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.
- molar partial volumes
- molar partial adiabatic compressibilities
- molar partial enthalpies
- sound velocity
- isothermal titration calorimetry
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:
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  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:
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:
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
where θ is the colatitude (the complement of the latitude) and ϕ the longitude.
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  and Klotz and Rosenberg .
2.1. Changes of variable
The gradient of
If we consider the change of variable:
where its gradient will be:
Our interest is to relate the partial derivatives with respect to the variables
where the matrix
From (6) and using (7), the total differential of
Eq. (17) relates the vector gradient with respect to the variables (
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.
From Figure 2, it is clear that being
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 (
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
If , the function will behave as an extensive property. This concept is mathematically implemented as:
In this case,
2.3. Euler´s theorem
The demonstration is as follows. The differential with respect to λ in the left side of (24) is:
For the sets of variables
The differential of
Eq. (25) is obtained by substituting Eqs. (27), (28), (30), (31) in Eq. (26). In addition, it is interesting to see that, defining
3. Thermodynamic descriptions
3.1. Description by components
From the above section, we know that
where we have considered that
The Euler equation is obtained by considering that
If we consider that partial molar properties are function of
Calculating the differentials by considering that partial molar properties depend on
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
The Euler equation is obtained by remembering that
It is interesting to observe that considering constant composition (
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.
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.
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:
it is obtained that:
and then (57) yields:
Because the first term depends only on
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:
In addition to this
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:
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
From Eq. (74), we have for diluted solutions:
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),
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:
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
where we are considering the interaction between components 2 and 3 since
In a diluted solution without interaction between 2 and 3, the property
In this way, we can calculate the interaction contributions to
is the partial molar property of interaction of the complex solute and
In Eq. (85), Δ
Considering the Gibbs-Duhem equation for a two-component system:
in Eq. (87) and bearing in mind that solutions are diluted,
Similarly for component 3,
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:
and the partial properties of interaction of components 2 and 3 can be obtained from (92) using the equations:
Eq. (93) is obtained by differentiating in Eq. (92) with respect to
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 . 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]:
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:
The partial specific volume of interaction of the polymeric particles (Δ
which are shown in Figure 6. The atomic volume contribution (
For the complex solute, we can write a similar breakdown:
One arrives at the following result:
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
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 . Partial specific enthalpy of interaction of polymeric particles is:
and the partial specific enthalpy interaction of the electrolyte is:
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:
and the values of Δ
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,
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,
and using Eq. (117), we have:
For the solute, we have:
We can obtain experimentally the value of as follows. The Taylor’s expansion of
For this reason, we can obtain experimentally from a linear fit in a plot of
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,
For the solute, it is obtained that:
where we have used Eq. (48). Similar to the case of two-component systems, the amount can be obtained experimentally by using the equation:
This equation is obtained by using the first-order Taylor’s expansion of
5.2.2. Limits of type II
Similarly to the above cases, at infinite dilution we have for the solvent:
Accordingly to case of the two-component system, one gets for the complex solute:
and in a similar way than for the type I limits, can be calculated as
In order to study the contributions of components 2 and 3 to , we define the following limits an infinite dilution:
Now, we will see some mathematical properties of limits of type II. One of them is for example:
This property is demonstrated by using iterated limits:
The other mathematical property is:
where its demonstration is as follows:
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 xB represents the molar fraction of B and
Other interesting property of the limits of type II is that they are related to each other by the following equation:
The demonstration of this equation is as follows. Both sides of the following equation:
Eq. (142) is obtained.
From values of it is possible to obtain and by using the following equations:
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 . 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
In this case, Eq. (133) will take the form:
and considering that
Using Eq. (148) as a fit function in Figure 8C and D, the partial specific volume at infinite dilution of the particles () and the partial specific adiabatic compressibility at infinite dilution of the particles ()were obtained from the independent term of Eq. (148) and the results are shown in Figure 9A and B as functions of
The partial specific properties of polar () and non-polar () 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 (
With similar arguments than in Section 4.4.1, we can get the following equations for the volumes:
and for the adiabatic compressibilities:
where similar equations can be obtained for the adiabatic compressibilities. Figure 9A and B shows that and 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 and decrease from 0 to 15% of polar groups, while Figure 9E and F shows that and 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.
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.