the thermodynamics of fractions is developed. The Kern-Weisbrod method has been proposed for two-component systems, but by considering the system as fractionalized, this method can also be employed to calculate the specific partial quantities of the components of a three-component system. This method is used to study the swelling process of functionalized polymeric particles. A series of copolymers of butyl acrylate (BuA) and methyl methacrylate (MMA) functionalized with 0, 5, 10, 15, and 20 wt % of acrylic acid (AA) was synthesized by emulsion polymerization. A thermodynamic study of the polymeric particle swelling as a function of the acrylic acid content and the temperature was carried out. Specific partial volumes, compressibilities, and expansibilities were calculated using density and speed of sound measurements. It was found that the hydration behavior of carboxylic and nonpolar groups of the polymeric particles is thermodynamically driven in accordance with the Gibbs-Duhem equation.

Study of the binding between lysozyme and C10-TAB: Determination and interpretation of the partial properties of protein and surfactant at infinite dilution

“…Defining a fraction of the system [25] as a thermodynamic entity with an internal composition that groups several components, the volume of the system can be written as V = V(m 1 , m F , t f3 ), where m F = m 2 + m 3 is the mass of the fraction F composed of components 2 and 3, and t f3 is the composition of the fraction (t f 3 = m 3 / (m 2 + m 3 )). With this variable change, the partial property of the fraction F in the presence of component 1 is [25]:…”

“…This region wherein the measured properties as functions of t F are linear is the so-called high dilution region [25]. In addition to this, the amount of C 10 -TAB was chosen in order to obtain values of t F below its critical micelle concentration [33] (66.1 mol m −3 ).…”

“…In this way, it is possible to calculate the partial volume of protein and ligand at infinite dilution. In a previous work [25], we employed this method to study the swelling process of functionalized polymer particles. However the narrow interval of ligand concentrations limits the possibility to study the binding process.…”

On the Physical Meaning of the Isothermal Titration Calorimetry Measurements in Calorimeters with Full Cells

“…Above the value xc f3 , j o F;1 can be written as: ${\text{j}}_{\text{F};1}^{\text{o}}={\text{X}}_{\text{f}2}{\text{j}}_{\text{F};1}^{\text{o}}false({\text{x}}_{\text{f}3}^{\text{c}}false)+{\text{X}}_{\text{f}3}{\text{j}}_{3;1}^{\text{o}}$where: ${\text{X}}_{\text{f}3}=\frac{{\text{x}}_{\text{f}3}-{\text{x}}_{\text{f}3}^{\text{c}}}{{\text{x}}_{\text{f}2}^{\text{c}}}$and X f2 = 1-X f3 . When we write jΔ 2;1,3 and j Δ 3;1,2 , we assume [4–6] that concomitantly component 2 is in the presence of components 1 and 3 and component 3 is in the presence of components 1 and 2. Thus the notation j o F;1 = x f2 j Δ 2;1,3 + x f3 j Δ 3;1,2 indicates that, F is composed of components 2 and 3, which are interacting in a medium (component 1).…”

“…By notation, we understand that j 1;2,3 means “the partial property of component 1 in the presence of components 2 and 3”. The notations j 2;1,3 and j 3;1,2 are interpreted in the same way.A fraction of a system [4–6] is defined as a thermodynamic entity with an internal composition that groups several components. If we suppose a fraction F is composed of components 2 and 3, the property J can be written as a “description by fractions” as: $\text{J}=\text{J}false({\text{n}}_1,{\text{n}}_{\text{F}},{\text{x}}_{\text{f}3}false)$where the new variables (fraction variables) are the total number of moles of the fraction F, n F = n 2 + n 3 , and x f3 = n 3 /(n 2 + n 3 ), which are related to the composition of F. The Gibbs equation for Equation (104) takes the form: $\text{dJ}={\text{j}}_{1;\text{F}}{\text{dn}}_1+{\text{j}}_{\text{F};1}{\text{dn}}_{\text{F}}+{\left(\frac{\partial \text{J}}{\partial {\text{x}}_{\text{f}3}}\right)}_{{\text{n}}_1,{\text{n}}_{\text{F}}}{\text{dx}}_{\text{f}3}$where j 1;F and j F;1 are respectively: $\begin{array}{c}left{\text{j}}_{1;\text{F}}({\text{x}}_{\text{F}},{\text{x}}_{\text{f}3})={\left(\frac{\partial \text{J}false({\text{n}}_1,{\text{n}}_{\text{F}},{\text{x}}_{\text{f}3}false)}{\partial {\text{n}}_1}\right)}_{{\text{n}}_{\text{F}},{\text{x}}_{\text{f}3}}\\ {\text{j}}_{\text{F};1}({\text{x}}_{\text{F}},{\text{x}}_{\text{f}3})={\left(\frac{\partial \text{J}false({\text{n}}_1,{\text{n}}_{\text{F}},{\text{x}}_{\text{f}3}false)}{\partial {\text{n}}_{\text{F}}}\right)}_{{\text{n}}_1,{\text{x}}_{\text{f}3}}\end{array}$…”