Determination of Thermodynamic Partial Properties in Multicomponent Systems by Titration Techniques

Abstract

Thermodynamic experimental techniques using titration are usually employed to study the interaction between solutes in a diluted solution. This chapter deals with the underlying thermodynamic framework when titration technique is applied with densimetry, sound speed measurement and isothermal titration calorimetry. In the case of partial volumes and partial adiabatic compressibilities, a physical interpretation is proposed based upon atomic, free volume and hydration contributions.

1. Introduction

The purposes of this chapter are twofold. First, the thermodynamics fundaments are studied in detail to determine experimentally, calculate and interpret thermodynamic partial molar properties using different titration techniques. Second, the postgraduate students are provided with the necessary thermodynamic background to extract behavioural trends from experimental techniques including densimetry, sound speed measurement and isothermal titration calorimetry.

The first concept introduced in this chapter is “thermodynamic description”. It is defined as a set of variables employed to define thermodynamically the studied system. For example, a description by components of a multicomponent system is:

$J=J(n_1,n_2,n_3)$E1

where J is an extensive thermodynamic property; n₁, n₂ and n₃ are the number of moles of components 1, 2 and 3. Other type of thermodynamic description is in terms of the concept of “fraction of a system”. A fraction of a system is a thermodynamic entity, with internal composition, which groups several components. For example, the above-mentioned system can be considered as being composed of the component 1, and a fraction F grouping components 2 and 3. In this way, J can be written as:

$J=J(n_1,n_F,x_{f3})$E2

where n_F is the total number of moles of the fraction F and x_f3 is a variable related to the composition of the fraction. Depending on the system, one can choose the more adequate description. For example, in a liquid mixture, a description by components (Eq. (1)) can be suitable. Other systems as those shown in Figure 1 could be better described in terms of fractions.

Figure 1.

Examples of different descriptions in two systems. (A) and (B) are several solutes in a solvent. (C) and (D) are a functionalized latex with polar groups.

Figure 1A shows a system composed of the solvent (component 1), solute A (component 2) and solute B (component 3). This system will be described in this chapter using a description by fractions representing a “complex solute” composed of solutes A and B (see Figure 1B). This description is appropriate to use in conditions of infinite dilution and dilute solutions. Other example (see Figure 1C and D) is a functionalized latex particle. A latex is a system composed of polymeric particles dispersed in a solvent. In a functionalized latex, particles are composed of non-polar groups and of functional groups (usually polar groups). In this case, a description by components expressed in Eq. (1) and visualized in Figure 1C is very difficult to use and it is more convenient to consider a fraction (polymeric particle) composed of non-polar groups (component 2) and of polar groups (component 3). Figure 1D shows a sketch of this description.

When different descriptions are considered for a system, we have to reconsider the relation between the description and the thermodynamic object studied. In principle, one might think that all descriptions are equivalent. But this is not true because not all descriptions can retain all features of a thermodynamic system. For example, it is not possible to speak about thermodynamic partial properties at infinite dilution in multicomponent systems. This fact should not be surprising because in differential geometry [1] there is the same problem associated with the relation between a parametrization and a geometric object. Let’s consider, for example, the sphere of radius equal to one, and a parametrization is:

$X_1(x,y)=\left(x,y,\sqrt{1-(x^2+y^2)}\right)$E3

The problem with this parametrization is that it only covers the top half of the sphere. In addition, it is not differentiable in the points of the sphere’s equator. Other possibility is:

$X_2(x,y)=\left(x,y,-\sqrt{1-(x^2+y^2)}\right)$E4

But, it only covers the lower half of the sphere and neither is differentiable in points of the sphere’s equator. Even if we consider a combination of X₁ and X₂, we have the problem of the lack of differentiability in the points of the sphere’s equator. Another possible parametrization is:

$X_3(\text{θ},\text{ϕ})=\left(\sin \text{θ}\cos \text{ϕ},\sin \text{θ}\sin \text{ϕ},\cos \text{θ}\right)$E5

where θ is the colatitude (the complement of the latitude) and ϕ the longitude. X₃ covers the whole surface of the sphere and it is also differentiable in all points. For this reason, it contains more information about the sphere (geometric object) than X₁ and X₂. Backing to thermodynamics, in the same case than for X₃, the partial molar properties at infinite dilution cannot be obtained and manipulated using the description by components, and it is necessary to use the description by fractions.

The other concept also introduced in this chapter is the “interaction between components of a system”. The first principle of thermodynamics establishes the way, in which systems interact between them and/or with surroundings. In this case, we are interested in the interaction inside the systems and this cannot be interpreted macroscopically using the first principle of thermodynamics. With the concept of interaction between components, we can define mathematically a dilute solution and characterize its thermodynamic behaviour in terms of molar partial properties. In addition to this, we will consider the partial molar properties at infinite dilution. These properties are essential in studies of polymeric particles because they contain the information about the interactions inside the particles. These interactions determine the architecture and final application of the particle.

2. Mathematical fundaments

In this section, some mathematical tools are presented such as changes of variable, changes of size, the Euler theorem and limits in multivariable functions. Variable changes will allow us to relate partial properties of different descriptions. Changes of size are the processes underlying the extensivity and non-extensivity of thermodynamic properties, which will be mathematically implemented by the concept of homogeneity. The Euler’s theorem will be treated in the more general form, and in its demonstration we will avoid some aspects, which remain unclear in the versions of the textbooks of Callen [2] and Klotz and Rosenberg [3].

2.1. Changes of variable

Let f be the function defined as:

$f=f(x_1,x_2,x_3)$E6

The gradient of f with respect to the variables x₁, x₂ and x₃ is the vector:

$\nabla f(x_1,x_2,x_3)=\left[\begin{array}{c}{\left(\frac{\partial f}{\partial x_1}\right)}_{x_2,x_3}\\ {\left(\frac{\partial f}{\partial x_2}\right)}_{x_1,x_3}\\ {\left(\frac{\partial f}{\partial x_3}\right)}_{x_1,x_2}\end{array}\right]$E7

If we consider the change of variable:

$\{\begin{array}{l}{x}_1=x_1(y_1,y_2,y_3)\\ x_2=x_2(y_1,y_2,y_3)\\ x_3=x_3(y_1,y_2,y_3)\end{array}$E8

the function f will take the form:

$f=f(y_1,y_2,y_3)$E9

where its gradient will be:

$\nabla f(y_1,y_2,y_3)=\left[\begin{array}{c}{\left(\frac{\partial f}{\partial y_1}\right)}_{y_2,y_3}\\ {\left(\frac{\partial f}{\partial y_2}\right)}_{y_1,y_3}\\ {\left(\frac{\partial f}{\partial y_3}\right)}_{y_1,y_2}\end{array}\right]$E10

Our interest is to relate the partial derivatives with respect to the variables x₁, x₂ and x₃ given in Eq. (7) with the partial properties with respect to y₁, y₂ and y₃ given in Eq. (10). From Eq. (8), the total differential of x₁ is:

$d{x}_1={\left(\frac{\partial x_1}{\partial y_1}\right)}_{y_2,y_3}d{y}_1+{\left(\frac{\partial x_1}{\partial y_2}\right)}_{y_1,y_3}d{y}_2+{\left(\frac{\partial x_1}{\partial y_3}\right)}_{y_1,y_2}d{y}_3$E11

Using dx₁ given by Eq. (11) and similarly with equations for dx₂ and dx_3, we can write:

$\left[\begin{array}{c}d{x}_1\\ d{x}_2\\ d{x}_3\end{array}\right]=T\left(\begin{array}{c}{x}_1{x}_{\text{2}}{x}_{\text{3}}\\ y_1{\text{ y}}_{\text{2}}{\text{ y}}_{\text{3}}\end{array}\right)\left[\begin{array}{c}d{y}_1\\ d{y}_2\\ d{y}_3\end{array}\right]$E12

where the matrix T is:

$T\left(\begin{array}{c}{x}_1{x}_{\text{2}}{x}_{\text{3}}\\ y_1{\mathrm{ y}}_{\text{2}}{\mathrm{ y}}_{\text{3}}\end{array}\right)=\left[\begin{array}{ccc}{\left(\frac{\partial x_1}{\partial y_1}\right)}_{y_2,y_3}& {\left(\frac{\partial x_1}{\partial y_2}\right)}_{y_1,y_3}& {\left(\frac{\partial x_1}{\partial y_3}\right)}_{y_1,y_2}\\ {\left(\frac{\partial x_2}{\partial y_1}\right)}_{y_2,y_3}& {\left(\frac{\partial x_2}{\partial y_2}\right)}_{y_1,y_3}& {\left(\frac{\partial x_2}{\partial y_3}\right)}_{y_1,y_2}\\ {\left(\frac{\partial x_3}{\partial y_1}\right)}_{y_2,y_3}& {\left(\frac{\partial x_3}{\partial y_2}\right)}_{y_1,y_3}& {\left(\frac{\partial x_3}{\partial y_3}\right)}_{y_1,y_2}\end{array}\right]$E13

From (6) and using (7), the total differential of f can be expressed as:

$df={\left(\frac{\partial f}{\partial x_1}\right)}_{x_2,x_3}d{x}_1+{\left(\frac{\partial f}{\partial x_2}\right)}_{x_1,x_3}d{x}_2+{\left(\frac{\partial f}{\partial x_3}\right)}_{x_1,x_2}d{x}_3={\left[\nabla f(x_1,x_2,x_3)\right]}^{\text{T}}\left[\begin{array}{c}d{x}_1\\ d{x}_2\\ d{x}_3\end{array}\right]$E14

where the symbol “T” indicates “transpose”. From Eq. (9) using Eq. (10), the differential of f can be written as:

$df={\left(\frac{\partial f}{\partial y_1}\right)}_{y_2,y_3}d{y}_1+{\left(\frac{\partial f}{\partial y_2}\right)}_{y_1,y_3}d{y}_2+{\left(\frac{\partial f}{\partial y_3}\right)}_{y_1,y_2}d{y}_3={\left[\nabla f(y_1,y_2,y_3)\right]}^{\text{T}}\left[\begin{array}{c}d{y}_1\\ d{y}_2\\ d{y}_3\end{array}\right]$E15

Equaling (15) to (14) and using (12):

${\left[\nabla f(y_1,y_2,y_3)\right]}^{\text{T}}={\left[\nabla f(x_1,x_2,x_3)\right]}^{\text{T}}T\left(\begin{array}{c}{x}_1{x}_{\text{2}}{x}_{\text{3}}\\ y_1{\text{ y}}_{\text{2}}{\text{ y}}_{\text{3}}\end{array}\right)$E16

Remembering that x being a vector and A a matrix, then (x^T A)^T = A^T x, and taking the transpose in both sides of (16):

$\nabla f(y_1,y_2,y_3)={\left[T\left(\begin{array}{c}{x}_1{x}_{\text{2}}{x}_{\text{3}}\\ y_1{\text{ y}}_{\text{2}}{\text{ y}}_{\text{3}}\end{array}\right)\right]}^{\text{T}}\nabla f(x_1,x_2,x_3)$E17

Eq. (17) relates the vector gradient with respect to the variables (x₁, x₂, x₃) to the vector gradient with respect to the variables (y₁, y₂, y₃), and it will allow us to express the partial properties in two different descriptions.

2.2. Changes of size

In this paragraph, the process of size change in thermodynamic systems is analyzed. The behaviour of systems in a size change has consequences on the behaviour or nature of the thermodynamic properties as well as on the form of the thermodynamic equations of the system. Figure 2 shows a visualization of this process in both directions: increasing and reduction.

Figure 2.

Sketch of the change of size (increasing and reduction) of a system with volume V.

From Figure 2, it is clear that being V the volume, N the number of moles and U the internal energy, the configuration of this system is, under increasing size λ times:

$\begin{array}{l}V{\displaystyle \stackrel{\text{λ}-times}{\to }}{V}^{\prime }=\text{λ}V\\ N{\displaystyle \stackrel{\text{λ}-times}{\to }}{N}^{\prime }=\text{λ}N\\ U{\displaystyle \stackrel{\text{λ}-times}{\to }}{U}^{\prime }=\text{λ}U\end{array}$E18

Thermodynamic properties, which transform accordingly to (18), depend on the size of the system and are named extensive variables. Not all thermodynamic variables transform according to Eq. (18). An example is the molar fraction of the component 2 (x₂) in a two-component system. We can see this formally in the following way. For a two-component system:

$\begin{array}{l}{N}_1{\displaystyle \stackrel{\text{λ}-times}{\to }}{N^{\prime }}_1=\text{λ}{N}_1\\ N_2{\displaystyle \stackrel{\text{λ}-times}{\to }}{N^{\prime }}_2=\text{λ}{N}_2\end{array}$E19

and x₂ transforms as:

$x_2{\displaystyle \stackrel{\text{λ}-times}{\to }}{x^{\prime }}_2=\frac{{N^{\prime }}_2}{{N^{\prime }}_1+{N^{\prime }}_2}=\frac{\text{λ}{N}_2}{\text{λ}{N}_1+\text{λ}{N}_2}=\frac{N_2}{N_1+N_2}=x_2$E20

That is, the molar fraction of the component 2 is independent of the system size. Properties, which remain constant upon size change, are named intensive properties. Other thermodynamic properties with such characteristics are temperature, pressure, pH and concentration c₂ (c₂ = N₂/V). It is also interesting to look at the behaviour of functions, which depend on thermodynamic variables (intensive and/or extensive), in a size change. Let, for example, the function f be given by f = f(T, P, N₁, N₂, …). For particular values of the variables T₀, P₀, N₀₁, N₀₂, …, the function f takes the value f₀, and in a change of size:

$\begin{array}{l}{T}_0{\displaystyle \stackrel{\text{λ}-times}{\to }}{T^{\prime }}_0=T_0\\ P_0{\displaystyle \stackrel{\text{λ}-times}{\to }}{P^{\prime }}_0=P_0\\ N_{01}{\displaystyle \stackrel{\text{λ}-times}{\to }}{N^{\prime }}_{01}=\text{λ}{N}_{01}\\ N_{02}{\displaystyle \stackrel{\text{λ}-times}{\to }}{N^{\prime }}_{02}=\text{λ}{N}_{02}\\ \dots \\ f_0{\displaystyle \stackrel{\text{λ}-times}{\to }}{f^{\prime }}_0\end{array}$E21

If ${f^{\prime }}_0=l{f}_0$, the function will behave as an extensive property. This concept is mathematically implemented as:

$f(T,P,\text{λ}{N}_1,\text{λ}{N}_2,\dots )=\text{λ}f(T,P,N_1,N_2,\dots )$E22

In this case, f is a homogeneous function of one degree. If ${f^{\prime }}_0=f_0$, f will behave as an intensive property. Mathematically, f is expressed as a homogeneous function of zero degree as:

$f(T,P,\text{λ}{N}_1,\text{λ}{N}_2,\dots )=f(T,P,N_1,N_2,\dots )$E23

2.3. Euler´s theorem

Let f = f(x₁, x₂,…; y₁, y₂,…) be a function, which is a homogeneous function of one degree with respect to the variables y₁, y₂,…:

$f(x_1,x_2,\dots ;\text{λ}{y}_1,\text{λ}{y}_2,\dots )=\text{λ}f(x_1,x_2,\dots ;y_1,y_2,\dots )$E24

Then,

$f={\left(\frac{\partial f}{\partial y_1}\right)}_{x_1,x_2,\dots ;y_2,y_3,\mathrm{\dots .}}{y}_1+{\left(\frac{\partial f}{\partial y_2}\right)}_{x_1,x_2,\dots ;y_1,y_3,\mathrm{\dots .}}{y}_2+\mathrm{\dots .}$E25

The demonstration is as follows. The differential with respect to λ in the left side of (24) is:

$\begin{array}{c}\frac{df(x_1,x_2,\dots ;\text{λ}{y}_1,\text{λ}{y}_2,\dots )}{d\text{λ}}={\left(\frac{\partial f(x_1,x_2,\dots ;\text{λ}{y}_1,\text{λ}{y}_2,\dots )}{\partial x_1}\right)}_{x_2,x_3,\dots ;\text{λ}{y}_1,\text{λ}{y}_2,\dots }\\ \frac{d{x}_1}{d\text{λ}}+{\left(\frac{\partial f(x_1,x_2,\dots ;\text{λ}{y}_1,\text{λ}{y}_2,\dots )}{\partial x_2}\right)}_{x_1,x_3,\dots ;\text{λ}{y}_1,\text{λ}{y}_2,\dots }\\ \frac{d{x}_2}{d\text{λ}}+{\left(\frac{\partial f(x_1,x_2,\dots ;\text{λ}{y}_1,\text{λ}{y}_2,\dots )}{\partial (\text{λ}{y}_1)}\right)}_{x_1,x_2,\dots ;\text{λ}{y}_2,\text{λ}{y}_3,\dots }\\ \frac{d(\text{λ}{y}_1)}{d\text{λ}}+{\left(\frac{\partial f(x_1,x_2,\dots ;\text{λ}{y}_1,\text{λ}{y}_2,\dots )}{\partial (\text{λ}{y}_2)}\right)}_{x_1,x_2,\dots ;\text{λ}{y}_1,\text{λ}{y}_3,\dots }\frac{d(\text{λ}{y}_2)}{d\text{λ}}+\dots \end{array}$E26

For the sets of variables x₁, x₂,… and y₁,y₂,…, we obtain respectively that:

$\frac{d{x}_1}{d\text{λ}}=\frac{d{x}_2}{d\text{λ}}=\dots =0$E27

$\frac{d(\text{λ}{y}_1)}{d\text{λ}}=y_1,\frac{d(\text{λ}{y}_2)}{d\text{λ}}=y_2,\dots$E28

The following step in this demonstration is different from the step proposed in other textbooks [2, 3]. The partial derivative of f with respect to (λy₁) can be expressed as:

$\begin{array}{c}{\left(\frac{\partial f(x_1,x_2,\dots ;\text{λ}{y}_1,\text{λ}{y}_2,\dots )}{\partial (\text{λ}{y}_1)}\right)}_{x_1,x_2,\dots ;\text{λ}{y}_2,\text{λ}{y}_3,\dots }\\ ={\lim }_{\text{Δ}\to 0}\frac{f(x_1,x_2,\dots ;\text{λ}{y}_1+\text{Δ},\text{λ}{y}_2,\dots )-f(x_1,x_2,\dots ;\text{λ}{y}_1,\text{λ}{y}_2,\dots )}{\text{Δ}}\end{array}$E29

Considering that f is a homogeneous function of one degree with respect to the variables y₁,y₂,… and making Δ’= Δ/λ in (29),

$\begin{array}{c}{\left(\frac{\partial f(x_1,x_2,\dots ;\text{λ}{y}_1,\text{λ}{y}_2,\dots )}{\partial (\text{λ}{y}_1)}\right)}_{x_1,x_2,\dots ;\text{λ}{y}_2,\text{λ}{y}_3,\dots }\\ ={\lim }_{{\Delta }^{\prime }\to 0}\frac{f(x_1,x_2,.\dots ;y_1+{\Delta }^{\prime },y_2,\dots )-f(x_1,x_2,.\dots ;y_1,y_2,\dots )}{{\Delta }^{\prime }}={\left(\frac{\partial f(x_1,x_2,\dots ;y_1,y_2,\dots )}{\partial y_1}\right)}_{x_1,x_2,\dots ;y_2,y_3,\dots }\end{array}$E30

The differential of f with respect to λ in the right side of (24) is:

$\frac{d\left[\text{λ}f(x_1,x_2,\dots ;y_1,y_2,\dots )\right]}{d\text{λ}}=f(x_1,x_2,\dots ;y_1,y_2,\dots )$E31

Eq. (25) is obtained by substituting Eqs. (27), (28), (30), (31) in Eq. (26). In addition, it is interesting to see that, defining f₁ as f₁ = (∂f/∂x₁) and using (30), f₁ is a homogeneous function of zero degree with respect to the variables y₁, y₂,…:

$f_1(x_1,x_2,\dots ;\text{λ}{y}_1,\text{λ}{y}_2,\dots )=f_1(x_1,x_2,\dots ;y_1,y_2,\dots )$E32

3. Thermodynamic descriptions

3.1. Description by components

Let it be a three-component system (e.g., as those of Figure 1A and C). Being J an extensive property, a description by components is:

$J=J(n_1,n_2,n_3)$E33

where n₁, n₂ and n₃ are the number of moles of components 1, 2 and 3. The partial property of 1 is defined as:

$j_{1;2,3}(n_1,n_2,n_3)={\left(\frac{\partial J(n_1,n_2,n_3)}{\partial n_1}\right)}_{n_2,n_3}$E34

From the above section, we know that j_1;2,3 is homogeneous function of zero degree with respect to n₁, n₂ and n₃. With this and considering λ = 1/(n₁+n₂+n₃),

$\begin{array}{c}{j}_{1;2,3}(n_1,n_2,n_3)=j_{1;2,3}(\frac{n_1}{n_1+n_2+n_3},\frac{n_2}{n_1+n_2+n_3},\frac{n_3}{n_1+n_2+n_3})=\\ =j_{1;2,3}(x_1,x_2,x_3)=j_{1;2,3}(x_2,x_3)\end{array}$E35

where we have considered that x₁ is a function of x₂ and x₃ because x₁ = 1−x₂−x₃. From (35), we see that the partial molar properties depend only on the composition of the system. Alternatively to (35), we could use other scales of composition/concentration to express j_1;2,3.

The equation of Gibbs is obtained by differentiating J in (33) and using Eq. (34) and similar definitions for components 2 and 3:

$dJ=j_{1;2,3}d{n}_1+j_{2;1,3}d{n}_2+j_{3;1,2}d{n}_3$E36

The Euler equation is obtained by considering that J is a homogeneous function with respect to n₁, n₂ and n₃ and applying the Euler’s theorem:

$J=n_1{j}_{1;2,3}+n_2{j}_{2;1,3}+n_3{j}_{3;1,2}$E37

The Gibbs-Duhem equation is obtained by differentiating in Eq. (37), equalling to Eq. (36) and cancelling common terms:

$0=n_{1}d{j}_{1;2,3}+n_{2}d{j}_{2;1,3}+n_{3}d{j}_{3;1,2}$E38

If we consider that partial molar properties are function of n₁, n₂ and n₃, Eq. (38) would be the Gibbs-Duhem equation in the representation of variables n₁, n₂ and n₃. The representation in the variables x₂ and x₃ is as follows. Dividing (38) by the total number of moles,

$0=x_{1}d{j}_{1;2,3}+x_{2}d{j}_{2;1,3}+x_{3}d{j}_{3;1,2}$E39

Calculating the differentials by considering that partial molar properties depend on x₂ and x₃, and bearing in mind that x₂ and x₃ are independent variables, (38) can be written in an alternative way as:

$\{\begin{array}{c}{x}_1{\left(\frac{\partial j_{1;2,3}}{\partial x_2}\right)}_{x_3}+x_2{\left(\frac{\partial j_{2;1,3}}{\partial x_2}\right)}_{x_3}+x_3{\left(\frac{\partial j_{3;1,2}}{\partial x_2}\right)}_{x_3}=0\\ x_1{\left(\frac{\partial j_{1;2,3}}{\partial x_3}\right)}_{x_2}+x_2{\left(\frac{\partial j_{2;1,3}}{\partial x_3}\right)}_{x_2}+x_3{\left(\frac{\partial j_{3;1,2}}{\partial x_3}\right)}_{x_2}=0\end{array}$E40

3.2. Description by fractions

In a description by fractions, we consider the three-component system as composed of a component 1 and a group (or fraction) composed of components 2 and 3. Figure 1B shows the example when two solutes are grouped in a “complex solute”, and Figure 1D shows the example in which a polymeric particle composed of polar and non-polar groups is considered as a fraction of the system. In this case, the extensive property J is expressed as:

$J=J(n_1,n_F,x_{f3})$E41

where

$n_F=n_2+n_3$E42

$x_{f3}=\frac{n_3}{n_2+n_3}$E43

The variable n_F is the total number of moles of the fraction F, and x_f3 is a variable related to its internal composition. The partial molar properties of J in this description are:

$j_{1;F}(n_1,n_F,x_{f3})={\left(\frac{\partial J(n_1,n_F,x_{f3})}{\partial n_1}\right)}_{n_F,x_{f3}}$E44

$j_{F;1}(n_1,n_F,x_{f3})={\left(\frac{\partial J(n_1,n_F,x_{f3})}{\partial n_F}\right)}_{n_1,x_{f3}}$E45

Because J is a homogeneous function of n₁ and n_F, the partial properties j_1;F and j_F;1 will be homogeneous functions of zero degree with respect to the variables n₁ and n_F. In this way and similarly to Eq. (35):

$j_{1;F}(n_1,n_F,x_{f3})=j_{1;F}(x_F,x_{f3})$E46

where x_F = n_F/(n₁+n_F). Now, we will see the relation between both descriptions. From (42) and (43), the change of variable of Eq. (8) is in this case:

$\{\begin{array}{l}{n}_1(n_1,n_F,x_{f3})=n_1\\ n_2(n_1,n_F,x_{f3})=(1-x_{f3})n_F\\ n_3(n_1,n_F,x_{f3})=x_{f3}{n}_F\end{array}$E47

Substituting (47) in (13) and the result in (17), one obtains that:

$j_{1;F}=j_{1;2,3}$E48

$j_{F;1}=x_{f2}{j}_{2;1,3}+x_{f3}{j}_{3;1,2}$E49

${\left(\frac{\partial J}{\partial x_{f3}}\right)}_{n_1,n_F}=n_F(j_{3;1,2}-j_{2;1,3})$E50

The equations of Gibbs, Euler and Gibbs-Duhem in this description are as follows. The Gibbs equation is obtained by differentiating in (41) and considering the definitions given in (44) and (45):

$dJ=j_{1;F}d{n}_1+j_{F;1}d{n}_F+{\left(\frac{\partial J}{\partial x_{f3}}\right)}_{n_1,n_F}d{x}_{f3}$E51

The Euler equation is obtained by remembering that J is a homogeneous function of degree one of n₁ and n_F and using the Euler’s theorem:

$J=n_1{j}_{1;F}+n_F{j}_{F;1}$E52

The Gibbs-Duhem equation in the representation of variables x_F and x_f3 is obtained by differentiating in (52), equalling to (51) and cancelling common terms, and dividing by the total number of moles:

$x_{1}d{j}_{1;F}+x_{F}d{j}_{F;1}=x_F(j_{3;1,2}-j_{2;1,3})d{x}_{f3}$E53

Considering that j_1;F and j_F;1 are functions of the independent variables x_F and x_f3, then (53) will take the form:

$\{\begin{array}{l}{x}_1{\left(\frac{\partial j_{1;F}}{\partial x_F}\right)}_{x_{f3}}+x_F{\left(\frac{\partial j_{F;1}}{\partial x_F}\right)}_{x_{f3}}=0\\ x_1{\left(\frac{\partial j_{1;F}}{\partial x_{f3}}\right)}_{x_F}+x_F{\left(\frac{\partial j_{F;1}}{\partial x_{f3}}\right)}_{x_F}=x_F(j_{3;1,2}-j_{2;1,3})\end{array}$E54

Calculating the partial derivative of j_F;1 with respect to x_f3 in Eq. (49) and substituting in Eq. (54), we obtain:

$\{\begin{array}{l}{x}_1{\left(\frac{\partial j_{1;F}}{\partial x_F}\right)}_{x_{f3}}+x_F{\left(\frac{\partial j_{F;1}}{\partial x_F}\right)}_{x_{f3}}=0\\ x_1{\left(\frac{\partial j_{1;F}}{\partial x_{f3}}\right)}_{x_F}+x_F{\left[x_{f2}{\left(\frac{\partial j_{2;1,3}}{\partial x_{f3}}\right)}_{x_F}+x_{f3}{\left(\frac{\partial j_{3;1,2}}{\partial x_{f3}}\right)}_{x_F}\right]}_{x_F}=0\end{array}$E55

It is interesting to observe that considering constant composition (dx_f3 = 0) in Eqs. (51)–(53), then the system behaves as a two-component system. This fact cannot be obtained using the description by components.

4. Partial properties in diluted solutions of multicomponent systems

We consider intuitively a diluted solution when the properties of the solution are similar to those of its solvent in pure state. In this section, we will study the thermodynamic behaviour of the partial molar properties in this region of concentrations.

4.1. Thermodynamic concept of interaction between components

In this paragraph, we will define the concept of non-interaction and prove that when applying it to a system, the system behaves as an ideal mixing. From a thermodynamic point of view, the components of a system are not interacting if both following points hold simultaneously.

1. The state of each component in the system, expressed in terms of its partial molar properties, does not vary by changes of composition of the other components. It means each component does not detect the presence of the other components.

2. The formation of the system from its pure components is carried out with any cost of energy, neither for the system nor for the surroundings.

Mathematically, the first point can be written as:

${\left(\frac{\partial j_{1;2,3}(x_2,x_3)}{\partial x_2}\right)}_{x_3}={\left(\frac{\partial j_{1;2,3}(x_2,x_3)}{\partial x_3}\right)}_{x_2}=0\iff j_{1;2,3}=j_{1;2,3}(x_1)$E56

Substituting (56) in (40) and considering also that:

$\{\begin{array}{c}{\left(\frac{\partial j_{1;2,3}}{\partial x_2}\right)}_{x_3}={\left(\frac{\partial j_{1;2,3}}{\partial x_1}\right)}_{x_3}{\left(\frac{\partial x_1}{\partial x_2}\right)}_{x_3}=-{\left(\frac{\partial j_{1;2,3}}{\partial x_1}\right)}_{x_3}\\ {\left(\frac{\partial j_{1;2,3}}{\partial x_3}\right)}_{x_2}={\left(\frac{\partial j_{1;2,3}}{\partial x_1}\right)}_{x_2}{\left(\frac{\partial x_1}{\partial x_3}\right)}_{x_2}=-{\left(\frac{\partial j_{1;2,3}}{\partial x_1}\right)}_{x_2}\end{array}$E57

it is obtained that:

$\{\begin{array}{c}-x_1{\left(\frac{\partial j_{1;2,3}}{\partial x_1}\right)}_{x_3}+x_2{\left(\frac{\partial j_{2;1,3}}{\partial x_2}\right)}_{x_3}=0\\ -x_1{\left(\frac{\partial j_{1;2,3}}{\partial x_1}\right)}_{x_2}+x_3{\left(\frac{\partial j_{2;1,3}}{\partial x_3}\right)}_{x_2}=0\end{array}$E58

Because j_1;2,3 depends only on x₁:

${\left(\frac{\partial j_{1;2,3}}{\partial x_1}\right)}_{x_2}={\left(\frac{\partial j_{1;2,3}}{\partial x_1}\right)}_{x_3}$E59

and then (57) yields:

$x_1{\left(\frac{\partial j_{1;2,3}(x_1)}{\partial x_1}\right)}_{x_3}=x_2{\left(\frac{\partial j_{2;1,3}(x_2)}{\partial x_2}\right)}_{x_3}=x_3{\left(\frac{\partial j_{3;1,2}(x_3)}{\partial x_3}\right)}_{x_3}$E60

Because the first term depends only on x₁ and the second and third terms depend only on x₂ and x₃, respectively, from (60), we have that:

$x_1\frac{d{j}_{1;2,3}}{d{x}_1}=k{}_J(T,P)$E61

where k_J is a function, which only depends on temperature T and pressure P. Similar equations to (61) are obtained for j_2;1,3 and j_3;1,2. Integrating in (61) between ${x^{\prime }}_1=1$ and x₁,

$j_{1;2,3}(x_1)=j_1+k_J(T,P)ln(x_1)$E62

For components 2 and 3, similar equations to (62) are obtained. Now, we will apply the second point of the above definition of non-interaction. The zero cost of energy for the system and surroundings is equivalent to:

$Q_{mix}=W_{mix}=0\Rightarrow \text{Δ}{U}_{mix}=0\Rightarrow \{\begin{array}{c}{u}_{1;2,3}=u_1\\ h_{1;2,3}=h_1\\ v_{1;2,3}=v_1\end{array}$E63

Considering u_{1; 2,3}, h_{1; 2,3} and v_{1; 2,3} as (62) and bearing in mind (63):

$k_U(T,P)=k_H(T,P)=k_V(T,P)=0$E64

In addition to this g_1;2,3 (free energy of Gibbs),

$v_{1;2,3}={\left(\frac{\partial g_{1;2,3}}{\partial P}\right)}_{T,x_2,x_3}\Rightarrow k_V(T,P)={\left(\frac{\partial k_G(T,P)}{\partial P}\right)}_T$E65

$\frac{h_{1;2,3}}{T^2}=-{\left(\frac{\partial }{\partial T}\left(\frac{g_{1;2,3}}{T}\right)\right)}_{P,x_2,x_3}\Rightarrow \frac{k_H(T,P)}{T^2}=-{\left(\frac{\partial }{\partial T}\left(\frac{k_G(T,P)}{T}\right)\right)}_T$E66

Combining Eqs. (64)–(66), we have that:

$k_G(T,P)=kT$E67

where k is a constant. For the entropy, one gets:

$-s_{1;2,3}={\left(\frac{\partial g_{1;2,3}}{\partial T}\right)}_{P,x_2,x_3}\Rightarrow -k_S(T,P)={\left(\frac{\partial k_G(T,P)}{\partial T}\right)}_P$E68

With this,

$g_{1;2,3}=g_1+kTln(x_1)$E69

$s_{1;2,3}=s_1-kln(x_1)$E70

and we have demonstrated that a system holding the non-interaction definition proposed is an ideal mixing.

4.2. Diluted solutions

In this section, we will define the thermodynamic concept of diluted solutions and study the behaviour of the partial molar properties of these solutions. Commonly and intuitively, we consider a solution as diluted when its properties are similar to those of the pure solvent. We can implement mathematically this concept in the following way. When we remove all solutes from a solution, we have that:

${\lim }_{x_2+x_3\to 0}j(x_2,x_3)=j_1$E71

where j is the molar property of the extensive thermodynamic property J. In addition, the partial derivatives must vanish:

${\lim }_{x_2+x_3\to 0}{\left(\frac{\partial j_{1;2,3}(x_2,x_3)}{\partial x_2}\right)}_{x_3}={\lim }_{x_2+x_3\to 0}{\left(\frac{\partial j_{1;2,3}(x_2,x_3)}{\partial x_3}\right)}_{x_2}=0$E72

Otherwise, we would have memory effects and we can see this with an example. If we purify water, the pure substance obtained does not depend on the initial diluted solution employed. Actually, pure water is commonly used as a standard because it does not depend on the part of world, in which it is obtained. The Taylor’s expansion of j_1;2,3 is:

$j_{1;2,3}(x_2,x_3)=j_1(0,0)+{\left[\nabla j_{1;2,3}(0,0)\right]}^{\text{T}}\left[\begin{array}{c}{x}_2\\ x_3\end{array}\right]+\frac{1}{2}\left(x_2,x_3\right)H{j}_{1;2,3}(0,0)\left[\begin{array}{c}{x}_2\\ x_3\end{array}\right]+\dots$E73

where ∇j_1;2,3(0,0) and Hj_1;2,3(0,0) are, respectively, the vector gradient and the Hessian of j_1;2,3 matrix at (0,0). Considering (71) and (72) in (73) and that all partial derivatives mush vanish at (0,0), we have that for diluted solutions:

$j_{1;2,3}(x_2,x_3)\approx j_1+\dots$E74

From Eq. (74), we have for diluted solutions:

${\left(\frac{\partial j_{1;2,3}}{\partial x_2}\right)}_{x_3}\approx {\left(\frac{\partial j_{1;2,3}}{\partial x_3}\right)}_{x_2}\approx 0$E75

The behaviour of molar partial properties of solutes is as follows. Considering a “complex solute” S composed of 2 and 3 (as in Figure 1B),

$j_{1;S}(x_S,x_{s3})\approx j_1$E76

and substituting Eq. (74) in the first equation of (55),

${\left(\frac{\partial j_{S;1}}{\partial x_S}\right)}_{x_{s3}}\approx 0\iff j_{S;1}(x_S,x_{s3})\approx j_{S;1}(x_{s3})$E77

Inserting Eq. (76) in the second equation of (55), it is obtained that:

$x_{s2}{\left(\frac{\partial j_{2;1,3}}{\partial x_{s3}}\right)}_{x_S}+x_{s3}{\left(\frac{\partial j_{3;1,2}}{\partial x_{s3}}\right)}_{x_S}\approx 0$E78

Until now, we have seen the effect of the dilution in the capacity of detecting the presence of other components in a diluted solution. In order to gain an insight into the interactions, we have to study the process of mixing in diluted solutions. From (71), we can write:

$\{\begin{array}{l}{\lim }_{x_2+x_3\to 0}h=h_1\Rightarrow {\lim }_{x_2+x_3\to 0}{q}_{mix}={\lim }_{x_2+x_3\to 0}{\text{Δ}}_{mix}h=0\\ {\lim }_{x_2+x_3\to 0}v=v_1\Rightarrow {\lim }_{x_2+x_3\to 0}{w}_{mix}={\lim }_{x_2+x_3\to 0}-P{\text{Δ}}_{mix}v=0\end{array}$E79

It indicates that in the limit of infinite dilution, components do not interact because the process of mixture does not have any energy cost. This result implicates that in diluted solutions, according to the asymptotic approach given by Eq. (74), the interaction between solvent and solutes is weak and it can be neglected.

4.3. Partial molar properties of interaction in diluted solutions

The molar property j of a diluted solution can be written as:

$j\approx x_1{j}_1+x_S{j}_{S;1}$E80

where we are considering the interaction between components 2 and 3 since

$j_{S;1}=x_{s2}{j}_{2;1,3}+x_{s3}{j}_{3;1,2}$E81

In a diluted solution without interaction between 2 and 3, the property j^Ø can be written as:

$j^{\varnothing }=x_1{j}_1+x_S(x_{s2}{j}_{2;1}+x_{s3}{j}_{3;1})$E82

In this way, we can calculate the interaction contributions to j as:

$\text{Δ}{j}^{\mathrm{int}}=j-j^{\varnothing }\approx x_S\text{Δ}{j}_{S;1}$E83

where

$\text{Δ}{j}_{S;1}=x_{s2}\text{Δ}{j}_{2;1,3}+x_{s3}\text{Δ}{j}_{3;1,2}$E84

is the partial molar property of interaction of the complex solute and

$\{\begin{array}{l}\text{Δ}{j}_{2;1,3}=j_{2;1,3}-j_{2;1}\\ \text{Δ}{j}_{3;1,2}=j_{3;1,2}-j_{3;1}\end{array}$E85

are the partial molar properties of interaction of the components 2 and 3, respectively. These properties are not independent as we will see as follows. Combining (78) and (85),

$x_{s2}{\left(\frac{\partial \text{Δ}{j}_{2;1,3}}{\partial x_{s3}}\right)}_{x_S}+x_{s3}{\left(\frac{\partial \text{Δ}{j}_{3;1,2}}{\partial x_{s3}}\right)}_{x_S}+x_{s2}{\left(\frac{\partial j_{2;1}}{\partial x_{s3}}\right)}_{x_S}+x_{s3}{\left(\frac{\partial j_{3;1}}{\partial x_{s3}}\right)}_{x_S}\approx 0$E86

In Eq. (85), Δj_2;1,3 and Δj_3;1,2 are evaluated when using concentrations x_S and x_s3. Accordingly, j_2,1 is evaluated using the concentration x₂ given by x₂ = x_S (1−x_s3), and then,

${\left(\frac{\partial j_{2;1}}{\partial x_{s3}}\right)}_{x_S}=\frac{d{j}_{2;1}}{d{x}_2}{\left(\frac{\partial x_2}{\partial x_{s3}}\right)}_{x_S}=-x_S\frac{d{j}_{2;1}}{d{x}_2}$E87

Considering the Gibbs-Duhem equation for a two-component system:

$x_1\frac{d{j}_{1;2}}{d{x}_2}+x_2\frac{d{j}_{2;1}}{d{x}_2}=0$E88

in Eq. (87) and bearing in mind that solutions are diluted,

$x_{s2}{\left(\frac{\partial j_{2;1}}{\partial x_{s3}}\right)}_{x_S}=x_1\frac{d{j}_{1;2}}{d{x}_2}\approx 0$E89

Similarly for component 3,

$x_{s3}{\left(\frac{\partial j_{3;1}}{\partial x_{s3}}\right)}_{x_S}=x_1\frac{d{j}_{1;3}}{d{x}_3}\approx 0$E90

Substituting (89) and (90) in (86), we obtain:

$x_{s2}{\left(\frac{\partial \text{Δ}{j}_{2;1,3}}{\partial x_{s3}}\right)}_{x_S}+x_{s3}{\left(\frac{\partial \text{Δ}{j}_{3;1,2}}{\partial x_{s3}}\right)}_{x_S}\approx 0$E91

Eq. (91) indicates that in a diluted solution, the interaction between components 2 and 3 is not vanished. The partial molar property of interaction of the complex solute can be calculated experimentally as:

$\text{Δ}{j}_{S;1}\approx j_{S;1}-(x_{s2}{j}_{2;1}+x_{s3}{j}_{3;1})$E92

and the partial properties of interaction of components 2 and 3 can be obtained from (92) using the equations:

$\{\begin{array}{l}\text{Δ}{j}_{2;1,3}\approx \text{Δ}{j}_{S;1}-x_{s3}\frac{d\text{Δ}{j}_{S;1}}{d{x}_{s3}}\hfill \\ \text{Δ}{j}_{3;1,2}\approx \text{Δ}{j}_{S;1}+(1-x_{s3})\frac{d\text{Δ}{j}_{S;1}}{d{x}_{s3}}\hfill \end{array}$E93

Eq. (93) is obtained by differentiating in Eq. (92) with respect to x_s3, using Eq. (91) and combining the result with Eq. (92). As we will see below, Eq. (93) will allow us to obtain the interaction partial properties of 2 and 3 from experimental data.

4.4. Experimental determination of partial molar properties of interaction in diluted solutions

4.4.1. Partial specific volumes of interaction and partial specific adiabatic compressibility of interaction

As an example, we will consider the interaction between functionalized polymeric particles and an electrolyte at 30°C [4]. For that, polymeric particles synthesized of poly(n-butyl acrylate-co-methyl methacrylate) functionalized with different concentrations of acrylic acid were used in this study. The electrolyte was NaOH. Similarly to Figure 1A, water (solvent) was considered as component 1, polymeric particles as component 2 and electrolyte as component 3. And similarly to Figure 1B, the system was fractionalized in component 1 and a complex solute composed of polymeric particles and electrolyte. The experimental measurements were carried out using a Density and Sound Analyzer DSA 5000 from Anton-Paar connected to a titration cell. It is of full cell type, which is usually employed in isothermal titration calorimetry. Polymeric particles were located in the titration cell, and electrolyte was located in the syringe. Concentrations of polymeric particles (c₂) and electrolyte (c₃) after each titration were calculated as [4, 5]:

$\{\begin{array}{l}{c}_2^{i+1}=c_2^i{e}^{-\frac{v}{V}}\hfill \\ c_3^{i+1}=c_3^s-(c_3^s-c_3^i)e^{-\frac{v}{V}}\hfill \end{array}$E94

where V is the effective volume of the titration cell, v is the titration volume and $c_3^s$ is the stock concentration of electrolyte in the syringe. Figure 3A and B shows, respectively, data of density (ϱ) and sound speed (u) as function of the electrolyte concentration. The specific volume (v) and the specific adiabatic compressibility (ks) were calculated as:

Figure 3.

(A) Density as function of the electrolyte concentration c₃ (g/L). (B) Sound speed as function of the electrolyte concentration. (C) Partial specific volume of the complex solute composed of polymeric particles and electrolyte as function of the mass fraction of the electrolyte in the complex solute (t_s3). (D) Partial specific adiabatic compressibility of the complex solute as function of t_f3.

$v=\frac{1}{\rho }$E95

$k_s={\left(\frac{10}{\rho u}\right)}^2$E96

Considering the solution in the cell as diluted, the partial specific volume (and similarly the partial specific adiabatic compressibility) of the complex solute can be calculated as:

$v_{S;1}=\frac{v-t_1{v}_1}{t_S}$E97

where t₁ and t_S are the mass fraction of the water and of the complex solute, respectively. Figure 3C and D shows the partial specific volume and partial specific adiabatic compressibility as function of t_f3 (mass fraction of the electrolyte in the complex solute). The term of interaction Δv_S;1 is calculated by Eq. (92), where v_2;1 is obtained by considering that:

$v_{2;1}={\lim }_{t_{s3}\to 0}{v}_{S;1}$E98

in Figure 3C. The term v_3;1 is calculated by extrapolating the linear part of v_S;1 in Figure 3C as:

$v_{3;1}={\lim }_{t_{s3}\to 1}{v}_{S;1}$E99

The partial specific volume of interaction of the polymeric particles (Δv_2;1,3) and the partial specific volume of interaction of the electrolyte (Δv_3;1,2) were obtained using Eq. (93). The numerical method employed to calculate the derivatives is shown elsewhere [4]. Figure 4 shows the values of Δv_S;1, Δv_2;1,3 and Δv_3;1,2, and Figure 5 shows the values of Δk_{s S;1}, Δk_{s 2;1,3} and Δk_{s 3;1,2} obtained in a similar way than for volumes.

Figure 4.

(A) Partial specific volume of interaction of the complex solute (polymeric particles + electrolyte) as function of the mass fraction of the electrolyte in the complex solute (t_s3). (B) Partial specific volume of interaction of the polymeric particles as function of t_s3. (C) Partial specific volume of interaction of the electrolyte as function of t_s3.

Partial volume of polymeric particles (v_2;1) can be broken down in the following contributions [6–11]:

$v_{2;1}=v_{2;1/atom}+v_{2;1/free}+v_{2;1/hyd}$E100

which are shown in Figure 6. The atomic volume contribution (v_2;1/atom) is the sum of all volumes of the atoms, which make up polymeric chains. The free volume contribution (v_2;1/free) is consequence of the imperfect packing of the polymeric chains. The atomic volume contribution and free volume contribution are both positive contributions. The hydration contribution (v_2;1/hyd) is negative, as a consequence of that the specific volume of water molecules in bulk is larger than the specific volume in the hydration shell. The contributions to the partial specific adiabatic compressibility are the free volume and hydration because the effect of the pressure on the atomic volume is neglected [10, 12–21]:

Figure 5.

(A) Partial specific adiabatic compressibility of interaction of the complex solute (polymeric particles + electrolyte) as function of the mass fraction of the electrolyte in the complex solute (t_s3). (B) Partial specific adiabatic compressibility of interaction of the polymeric particles as function of t_s3. (C) Partial specific adiabatic compressibility of interaction of the electrolyte as function of t_s3.

Figure 6.

Contributions to the partial volume in a polymeric particle.

$k_{T\text{ 2;1}}=-{\left(\frac{\partial v_{2;1}}{\partial P}\right)}_T=-{\left(\frac{\partial v_{2;1/free}}{\partial P}\right)}_T-{\left(\frac{\partial v_{2;1/hyd}}{\partial P}\right)}_T=k_{T2;1/free}+k_{T2;1/hyd}$E101

The contribution k_{T 2;1/free} is positive, and the contribution k_{T 2;1/hyd} is negative [4, 8]. In this chapter, we will take the adiabatic compressibility as an approximation of the isothermal compressibility. For the electrolyte, the free volume contribution is null, and then, v_3;1 and k_{T 3;1} will take the following form:

$v_{3;1}=v_{3;1/atom}+v_{3;1/hyd}$E102

$k_{T3;1}=-{\left(\frac{\partial v_{3;1}}{\partial P}\right)}_T=-{\left(\frac{\partial v_{3;1/hyd}}{\partial P}\right)}_T=k_{T3;1/hyd}$E103

For the complex solute, we can write a similar breakdown:

$v_{S;1}=v_{S;1/atom}+v_{S;1/free}+v_{S;1/hyd}$E104

Inserting Eqs. (100), (102) and (104) in Eq. (92) and neglecting the variation in the atomic contributions, the following equation for the interaction specific partial volume is obtained:

$\text{Δ}{v}_{S,1}=\text{Δ}{v}_{S,1/free}+\text{Δ}{v}_{S,1/hyd}$E105

where

$\text{Δ}{v}_{S;1/free}=v_{S;1/free}-t_{s2}{v}_{2;1/free}$E106

$\text{Δ}{v}_{S;1/hyd}=v_{S;1/hyd}-(t_{s2}{v}_{2;1/hyd}+t_{s3}{v}_{3;1/hyd})$E107

Substituting (105) in (93), we get

$\{\begin{array}{l}\text{Δ}{v}_{2;1,3}=\left(\text{Δ}{v}_{S;1/free}-t_{s3}\frac{d\text{Δ}{v}_{S;1/free}}{d{t}_{s3}}\right)+\left(\text{Δ}{v}_{S;1/hyd}-t_{s3}\frac{d\text{Δ}{v}_{S;1/hyd}}{d{t}_{s3}}\right)\hfill \\ \text{Δ}{v}_{3;1,2}=\left(\text{Δ}{v}_{S;1/free}+(1-t_{s3})\frac{d\text{Δ}{v}_{S;1/free}}{d{t}_{s3}}\right)+\left(\text{Δ}{v}_{S;1/hyd}+(1-t_{s3})\frac{d\text{Δ}{v}_{S;1/hyd}}{d{t}_{s3}}\right)\hfill \end{array}$E108

Defining now:

$\begin{array}{c}\begin{array}{l}\text{Δ}{v}_{2;1,3/free}=\text{Δ}{v}_{S;1/free}-t_{s3}\frac{d\text{Δ}{v}_{S;1/free}}{d{t}_{s3}}\\ \text{Δ}{v}_{2;1,3/hyd}=\text{Δ}{v}_{S;1/hyd}-t_{s3}\frac{d\text{Δ}{v}_{S;1/hyd}}{d{t}_{s3}}\end{array}\\ \begin{array}{l}\text{Δ}{v}_{3;1,2/free}=\text{Δ}{v}_{S;1/free}+(1-t_{s3})\frac{d\text{Δ}{v}_{S;1/free}}{d{t}_{s3}}\\ \text{Δ}{v}_{3;1,2/hyd}=\text{Δ}{v}_{S;1/hyd}+(1-t_{s3})\frac{d\text{Δ}{v}_{S;1/hyd}}{d{t}_{s3}}\end{array}\end{array}$E109

One arrives at the following result:

$\begin{array}{l}\text{Δ}{v}_{2;1,3}=\text{Δ}{v}_{2;1,3/free}+\text{Δ}{v}_{S;1/free}\\ \text{Δ}{v}_{3;1,2}=\text{Δ}{v}_{3;1,2/free}+\text{Δ}{v}_{3;1,2/hyd}\end{array}$E110

where similar equations are obtained for the interaction partial specific compressibilities.

Considering these contributions, the interpretation of the partial specific volumes of interaction of the particle as function of the electrolyte concentration is as follows. From t_f3 = 0 to around 0.05 (see Figure 4B), there is an increment in Δv_2,1,3 which can be interpreted as a gain of free volume by the disentanglement of the polymeric chains. This increment of free volume is accompanied by an increment in the hydrodynamic radius [4]. From around t_f3 = 0.05 to around 0.1, there is a decrement in Δv_2,1,3 due to hydration. In this region of compositions, the separation of polymeric chains allows the entrance of water molecules in the polymeric particle. As a result, the hydrodynamic radius of the particle increases [4]. From around t_s3 = 0.1 to 0.15, Δv_2,1,3 increases sharply. This fact can be interpreted as an increment of the dehydration. Beyond t_s3 = 0.15, Δv_2,1,3 becomes constant, indicating that the interaction between particles and the electrolyte is saturated. Similar regions with similar interpretations are obtained for the partial specific adiabatic compressibility (see Figure 5B).

4.4.2. Partial specific enthalpies of interaction

This section deals with the determination of the partial specific enthalpies of interaction of the same system than in the latest example [4]. Partial specific enthalpy of interaction of polymeric particles is:

$\text{Δ}{h}_{2;1,3}=h_{2;1,3}-h_{2;1}$E111

and the partial specific enthalpy interaction of the electrolyte is:

$\text{Δ}{h}_{3;1,2}=h_{3;1,2}-h_{3;1}$E112

The partial specific enthalpy of interaction of the electrolyte can be measured by isothermal titration calorimetry using the combination of two experiments [6, 7]. The first experiment is locating the polymeric particles in the cell and the electrolyte in the syringe. The heat per unit of titration volume in an infinitesimal titration is:

$\frac{d{Q}^{cd}}{dv}=({\rho }^s-c_3^s)h_{1;2,3}+c_3^s{h}_{3;1,2}-h_v(c_3^s)$E113

where ρ^s is the density of the stock solution and $h_v\left(c_3^s\right)$ is the enthalpy of the stock solution per unit volume. The second experiment consists of titrating water with the above stock solution, and its heat per unit of titration volume in an infinitesimal titration is:

$\frac{d{Q}^c}{dv}=({\rho }^s-c_3^s)h_{1;3}+c_3^s{h}_{1;3}-h_v(c_3^s)$E114

The partial specific enthalpy of interaction of the electrolyte is obtained by subtracting (114) from (113), considering Eq. (112), diluted solutions and bearing in mind that $d{n}_3^2=c_3^{s}dv$:

$\frac{d{Q}^{cd}}{d{n}_3^s}-\frac{d{Q}^c}{d{n}_3^s}=\text{Δ}{h}_{3;1,2}$E115

Figure 7A shows the experimental values Δh_3;1,2. The partial specific enthalpy of interaction of polymeric particles was calculated by integrating Eq. (91) [7]:

Figure 7.

(A) Partial specific enthalpy of interaction of the electrolyte as function of the mass fraction of the electrolyte in the complex solute (t_s3). (B) Partial specific enthalpy of interaction of the polymeric particles as function of t_s3.

$\text{Δ}{h}_{2;1,3}(t_{f3})=-{\displaystyle {\displaystyle \underset{0}{\overset{t_{f3}}{\int }}}\frac{{t^{\prime }}_{f3}}{1-{t^{\prime }}_{f3}}\left(\frac{d\text{Δ}{h}_{3;1,2}}{d{t^{\prime }}_{f3}}\right)}d{t^{\prime }}_{f3}$E116

and the values of Δh_2;1,3 are shown in Figure 7B. It is very interesting to observe in Figure 7B that Δh_2;1,3 is zero from t_s3 = 0 to around t_s3 = 0.1. This fact indicates that the changes, which take place in the first two regions in Figures 4B and 5B, are entropic in origin.

5. Partial molar properties at infinite dilution

First, we will discuss the case of the two-component system and then make the extension to three-component system. In this section, J can be U, H, V or their derivatives Cv = (∂H/∂T)_V, C_p = (∂H/∂T)_P or E = (∂V/∂T)_P, K_T = (∂V/∂P)_T and K_S = (∂V/∂P)_S.

5.1. Two-component systems

In a two-component system, we only have one way to calculate limits at infinite dilution and it is to take a component as solvent (component 1) and the other as solute (component 2). For a two-component system, j takes the form:

$j(x_2)=x_1{j}_{1;2}(x_2)+x_2{j}_{2;1}(x_2)$E117

Because

${\lim }_{x_2\to 0}j(x_2)=j_1$E118

and using Eq. (117), we have:

${\lim }_{x_2\to 0}{j}_{1;2}(x_2)=j_1$E119

For the solute, we have:

${\lim }_{x_2\to 0}{j}_{2;1}(x_2)=j_{2;1}^o$E120

We can obtain experimentally the value of $j_{2;1}^o$ as follows. The Taylor’s expansion of j(x₂) around x₂ = 0 is:

$j(x_2)=j(0)+\frac{dj(0)}{d{x}_2}{x}_2+\dots$E121

Differentiating (117) with respect to x₂, considering the Gibbs-Duhem equation for a two-component system, Eq. (117) again and Eq. (120) in (121):

$j(x_2)=j_1+(j_{2;1}^o-j_1)x_2$E122

For this reason, we can obtain experimentally $j_{2;1}^o$ from a linear fit in a plot of j(x₂) as function of x₂.

5.2. Three-component systems

In three-component systems, we have two ways to calculate limits at infinite dilution. The first way is to group two components in a “complex solvent” and to calculate the limit at infinite dilution of the other component in this complex solvent (type I). The other way is considering a component as solvent, to group the other two components in a complex solute, and to calculate the limit at infinite dilution of the complex solute in the solvent (type II).

5.2.1. Limits of type I

In this case, we consider a complex solvent B composed of components 1 and 2 and a solute (component 3). For this system,

$J=J(n_3,n_B,x_{b2})$E123

where n_B = n₁+n₂ and x_b2 = n₂/(n₁+n₂). With this, j can be written as:

$j(x_3,x_{b2})=x_B{j}_{B;3}(x_3,x_{b2})+x_3{j}_{3;B}(x_3,x_{b2})$E124

where x₃ is the mole fraction of the component 3. At infinite dilution, we have:

${\lim }_{\begin{array}{l}{x}_3\to 0\\ {\text{x}}_{\text{b2}}\text{ constant}\end{array}}j(x_3,x_{b2})=j_B(x_{b2})$E125

and then combining Eq. (124) with (125), one gets for the solvent:

${\lim }_{\begin{array}{l}{x}_3\to 0\\ {\text{x}}_{\text{b2}}\text{ constant}\end{array}}{j}_{B;3}(x_3,x_{b2})=j_B(x_{b2})$E126

For the solute, it is obtained that:

${\lim }_{\begin{array}{l}{x}_3\to 0\\ {\text{x}}_{\text{b2}}\text{ constant}\end{array}}{j}_{3;B}(x_3,x_{b2})=j_{3;B}^o(x_{b2})\equiv j_{3;1,2}^o(x_{b2})$E127

where we have used Eq. (48). Similar to the case of two-component systems, the amount $j_{3;1,2}^o$ can be obtained experimentally by using the equation:

$j(x_3,x_{b2})=j_B(x_{b2})+\left(j_{3;1,2}^o-j_B(x_{b2})\right)x_3$E128

This equation is obtained by using the first-order Taylor’s expansion of j(x₃,x_b2) around x₃ = 0, the partial derivative of j(x₃, x_b2) with respect to x₃, the Gibbs-Duhem equation of the fractionalized system considering the composition of the fraction as constant and Eqs. (126) and (127).

5.2.2. Limits of type II

In this case (see Figure 1A and B), we will consider the component 1 as solvent and a “complex solute” S composed of 2 and 3 and then:

$J=J(n_1,n_S,x_{s3})$E129

where n_S = n₂+n₃ and x_s3 = n₃/(n₂ + n₃). The molar property j is:

$j(x_S,x_{s3})=x_1{j}_{1;S}(x_S,x_{s3})+x_S{j}_{S;1}(x_S,x_{s3})$E130

Similarly to the above cases, at infinite dilution we have for the solvent:

${\lim }_{\begin{array}{l}{x}_S\to 0\\ {\text{x}}_{\text{s3}}\text{ constant}\end{array}}{j}_{1;S}(x_S,x_{s3})=j_1$E131

Accordingly to case of the two-component system, one gets for the complex solute:

${\lim }_{\begin{array}{l}{x}_S\to 0\\ {\text{x}}_{\text{s3}}\text{ constant}\end{array}}{j}_{S;1}(x_S,x_{s3})=j_{S;1}^o(x_{s3})$E132

and in a similar way than for the type I limits, $j_{S;1}^o$ can be calculated as

$j(x_S,x_{s3})=j_1+\left(j_{S;1}^o(x_{s3})-j_1\right)x_S$E133

In order to study the contributions of components 2 and 3 to $j_{S;1}^o$, we define the following limits an infinite dilution:

$\{\begin{array}{l}{\lim }_{\begin{array}{l}{x}_S\to 0\\ {\text{x}}_{\text{s3}}\text{ constant}\end{array}}{j}_{2;1,3}(x_S,x_{s3})=j_{2;1,3}^{\text{Δ}}(x_{s3})\\ {\lim }_{\begin{array}{l}{x}_S\to 0\\ {\text{x}}_{\text{s3}}\text{ constant}\end{array}}{j}_{3;1,2}(x_S,x_{s3})=j_{3;1,2}^{\text{Δ}}(x_{s3})\end{array}$E134

In this way, taking limits in both sides of Eq. (49), and bearing in mind Eqs. (132) and (134), we have that:

$j_{S;1}^o(x_{s3})=x_{s2}{j}_{2;1,3}^{\text{Δ}}(x_{s3})+x_{s3}{j}_{3;1,2}^{\text{Δ}}(x_{s3})$E135

Now, we will see some mathematical properties of limits of type II. One of them is for example:

${\lim }_{x_{s3}\to 0}{j}_{2;1,3}^{\text{Δ}}(x_{s3})=j_{2;1}^o$E136

This property is demonstrated by using iterated limits:

$\begin{array}{l}{\lim }_{x_{s3}\to 0}{j}_{2;1,3}^{\text{Δ}}(x_{s3})={\lim }_{\begin{array}{l}{x}_{s3}\to 0\\ x_S\text{ constant}\end{array}}\left[{\lim }_{\begin{array}{l}{x}_S\to 0\\ x_{s3}\text{ constant}\end{array}}{j}_{2;1,3}(x_S,x_{s3})\right]\\ ={\lim }_{\begin{array}{l}{x}_S\to 0\\ x_{s3}\text{ constant}\end{array}}\left[{\lim }_{\begin{array}{l}{x}_{s3}\to 0\\ x_S\text{ constant}\end{array}}{j}_{2;1,3}(x_S,x_{s3})\right]={\lim }_{x_S\to 0}{j}_{2;1,3}(x_S,0)={\lim }_{x_2\to 0}{j}_{2;1}(x_2)=j_{2;1}^o\end{array}$E137

The other mathematical property is:

${\lim }_{x_{s3}\to 1}{j}_{3;1,2}^{\text{Δ}}(x_{s3})=j_{2;1,3}^o(0)$E138

where its demonstration is as follows:

${\lim }_{x_{s3}\to 1}{j}_{3;1,2}^{\text{Δ}}(x_{s3})={\lim }_{\begin{array}{l}{x}_{s3}\to 1\\ x_S\text{ constant}\end{array}}\left[{\lim }_{\begin{array}{l}{x}_S\to 0\\ x_{s3}\text{ constant}\end{array}}{j}_{3;1,2}(x_S,x_{s3})\right]={\lim }_{\begin{array}{l}{x}_S\to 0\\ x_{s3}\text{ constant}\end{array}}\left[{\lim }_{\begin{array}{l}{x}_{s3}\to 1\\ x_S\text{ constant}\end{array}}{j}_{3;1,2}(x_S,x_{s3})\right]$E139

Now, it is necessary to consider other way to fractionalize the system. For convenience, we will consider a complex solvent B composed of 1 and 3, and a solute 2 where the variable x_B represents the molar fraction of B and x_b3 = n₃/(n₁+n₃). With this,

$\{\begin{array}{l}{\lim }_{\begin{array}{l}{x}_{s3}\to 1\\ x_{S}cons\tan t\end{array}}{x}_2={\lim }_{\begin{array}{l}{x}_{s3}\to 1\\ x_{S}cons\tan t\end{array}}{x}_S(1-x_{s3})=0\hfill \\ {\lim }_{\begin{array}{l}{x}_{s3}\to 1\\ x_{S}cons\tan t\end{array}}{x}_{b3}={\lim }_{\begin{array}{l}{x}_{s3}\to 1\\ x_{S}cons\tan t\end{array}}{x}_{s3}{x}_S\left[1-x_S(1-x_{s3})\right]=x_S\hfill \end{array}$E140

and considering Eq. (140), (139) transforms into:

$\begin{array}{c}{\lim }_{x_{s3}\to 1}{j}_{3;1,2}^{\text{Δ}}(x_{s3})={\lim }_{\begin{array}{l}{x}_S\to 0\\ x_{s3}\text{ constant}\end{array}}\left[{\lim }_{\begin{array}{l}{x}_2\to 0\\ x_{b2}\text{ constant}\end{array}}{j}_{3;1,2}(x_2,x_{b2})\right]=\\ ={\lim }_{\begin{array}{l}{x}_S\to 0\\ x_{s3}\text{ constant}\end{array}}{j}_{3;1,2}^o(x_{b2})=j_{3;1,2}^o(0)\end{array}$E141

Other interesting property of the limits of type II is that they are related to each other by the following equation:

$x_{s2}\frac{d{j}_{2;1,3}^{\text{Δ}}}{d{x}_{s3}}+x_{s3}\frac{d{j}_{3;1,2}^{\text{Δ}}}{d{x}_{s3}}=0$E142

The demonstration of this equation is as follows. Both sides of the following equation:

$x_{s2}{\left(\frac{\partial j_{2;1,3}}{\partial x_{s3}}\right)}_{x_S}+x_{s3}{\left(\frac{\partial j_{3;1,2}}{\partial x_{s3}}\right)}_{x_S}=x_S(1-x_S){\left(\frac{\partial (j_{3;1,2}-j_{2;1,3})}{\partial x_S}\right)}_{x_{s3}}$E143

are calculated in the following way. The left-hand side is obtained by deriving partially Eq. (49) with respect to x_s3. The right-hand side of (143) is calculated considering that:

${\left(\frac{\partial j_{S;1}}{\partial x_{s3}}\right)}_{x_S}=\frac{\partial J}{\partial x_{s3}\partial n_S}=\frac{\partial J}{\partial n_S\partial x_{s3}}$E144

Using (50) in (144) and cancelling common terms, Eq. (143) is obtained. Taking the limit when x_s approaches to zero when x_s3 is kept constant in both sides of Eq. (143) and considering that:

$\begin{array}{c}{\lim }_{\begin{array}{l}{x}_S\to 0\hfill \\ x_{s3}\text{ constant}\hfill \end{array}}{\left(\frac{\partial \left(j_{3;1,2}(x_S,x_{s3})-j_{2;1,3}(x_S,x_{s3})\right)}{\partial x_S}\right)}_{x_{s3}}\\ ={\left(\frac{\partial j_{3;1,2}(0,x_{s3})}{\partial x_S}\right)}_{x_{s3}}-{\left(\frac{\partial j_{2;1,3}(0,x_{s3})}{\partial x_S}\right)}_{x_{s3}}=f(x_{s3})\end{array}$E145

Eq. (142) is obtained.

From values of $j_{S;1}^o$ it is possible to obtain $j_{2;1,3}^{\text{Δ}}$ and $j_{3;1,2}^{\text{Δ}}$ by using the following equations:

$\{\begin{array}{l}{j}_{2;1,3}^{\text{Δ}}=j_{S;1}^o-x_{s3}\frac{d{j}_{S;1}^o}{d{x}_{s3}}\hfill \\ j_{3;1,2}^{\text{Δ}}=j_{S;1}^o+(1-x_{s3})\frac{d{j}_{S;1}^o}{d{x}_{s3}}\hfill \end{array}$E146

Eq. (146) was obtained by differentiating Eq. (135) with respect to x_s3, considering Eq. (142) and combining the result with Eq. (135).

5.2.3. Application of the limits of type II to the study of polymeric particles

The polymeric particles used were synthesized with a gradient of concentration of functional groups (acrylic acid) inside the particle [9]. In this system, the content of acrylic acid represents the polar groups, while poly(butyl acrylate-co-methylmethacrylate) is the non-polar groups. As seen in Figure 1C and D, component 1 is water, component 2 is non-polar groups and component 3 is polar groups. The polymeric particle (composed of polar and non-polar groups) is taken as a fraction “P” of the system where the variable t_p3 = n₃/(n₂+n₃) will be the mass fraction of polar groups in the particle. In this study [9], the same experimental equipment than in Section 4.4.1 was used and measurements of density and sound speed were carried out by titrating water (in the cell) with latex of polymeric particles (in the syringe). Figure 7A and B shows the density ρ and u as functions of the concentration for several values of t_p3. The density and sound speed were transformed into specific volumes and specific adiabatic compressibilities by using Eqs. (95) and (96), and results are shown in Figure 1C and D.

In this case, Eq. (133) will take the form:

$j=j_1+(j_{P;1}^o-j_1)t_P$E147

and considering that t₁ = 1 – t_P, Eq. (147) transforms into:

$j=j_{P;1}^o+(j_1-j_{P;1}^o)t_1$E148

Using Eq. (148) as a fit function in Figure 8C and D, the partial specific volume at infinite dilution of the particles ($v_{P;1}^o$) and the partial specific adiabatic compressibility at infinite dilution of the particles ($k_{SP;1}^o$)were obtained from the independent term of Eq. (148) and the results are shown in Figure 9A and B as functions of t_p3. In this case, Eqs. (100) and (101) take the form:

Figure 8.

(A) Density of latex as function of polymeric particles concentration. (B) Sound speed as function of polymeric particles concentration. (C) Specific volume of latex as function of mass fraction of solvent (water). (D) Specific adiabatic compressibility as function of the mass fraction of solvent (water). In all figures (□) 0 wt%, (○) 5wt%, (∆) 10 wt%, (◊) 15 wt%, (◃) 20 wt%, (⬢) 25wt%.

Figure 9.

(A) Partial specific volume of the polymeric particles at infinite dilution as function of the polar group content. (B) Partial specific adiabatic compressibility of particles at infinite dilution as function of the polar group content. (C) Partial specific volume of non-polar groups at infinite dilution as function of the polar group content. (D) Partial specific adiabatic compressibility of non-polar groups at infinite dilution as function of the polar group content. (E) Partial specific volume of polar groups at infinite dilution as function of the polar group content. (F) Partial specific adiabatic compressibility of polar groups at infinite dilution as function of the polar group content.

$v_{P;1}^o=v_{P;1/atom}^o+v_{P;1/free}^o+v_{P;1/hyd}^o$E149

$k_{SP;1}^o=k_{SP;1/free}^o+k_{SP;1/hyd}^o$E150

The partial specific properties of polar ($j_{3;1,2}^{\text{Δ}}$) and non-polar ($j_{2;1,3}^{\text{Δ}}$) groups were calculated by using Eq. (146). The derivatives of Eq. (146) were calculated numerically by using schemes of finite differences. Figure 9C and D shows, as functions of the amount of polar groups (t_p3), the values of specific partial volumes of non-polar and polar groups, respectively. Figure 9D and F shows, respectively, the specific partial adiabatic compressibility of non-polar and polar groups.

With similar arguments than in Section 4.4.1, we can get the following equations for the volumes:

$v_{2;1,3}^{\text{Δ}}=v_{2;1,3/atom}^{\text{Δ}}+v_{2;1,3/free}^{\text{Δ}}+v_{2;1,3/hyd}^{\text{Δ}}$E151

$v_{3;1,2}^{\text{Δ}}=v_{3;1,2/atom}^{\text{Δ}}+v_{3;1,2/free}^{\text{Δ}}+v_{3;1,2/hyd}^{\text{Δ}}$E152

and for the adiabatic compressibilities:

$k_{T2;1,3}^{\text{Δ}}=k_{T2;1,3/free}^{\text{Δ}}+k_{T2;1,3/hyd}^{\text{Δ}}$E153

$k_{T3;1,2}^{\text{Δ}}=k_{T3;1,2/free}^{\text{Δ}}+k_{T3;1,2/hyd}^{\text{Δ}}$E154

In addition to this, by combining Eqs. (135), (149), (151) and (152), one gets the following equations:

$v_{P;1/atom}^o=t_{p2}{v}_{2;1,3/atom}^{\text{Δ}}+t_{p3}{v}_{3;1,2/atom}^{\text{Δ}}$E155

$v_{P;1/free}^o=t_{p2}{v}_{2;1,3/free}^{\text{Δ}}+t_{p3}{v}_{3;1,2/free}^{\text{Δ}}$E156

$v_{P;1/hyd}^o=t_{p2}{v}_{2;1,3/hyd}^{\text{Δ}}+t_{p3}{v}_{3;1,2/hyd}^{\text{Δ}}$E157

where similar equations can be obtained for the adiabatic compressibilities. Figure 9A and B shows that $v_{P;1}^o$ and $k_{sP;1}^o$ decrease when the amount of polar groups increases. This fact indicates an increment of the hydration in the interior of the particle when the amount of polar groups increases. The distribution of this hydration is as follows. Figure 9C and D shows that $v_{2;1,3}^{\text{Δ}}$ and $k_{s2;1,3}^{\text{Δ}}$ decrease from 0 to 15% of polar groups, while Figure 9E and F shows that $v_{3;1,2}^{\text{Δ}}$ and $k_{s3;1,2}^{\text{Δ}}$ increase. This fact can be interpreted in terms of that the hydration is redistributed from the polar groups to the non-polar groups. In the region of 15–25%, this behaviour is reversed.

6. Conclusions

In this chapter, we have developed common thermodynamic bases for isothermal titration calorimetry, densimetry and measurement of sound speed in terms of thermodynamic partial properties (interaction partial enthalpies, partial volumes and partial adiabatic compressibilities). To build these common thermodynamic bases, it is necessary to introduce new concepts, i.e., the concept of fraction of a system and the concept of thermodynamic interaction between components of a system. An advantage of the proposed thermodynamic scheme is the possibility of including new thermodynamic partial properties as partial heat capacities.