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:
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:
where nF is the total number of moles of the fraction F and xf3 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 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 X1 and X2, we have the problem of the lack of differentiability in the points of the sphere’s equator. Another possible parametrization is:
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  and Klotz and Rosenberg .
2.1. Changes of variable
Let f be the function defined as:
The gradient of f with respect to the variables x1, x2 and x3 is the vector:
If we consider the change of variable:
the function f will take the form:
where its gradient will be:
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:
Using dx1 given by Eq. (11) and similarly with equations for dx2 and dx3, we can write:
where the matrix T is:
From (6) and using (7), the total differential of f can be expressed as:
Remembering that x being a vector and A a matrix, then (xT A)T = AT x, and taking the transpose in both sides of (16):
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.
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:
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:
and x2 transforms as:
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:
If , the function will behave as an extensive property. This concept is mathematically implemented as:
In this case, f is a homogeneous function of one degree. If , f will behave as an intensive property. Mathematically, f is expressed as a homogeneous function of zero degree as:
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,…:
The demonstration is as follows. The differential with respect to λ in the left side of (24) is:
For the sets of variables x1, x2,… and y1,y2,…, we obtain respectively that:
Considering that f is a homogeneous function of one degree with respect to the variables y1,y2,… and making Δ’= Δ/λ in (29),
The differential of f with respect to λ in the right side of (24) is:
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,…:
3. Thermodynamic descriptions
3.1. Description by components
where n1, n2 and n3 are the number of moles of components 1, 2 and 3. The partial property of 1 is defined as:
From the above section, we know that j1;2,3 is homogeneous function of zero degree with respect to n1, n2 and n3. With this and considering λ = 1/(n1+n2+n3),
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 j1;2,3.
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:
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,
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:
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:
The variable nF is the total number of moles of the fraction F, and xf3 is a variable related to its internal composition. The partial molar properties of J in this description are:
Because J is a homogeneous function of n1 and nF, the partial properties j1;F and jF;1 will be homogeneous functions of zero degree with respect to the variables n1 and nF. In this way and similarly to Eq. (35):
The Euler equation is obtained by remembering that J is a homogeneous function of degree one of n1 and nF and using the Euler’s theorem:
The Gibbs-Duhem equation in the representation of variables xF and xf3 is obtained by differentiating in (52), equalling to (51) and cancelling common terms, and dividing by the total number of moles:
Considering that j1;F and jF;1 are functions of the independent variables xF and xf3, then (53) will take the form:
It is interesting to observe that considering constant composition (dxf3 = 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.
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:
Because j1;2,3 depends only on x1:
and then (57) yields:
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:
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 g1;2,3 (free energy of Gibbs),
where k is a constant. For the entropy, one gets:
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:
where j is the molar property of the extensive thermodynamic property J. In addition, the partial derivatives must vanish:
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 j1;2,3 is:
where ∇j1;2,3(0,0) and Hj1;2,3(0,0) are, respectively, the vector gradient and the Hessian of j1;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:
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 j of a diluted solution can be written as:
where we are considering the interaction between components 2 and 3 since
In a diluted solution without interaction between 2 and 3, the property jØ can be written as:
In this way, we can calculate the interaction contributions to j as:
is the partial molar property of interaction of the complex solute and
In Eq. (85), Δj2;1,3 and Δj3;1,2 are evaluated when using concentrations xS and xs3. Accordingly, j2,1 is evaluated using the concentration x2 given by x2 = xS (1−xs3), and then,
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 xs3, 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 . 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]:
where V is the effective volume of the titration cell, v is the titration volume and 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:
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:
where t1 and tS 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 tf3 (mass fraction of the electrolyte in the complex solute). The term of interaction ΔvS;1 is calculated by Eq. (92), where v2;1 is obtained by considering that:
The partial specific volume of interaction of the polymeric particles (Δv2;1,3) and the partial specific volume of interaction of the electrolyte (Δv3;1,2) were obtained using Eq. (93). The numerical method employed to calculate the derivatives is shown elsewhere . Figure 4 shows the values of ΔvS;1, Δv2;1,3 and Δv3;1,2, and Figure 5 shows the values of Δks S;1, Δks 2;1,3 and Δks 3;1,2 obtained in a similar way than for volumes.
which are shown in Figure 6. The atomic volume contribution (v2;1/atom) is the sum of all volumes of the atoms, which make up polymeric chains. The free volume contribution (v2;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 (v2;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]:
The contribution kT 2;1/free is positive, and the contribution kT 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, v3;1 and kT 3;1 will take the following form:
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 tf3 = 0 to around 0.05 (see Figure 4B), there is an increment in Δv2,1,3 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 . From around tf3 = 0.05 to around 0.1, there is a decrement in Δv2,1,3 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 . From around ts3 = 0.1 to 0.15, Δv2,1,3 increases sharply. This fact can be interpreted as an increment of the dehydration. Beyond ts3 = 0.15, Δv2,1,3 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 . 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:
where ρs is the density of the stock solution and 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:
and the values of Δh2;1,3 are shown in Figure 7B. It is very interesting to observe in Figure 7B that Δh2;1,3 is zero from ts3 = 0 to around ts3 = 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, Cp = (∂H/∂T)P or E = (∂V/∂T)P, KT = (∂V/∂P)T and KS = (∂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:
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 j(x2) around x2 = 0 is:
For this reason, we can obtain experimentally 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,
where nB = n1+n2 and xb2 = n2/(n1+n2). With this, j can be written as:
where x3 is the mole fraction of the component 3. At infinite dilution, we have:
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 j(x3,xb2) around x3 = 0, the partial derivative of j(x3, xb2) 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
where nS = n2+n3 and xs3 = n3/(n2 + n3). The molar property j is:
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 xb3 = n3/(n1+n3). With this,
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 tp3 = n3/(n2+n3) will be the mass fraction of polar groups in the particle. In this study , 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 tp3. 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:
and considering that t1 = 1 – tP, Eq. (147) transforms into:
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 tp3. In this case, Eqs. (100) and (101) take the form:
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 (tp3), 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:
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.