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We have performed a detailed study of the thermodynamics of the titration process in an isothermal titration calorimeter with full cells. We show that the relationship between the enthalpy and the heat measured is better described in terms of the equation Δ H = W_{inj} + Q (where W_{inj} is the work necessary to carry out the titration) than in terms of ΔH = Q. Moreover, we show that the heat of interaction between two components is related to the partial enthalpy of interaction at infinite dilution of the titrant component, as well as to its partial volume of interaction at infinite dilution.

Isothermal titration calorimetry [

When the liquid of the vessel interior (see

It is commonly accepted that with a suitable procedure involving simple titration experiments [

This experiment is carried out using the same conditions as the first experiment. In this case, the contributions to the heat measured are the heat of dilution of the component 3 and the interaction with the different parts of the experimental setup. In the third experiment the solution of component 2 in the solvent is titrated with the solvent and the heat of dilution of component 2 is the contribution to the heat measured. In the fourth experiment the solvent is titrated with the solvent.

The fourth experiment takes part in the protocol because its contribution appears also in experiments one, second and three. From a practical point of view, the heats of experiments 3 and 4 are negligible since they are usually insignificant [

Because all processes in the above protocol are carried out at constant pressure, the measured heat is usually interpreted in terms of the following Equation [

It is interesting to note that the origin of the protocol shown in

In this paper, we address the above problem and the physical meaning of the heat obtained from the given protocol based on the typical performance of an isothermal titration with full cells which is described in

We observed that the heat measured in an experiment where solvent is titrated with itself has its origin in the work required for to inject the volume of titrant. For this reason it can be named as “heat of injection”. In addition, we see that the heat involved per mol of titrant when the titration is infinitesimally small is related to its partial molar enthalpy of interaction at infinite dilution and its molar partial volume of interaction also at infinite dilution. That is, using the full-cell method, the heat measured by the calorimeter when the above protocol is employed is the partial molar enthalpy of interaction only when the variation in the molar partial volume of interaction can be neglected. This fact is true in binding events where protein unfolding is involved.

The calorimeter used was an ITC 4200 from CSC equipped to work with nanowatt sensitivity. The volume cell is 1,300 μL. The working temperature was in all cases 30 °C. The water used was bidestilated and the toluene (reagent grade) was obtained from Fermont.

From a thermodynamic point of view, the process of titration shown in

Thus, for a process in which the pressure and the volume are maintained constant, the variation in the enthalpy is:

Substituting

With _{inj}, will be named “injection work”; and the heat measured by the calorimeter is then:

Note that in

In this section, we will determinate the concentrations in experiments where a solution of components 2 and 3 in a solvent (component 1) is titrated with a stock solution of 3 in the same solvent. This titration experiment can be described as the combination of two simpler experiments. The first experiment that we will address is one in which a solution of component 3 in a solvent is titrated with a more concentrated stock solution of component 3 in the same solvent. Because the concentration of component 3 will increase in each titration, this type of experiment will be named “concentration experiment”. The other experiment is one in which a solution of component 2 in a solvent is titrated only with the solvent. In this case, the concentration of component 2 will decrease with each titration; for this reason this experiment will be named “dilution experiment”. The more complex experiment, in which a solution of components 2 and 3 in a solvent is titrated with a more concentrated stock solution of 3 in the same solvent, can be considered to be the combination of two simultaneous experiments: a dilution experiment component 2 and a concentration experiment for component 3. This experiment will named the “concentration-dilution experiment”.

The concentrations of 2 and 3 in component 1 are expressed as c_{2} = n_{2}/V and c_{3} = n_{3}/V, with n_{2} and n_{3} being the numbers of moles of components 2 and 3, respectively.

Let us now consider the system in ^{(i)}_{3}; and in the syringe, is present as a stock solution with a concentration c^{s}_{3}. We will consider the infinitesimal process with respect to the titration volume in which the solution of the vessel with concentration c_{3} is titrated with a volume dv of stock solution. The different steps of this infinitesimal process are shown in

In the first state, the number of moles of component 3, n_{3}, in the volume V is:

This solution (see State 1 of ^{s}_{3}. The number of moles of component 3 contained in the volume dv is:

In State 2, the volume dv of stock solution is introduced into the vessel. Because the volume of the vessel is constant, a similar volume with concentration c_{3} is removed from the vessel by the drainage capillary. The amount of moles of 3 that is pushed out is:

In state 2 (see _{3} and another solution of volume dv with concentration c^{s}_{3}. In the State 3, the above solutions are mixed, and the new concentration inside the vessel is c_{3} + dc_{3}, with (c_{3} + dc_{3})V being the final number of moles of component 3 in the vessel. Balancing the number of moles for the titration process, we have:
_{3} moles of component 3, dn^{s}_{3} moles were introduced into the vessel and dn_{3} moles were removed. Substituting the

_{3} = c_{3}(v), with the initial condition:
_{3} = c_{3}(v) can be written as:

In this experiment, we will consider that a solution of component 2 in component 1 is located in the vessel and that this solution is titrated with an amount of component 1. Assuming that there are similar states in this process as those presented in ^{s}_{3} = 0 because the syringe holds only component 1, we then obtain the equation:

We consider the case in which the vessel contains a solution of components 2 and 3 in component 1 which is titrated with a solution of component 3 in component 1. The initial concentrations of 2 and 3 are c^{(i)}_{2} and c^{(i)}_{3} respectively. This experiment can be considered as the sum of two experiments: the dilution the component 2 and the concentration of component 3. In the first, the concentration of 2 after titration is given by _{F} and t_{f3} as:

Upon substituting (16) and (17) into _{F}^{(i)} = c_{2}^{(i)} + c_{3}^{(i)}.

In this section, we will determinate the heats that are involved in the different titration experiments: the concentration experiment, the dilution experiment and the concentration-dilution experiment. The heat of stirring (homogenization) is the same in all cases (all States). Then it cancels into the thermomechanical balance.

In State 1 of _{3} in the interior of the vessel; before the titration, a volume dv of solution stock with concentration c^{s}_{3} is present at the end of the syringe needle. The enthalpy of the state 1, H_{1}, is:

In State 2 of ^{s}_{3} and a volume V-dv of solution with concentration c_{3}; outside the vessel, in the drainage capillary, we have a volume dv of concentration c_{3}. The enthalpy of state 2, H_{2}, is:

In State 3 of _{3} + dc_{3} and the drainage capillary has a volume dv of concentration c_{3}. The enthalpy of the state 3, H_{3}, is:

^{c}_{1–2}, for the process 1–2 between states 1 and 2 is defined as:
^{c}_{2–3}, for the process 2–3 between states 2 and 3 is:

The variation in enthalpy, dH^{c}, for the entire process of titration between states 1 and 3 is:

Applying the First Principle of Thermodynamics (

The value of dH^{c}_{1–2} can be calculated by substituting the values of H_{1} and H_{2} (^{c}_{1–2} (

Considering that H(c_{3},V) = h_{v}(c_{3})V (

By substituting (30) into (29), we obtain the value of dH^{c}_{1–2}:

The applying in that case the First Principle of Thermodynamics [

That is, the heat involved in the process 1–2, dQ^{c}_{1–2}, comes from the work applied in order to introduce a volume dv of stock solution into the interior of the vessel while an equal volume dv of solution with concentration c_{3} is pushed out from the vessel.

Applying the First Principle of Thermodynamics [

In process 2–3, only a homogenizing process occurs in the vessel; thus, the work of injection is zero and:

This process of homogenizing involves the interaction between components 2 and 3. It is possible to calculate dH^{c}_{2–3} by introducing the values of H_{2} and H_{3} (^{c}_{2–3} [

Again, by virtue of H(c_{3},V) = h_{v}(c_{3})V (

The heat involved in the process 2–3, dQ^{c}_{2–3}, is calculated by using (35) and (36) in (34):

Now, we can apply the First Principle of Thermodynamics [^{c} represents the heat measured by the isothermal titration calorimeter in the experiment of concentration. From ^{c}_{1–2} and dH^{c}_{2–3} calculated with respectively

Combining

Note that according to (41), dW^{c}_{1–2} is the work of injection in the process of concentration; because dQ^{c}_{1–2} = -dW^{c}_{1–2} (^{c}_{1–2} can be considered the “injection heat”. We name this heat dQ^{c}_{inj}; then (42) can take the following form:

Now, it is possible to obtain the heat involved in the infinitesimal process of concentration, dQ^{c}, inserting the value of dQ^{c}_{2–3} (

In this experiment we will consider similar states as those in the concentration process; because it is a dilution experiment, however, the change in composition from c_{3} to c_{3} + dc_{3} is produced by a titration with the solvent located in the syringe. The states in the titration process are:

The variation in enthalpy for the total process of titration is:

As in the concentration experiment presented in

The First Principle of Thermodynamics [

The value of dH^{d}_{1–2} is obtained by substituting the values of H^{d}_{1} and H^{d}_{2} (^{d}_{1–2} (_{2},V)=h_{v}(c_{2})V (

With this result, according to the First Principle of Thermodynamics [

For process 2–3, in which only a homogenizing process occurs, the work is zero and the First Principle of Thermodynamics [

From this equation it is possible to calculate the value of dQ^{d}_{2–3} by substituting the values H^{d}_{2} and H^{d}_{3} (^{d}_{2–3} [_{2},V)=h_{v}(c_{2})V:
_{1} and ρ_{1} are the enthalpy and the density, respectively, of component 1 in the pure state. Now, the First Principle of Thermodynamics (^{d}_{inj} is the work employed in the process of titration and dQ^{d} is the heat measured by the isothermal calorimeter in the experiment of dilution. ^{d} as the sum of the two contributions dH^{d}_{1–2} and dH^{d}_{2–3}. With the First Principle of Thermodynamics applied to the process 1–2 [

Putting (57) and (58) equal and reorganizing yields:
^{d}_{inj} = dQ^{d}_{1–2}= -dWd_{1–2}. Then substituting the value of dQd_{2–3} expressed by

In this experiment, a solution of component 2 in a solvent (component 1) is titrated with a stock solution of component 3 in the same solvent. For State 1 as in _{2} = n_{2}/V and c_{3} = n_{3}/V, respectively. We consider that the volume dv, before it is introduced into the vessel, has a concentration c^{s}_{3}. For convenience, we consider the 3-component system as fractionalized, being composed of component 1 and a fraction F containing components 2 and 3. The composition of the fraction F will be expressed as a function of the variables c_{F} and x_{f3}, as defined by _{1} of State 1 is:

In State 2, while a volume dv of stock solution with a concentration c^{s}_{3} is titrated, an equal volume dv of solution with the composition c_{F} and x_{f3}, is pushed out from the vessel. The enthalpy of this state is:

After homogenization, we have a volume V with composition c_{F} + dc_{F} and x_{f3} + dx_{f3} and a volume dv in the drainage capillary with the composition c_{F} and x_{f3}. In this way, the enthalpy of State 3 is:

Applying the First Principle of Thermodynamics (

Considering the processes 1–2 and 2–3 as in the above experiments, we arrive at the following equations:

Next, we will discuss the protocol for measuring the heat of interaction between two components in solution in the high dilution region (see “

The first experiment is the titration of a solution of component 2 with a stock solution of component 3. Initially, the concentration of component 2 in the vessel is c_{2}, and the concentration of component 3 in the stock solution is c^{s}_{3}, with dv being the volume of titration. The solvent in the two solutions is the same. The heat measured in this experiment is named dQ^{(3)} where the superindex (3) indicates that a 3-component system is considered. The second experiment is a concentration experiment, in which the solvent is titrated with a volume dv of a stock solution of component 3. As in the first experiment the titrated volume of the stock solution of concentration c^{s}_{3} is dv. In this case, the heat measured is dQ^{(2)c} where the superindex (2) indicates that a 2-component system is considered. The third experiment is a dilution experiment, in which a solution of component 2 is titrated with the solvent. Initially, the concentration of component 2 in the solvent is c_{2}. The heat measured in this case is dQ(2)d. The fourth experiment is the tritration of the solvent with itself. In this experiment the heat measured is dQ_{inj}(1) where the superindex (1) indicates that a 1-component system is considered in this experiment. We will define the following amounts:
^{(3)}_{inj}, dQ(2)c_{inj}, dQ(2)d_{inj} are the heats of titration in the three firsts experiments. We suppose that the heats of injection can be estimated by the titration of component 1 with itself (fourth experiment), dQ^{(1)}_{inj}:

The heat, dq_{3;1,2}, measured from the protocol with component 3 as the titrant is defined as:

The notation “dq_{3;1,2}” means that a solution of components 1 and 2 is titrated with a stock solution of component 3. By substituting the values of dQ^{(3)} (^{(2)c} (^{(2)d} [

Combining

For convenience, we define f_{v} as:
_{2} and c_{3} can be written as functions of c_{F} and x_{f3} as c_{2} = (1-x_{f3}) c_{F} and c_{2} = x_{f3} c_{F}.

We are interested in the following amount:

Substituting (72) and (73) into (74) yields:

Now, we assume that the solutions in the cell are diluted solutions. In general, a molar property depends on x_{F} (amount of fraction F) and x_{f3} (composition of F). In previous works [_{F} close to zero. The region of concentrations for which this approximation holds is a high dilution region. Function f_{v} in _{v}(c_{F},x_{f3}), h_{v}(c_{3}), h_{v}(c_{2}) and h_{1}ρ_{1}. From _{v} is a “volumetric enthalpy” since h_{v} = H/V, H being the total enthalpy of the system and V the total volume of the system. consequently, h_{v} is expressed in “units of enthalpy per unit of volume”. Furthermore, h_{v} = h/v, where h is the molar enthalpy and v the molar volume, we can thus consider dilute solutions in f_{v} by using the first order Taylor’s expansions of molar volumes and molar enthalpies. The details of our calculations are presented in “_{v}(x_{F},x_{f3}), h_{v}(c_{2}) and h_{v}(c_{3}) for their dilute solutions [

As indicated in “

The second is the contribution from the interactions between components of the fraction:

Using (77) and (78),

Therefore, if the solutions are sufficiently diluted, the function f_{v} shows the contribution of the interaction enthalpy and the interaction volume of the fraction F. The differential can be expressed as:

By combining the equation for df_{v}/dv [_{v} (_{F} = c_{F}(v) and t_{f3} = t_{f3}(v) given in (20) and (21), we obtain:

In the definition of dq_{3;1,2}/dv [_{F}/dv and dt_{f3}/dv in

Now, (81) takes the form:

Since dn^{s}_{3}=cs_{3} dv, then:

By combining

As it has been stated previously, a measured heat is obtained experimentally when a liquid is titrated with itself. _{i} of the titrant volume would be less than the final temperature T_{f}. According to equation:
_{inj} is the heat obtained from the injection, m the mass of the titrant volume, c_{p} the specific heat capacity and ΔT = T_{f} - T_{i} we would expect a heat positive. The heat shown on _{inj} + Q [_{inj} = -W_{inj}, the heat measured must to be negative. The heat shown in

Contributions to Q_{inj} can be several as for example the friction between liquids (relative viscosities) and the friction between the liquid and the narrow bore tube of the needle. Recently [^{3} min^{−1}, ρ is the fluid density in g cm^{3}, C is the fluid heat capacity in J g^{−1} K^{−1}, and d is the tube diameter in cm. For water flowing through a 0.4 mm diameter tube 30 cm long at 1 cm^{3} min^{−1}, ΔT = 0.002 K. As it is stated by _{inj}. This fact was shown experimentally in the Figure 2.7 of ref. [

Therefore we can state that a characteristic of isothermal titration calorimetry is the necessity of very small volume according to two considerations: first, with a large volume, the temperature of the experiment is not kept constant, second, the validity of

In ^{Δ}_{3;1,2}). The second contribution is – ρ_{1}h_{1}Δv^{Δ}_{3;1,2}. This term represents the enthalpy of a volume of solvent Δv^{Δ}_{3;1,2} as a consequence of the protocol employed. In addition to this, it is possible to demonstrate that when the interactions between two components are maximum, the heat dq_{3;1,2}/dns_{3} obtained is zero. In a previous work [^{o}_{F;1} as function of a variable of composition is linear for a range of compositions of F, then the interactions between the components of the fraction are maximum in that range. The composition variable employed was the mass fraction of component 3 in the fraction (t_{f3}). _{f3}, the behavior is non-linear. Considering that the value tc_{f3} in units of molar fractions is x^{c}_{f3}, at this composition the partial property of F takes the value:
_{f2}= 1-x_{f3}. Above the value xc_{f3}, j^{o}_{F;1} can be written as:
_{f2}= 1-X_{f3}. When we write jΔ_{2;1,3} and j^{Δ}_{3;1,2}, we assume [^{o}_{F;1} = x_{f2} j^{Δ}_{2;1,3} + x_{f3} j^{Δ}_{3;1,2} indicates that, F is composed of components 2 and 3, which are interacting in a medium (component 1). On the other hand, j^{o}_{2;1} and j^{o}_{3;1} indicate that component 2 is alone in component 1 and that component 3 is alone in component 1. Therefore, if we write j^{o}_{F;1} = x_{f2} j^{o}_{2;1} + x_{f3} j^{o}_{3;1} we assume that fraction F is composed of components 2 and 3, which are not interacting.

This is the case for ^{o}_{F;1}(xc_{f3})) and an amount of component 3 (with partial property j^{o}_{3;1}) and these components are not interacting. In other words [

By substituting the equation for j^{o}_{F;1} in the region of saturation (^{o}_{3;1,2} from j^{o}_{F;1} (

Substituting this result into

Another interesting problem in isothermal titration calorimetry is the following: is there a relationship between the experiments carried out when component 3 is the titrant and when component 2 is the titrant? We can answer this question as follows: the heat generated when component 3 is the titrant can be obtained from (86), dq_{3;1,2}/dns_{3}. In the same way, the heat obtained when component 2 is the titrant can be written as:

Next we can derivate dq_{3;1,2}/dns_{3} in _{f3} and multiply by x_{f2}, and we can also derivate dq_{2;1,3}/dns_{2} with respect to x_{f3} and multiply by x_{f3}. By adding the results and using

This is an equation of the Gibbs-Duhem type that relates the heats of interaction obtained when components 2 and 3 are the titrant components.

From equation ΔH = Q it is commonly assumed the heat measured by an ITC can be related to the variation of enthalpy; many papers and books in biochemistry and biophysics have reported results on this link. In this work, we have demonstrated that the equation ΔH = Q does not hold for isothermal titration calorimetry and that the true equation is ΔH = W_{inj} + Q, which involves a term of work. In addition, we have found that the heat obtained from the usual protocol employed in the determination of the heat of interaction dq_{3;1,2}/dns_{3} between two components (^{Δ}_{3;1,2} is not zero. As example of this, ^{Δ}_{F;1} can be neglected in the process of binding deciltrimethylammonium bromide to lysozyme [^{Δ}_{F;1} ≈ 0 in

Considering that _{3;1,2}/dns_{3} (

Another possibility is that for processes of biophysical interest, the approximation |ρ_{1}h_{1}Δv^{Δ}_{3;1,2}|≪ |Δh^{Δ}_{3;1,2}| holds.

In this work we have studied in detail the thermodynamics of the titration process in isothermal titration calorimeters with full cells. We have shown that the equation ΔH = Q does not hold for this type of calorimeters because it cannot explain the heat obtained when a liquid is titrated with itself. In its place, we propose the equation ΔH = W_{inj} + Q. The heat of interaction between two components is usually determined from a protocol composed of a number of simple titration experiments. Using the equation ΔH = W_{inj} + Q and the thermodynamic tools developed in our previous works for multicomponent systems at infinite dilution, we show that in an infinitesimal titration, the heat of interaction per mole of titrant component is related to the partial enthalpy of interaction at infinite dilution and to the partial volume of interaction of the titrant component also at infinite dilution. This information can be essential in order to link theoretical models to experimental measurements. Another interesting conclusion is that for this type of calorimeters the variation in enthalpy equals the variation in internal energy.

The authors are grateful to D. Hansen (Brigham Young University) for pointing out his paper “The art of calorimetry” (ref. [

An extensive thermodynamic property J at constant temperature and pressure can be written in a “description by components” as:
_{1}, n_{2} and n_{3} are the number of moles of the components 1, 2 and 3, respectively. The Gibbs Equation [_{1;2,3}, j_{2;1,3} and j_{3;1,2} are the partial properties of components 1, 2 and 3, respectively, defined as:
_{2} and x_{3} are the molar fraction of components 2 and 3, respectively. By notation, we understand that j_{1;2,3} means “the partial property of component 1 in the presence of components 2 and 3”. The notations j_{2;1,3} and j_{3;1,2} are interpreted in the same way.

A fraction of a system [_{F} = n_{2} + n_{3}, and x_{f3} = n_{3}/(n_{2} + n_{3}), which are related to the composition of F. The Gibbs equation for _{1;F} and j_{F;1} are respectively:

Again, by notation j_{1;F} means “the partial property of component 1 in the presence of fraction F” and in the same way, j_{F;1} means “the partial property of the fraction F in the presence of component 1”. By the technique of change of variable [_{1;F} and j_{F;1} as function of the partial properties j_{1;2,3}, j_{2;1,3}, j_{3;1,2}. The change of variable is:

By calculating the differentials of n_{1}, n_{2} and n_{3} in (107), substituting dn_{1}, dn_{2} and dn_{3} in (102), equaling the result to (105) and regrouping similar terms keeping in mind that n_{1}, n_{2} and n_{3} are independent variables, we have:

The limit of j_{F;1} at infinite dilution is defined as:

The limit in (111) is taken when the concentration of the fraction tends to zero while its composition is kept constant. Under these conditions, the limits at infinite dilution of j_{2;1,3} and j_{3;1,2} are defined as:

Taking the limit at infinite dilution on both sides of (109) and substituting

In our previous work we showed that:

Derivating in (114) with respect to x_{f3} and combining the result with

The partial properties of 2 and 3 contribute due to their interaction. This effect can be measured as the effect on the partial property of a component due to the presence of the other component. In this way, we define the terms of interaction as:

Reorganizing (118) and (119) and substituting the values of j^{Δ}_{2;1,3} and j^{Δ}_{3;1,2} in (114) we have:

Thus, from ^{∅︀}_{F;1}, which does not consider the interaction between components 2 and 3, and Δj^{o}_{F;1}, which contains all contributions from the interaction between 2 and 3.

It is possible to see that terms of interaction also hold in a Gibbs-Duhem type equation. Reorganizing in (118) and (119) and substituting the values of j^{Δ}_{2;1,3} and j^{Δ}_{3;12} in the Gibbs-Duhem type equation for the partial properties (

As in the case of the partial properties, derivating (121) with respect to x_{f3} and combining with

Let a 3-component be. The Euler equation of the system in the description of fractions [

Dividing both sides of (126) by the total mass of the systems and defining the intensive thermodynamic property j associate to the extensive thermodynamic property J as:
_{F} = n_{F}/(n_{1} + n_{F}), and x_{1} = 1 - x_{F}. The Taylor’s expansion of first order of j = j(x_{F},x_{f3}) with x_{F} close to zero is:

Using _{1} is the molar property of component 1 in the pure state. Using (128), we obtain:

In our previous paper [

Taking the limit of x_{F} tending to zero in (131) and including (130) and (132) we obtain:

The substitution of (129) and (133) in (129) yields:

The effect of work in the high dilution region of j with respect to the variable x_{F} is to replace the partial properties as follows:

In the more simple case of a 2-component system, it is easy to see that

In the high dilution region of j, the partial properties are replaced as:

In 3-component systems the enthalpy H is written as:
_{1;2,3}, h_{2;1,3} and h_{3;1,2} being the partial molar properties of components 1, 2 and 3, respectively, defined as:

Using the new variables c_{F}, x_{f3} and V, the enthalpy takes the form:

With the application of the Euler equation we have:

Setting (140) and (143) equal to each other and, considering the following relationship between the variables n_{1}, n_{2} and n_{3} and c_{F}, x_{f3} and V:

If instead we consider the system as to be composed of component 1 and the fraction F (composed of the component 2 and 3), then (146) takes the form:

Now we obtain an

The density can be written in terms of the molar volume:

Using

Substituting (150) in (149) and considering that c_{F} = x_{F} ρ:

The equation for h_{v} in the high dilution region can be obtained by substituting

In the more simple case of a 2-component system the enthalpy can be written as:
_{v} can be written as:

In the high dilution region

Typical performance of an isothermal titration calorimeter. The electronic details of the measurement of the calorimetric signal have been omitted for clarity.

Titration of water with water at 30 °C. Graph I shows the calorimetric signal as function of the time and graph II shows the heat involved in each titration. This heat is calculated by the integral of the calorimetric signal between the initial and final times for each peak. The volume titrated for each peak is 20 μL and the volume cell is 1,300 μL.

Different states to be considered during the titration process for an experiment of concentration of component 3. The first state (State 1) is a volume V (vessel volume) of solution with concentration c_{3}. The concentration of component 3 in the syringe is c^{s}_{3}. This state also includes a volume dv of stock solution with a concentration c^{s}_{3} at the end of the needle before the titration. In the second state (State 2), the volume dv of stock solution is introduced into the volume of the vessel while a volume dv with concentration c_{3} exits from the vessel volume by the drainage capillary. In the third state (State 3), the composition of the vessel interior is homogenized until it achieves the new concentration c_{3} + dc_{3}; the drainage capillary includes a volume dv of solution with concentration c_{3}.

Variation in enthalpy between the different states of a differential concentration experiment of titration.

Calorimetric signal of the titration of toluene with toluene at 30 °C. The volume of titration was 200 μL.

Specific partial volume at infinite dilution a) and specific partial adiabatic compressibility coefficient b) at infinite dilution in water at 30 °C, of a fraction F composed of non-charged polymeric particles (component 2) and decyltrimethyl-ammnonium bromide (component 3) as function of the mass fraction of component 3 in the fraction F. The solid line represents the region in which the interactions are saturated (data taken from ref. [

Partial volumes at infinite dilution of non-charged polymeric particles, v^{Δ}_{2;1,3}, and a cationic surfactant (C_{10}-TAB), vΔ_{3;1,2}, as function t_{f3} (data taken from ref. [

Specific partial volume at infinite dilution in water at 30 °C, of the fraction F composed of Lysozyme (component 2) and decyltrimethylammnonium bromide (component 3) as function of the mass fraction of component 3 in fraction F. Because the behavior of v^{o}_{F;1} is very close to linear, the interaction term Δv^{o}_{F;1} can be neglected (data taken from ref. [

Isothermal titration calorimeters that are currently manufactured and the method employed by each (full cell or half-full cell).

iTC_{200} (Microcal Inc.) |
Full Cell |

AUTO iTC_{200} (Microcal Inc.) |
Full Cell |

VP-ITC (Microcal Inc.) | Full Cell |

Nano ITC 2G (TA Instruments) | Both, but the full cell method is most often used and is the strongly recommended method |

TAM 2277 (TA Instruments) | Both, but the half-full cell method is most often used and is the strongly recommended method |

TAM III ITC (TA Instruments) | Both, but the half-full cell method is most often used and is the strongly recommended method |

Technical information supplied by MicroCal Inc.

Technical information supplied by TA Instruments.