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

^{1}

^{2}

^{*}

## Abstract

**:**

_{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.

## 1. Introduction

## 2. Experimental

## 3. Thermodynamics

#### 3.1. Application of the First Principle of Thermodynamics to the titration process with constant P and V

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

#### 3.2. Determination of the concentrations in the process of titration

_{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.

#### 3.2.1. Concentration experiment in 2-component systems

^{(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 Figure 3.

_{3}, in the volume V is:

^{s}

_{3}. The number of moles of component 3 contained in the volume dv is:

_{3}is removed from the vessel by the drainage capillary. The amount of moles of 3 that is pushed out is:

_{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 Equations (10)–(12) into (13) and reorganizing yields:

_{3}= c

_{3}(v), with the initial condition:

_{3}= c

_{3}(v) can be written as:

#### 3.2.2. Dilution experiment in 2-component systems

^{s}

_{3}= 0 because the syringe holds only component 1, we then obtain the equation:

#### 3.2.3. Concentration-dilution experiment in 3-component systems

^{(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 Equation (17). In the second, the concentration of 3 after the titration is given by Equation (16). For convenience we define the variables c

_{F}and t

_{f3}as:

_{F}

^{(i)}= c

_{2}

^{(i)}+ c

_{3}

^{(i)}.

#### 3.3. Determination of heats involved in the titration processes

#### 3.3.1. Concentration experiment in 2-component systems

_{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:

^{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:

_{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:

^{c}, for the entire process of titration between states 1 and 3 is:

^{c}

_{1–2}can be calculated by substituting the values of H

_{1}and H

_{2}(Equations (22) and (23)) for the definition of dH

^{c}

_{1–2}(Equation (25)):

_{3},V) = h

_{v}(c

_{3})V (Equation (153) in “Appendix 4: Basic equations”) one has:

^{c}

_{1–2}:

^{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.

^{c}

_{2–3}by introducing the values of H

_{2}and H

_{3}(Equations (23) and (24)) into the definition of dH

^{c}

_{2–3}[Equation (26)]:

_{3},V) = h

_{v}(c

_{3})V (Equation (153) in “Appendix 4: Basic equations”):

^{c}

_{2–3}, is calculated by using (35) and (36) in (34):

^{c}represents the heat measured by the isothermal titration calorimeter in the experiment of concentration. From Figure 3 and the values of dH

^{c}

_{1–2}and dH

^{c}

_{2–3}calculated with respectively Equations (28) and (34), we obtain:

^{c}

_{1–2}is the work of injection in the process of concentration; because dQ

^{c}

_{1–2}= -dW

^{c}

_{1–2}(Equation (32)), dQ

^{c}

_{1–2}can be considered the “injection heat”. We name this heat dQ

^{c}

_{inj}; then (42) can take the following form:

^{c}, inserting the value of dQ

^{c}

_{2–3}(Equation (37) ) into (43):

#### 3.3.2. Dilution experiment in 2-component systems

_{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:

^{d}

_{1–2}is obtained by substituting the values of H

^{d}

_{1}and H

^{d}

_{2}(Equation (45)) into the definition of dH

^{d}

_{1–2}(Equation (49)) and considering the property H(c

_{2},V)=h

_{v}(c

_{2})V (Equation (153) in “Appendix 4: Basic equations”):

^{d}

_{2–3}by substituting the values H

^{d}

_{2}and H

^{d}

_{3}(Equation (45)) into the definition of dH

^{d}

_{2–3}[Equation (48)] and considering the property H(c

_{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 (Equation (9)) for the complete titration process of dilution gives:

^{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. Equation (49) expresses dH

^{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 [Equation (50)] and to the process 2–3 [Equation (53)], we have:

^{d}

_{inj}= dQ

^{d}

_{1–2}= -dWd

_{1–2}. Then substituting the value of dQd

_{2–3}expressed by Equation (54) into (60) we obtain:

#### 3.3.3. Concentration-dilution experiment in 3-component systems

_{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 Equations (18) and (19). Thus, the enthalpy H

_{1}of State 1 is:

^{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:

_{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:

#### 3.4. Heats of interaction between 2 components in the high dilution region

_{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}:

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

_{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)}(Equation (67)), dQ

^{(2)c}(Equation (44)) and dQ

^{(2)d}[Equation (61)], we arrive at:

_{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}.

_{F}(amount of fraction F) and x

_{f3}(composition of F). In previous works [4–6] we have shown that a solution is diluted when its molar properties can be approximated by first order Taylor’s expansions for x

_{F}close to zero. The region of concentrations for which this approximation holds is a high dilution region. Function f

_{v}in Equation (73) is expressed in terms of h

_{v}(c

_{F},x

_{f3}), h

_{v}(c

_{3}), h

_{v}(c

_{2}) and h

_{1}ρ

_{1}. From Equations (143) or (153) (in “Appendix 4: Basic equations”), h

_{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 “Appendix 4: Basic equations”. By substituting the expressions of h

_{v}(x

_{F},x

_{f3}), h

_{v}(c

_{2}) and h

_{v}(c

_{3}) for their dilute solutions [Equations (152) and (155)] in (73), we obtain that:

_{v}shows the contribution of the interaction enthalpy and the interaction volume of the fraction F. The differential can be expressed as:

_{v}/dv [Equation (80)], f

_{v}(Equation (79)) and those for c

_{F}= c

_{F}(v) and t

_{f3}= t

_{f3}(v) given in (20) and (21), we obtain:

_{3;1,2}/dv [Equation (74)] the volume of tritration is considered to be infinitesimally small. Thus, in the calculations for dc

_{F}/dv and dt

_{f3}/dv in Equation (81), we assume that exp(-v/V) ≈ 1. From the Equation (125) (see “Appendix 2: Limits at infinite dilution in multicomponent systems”) it is possible to write:

^{s}

_{3}=cs

_{3}dv, then:

## 4. Discussion

_{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 Figure 2 is negative and therefore it is not possible to explain the heat observed on Figure 2 in terms of a “blank machine” or an “instrumental heat”. The merit of the equation ΔH = W

_{inj}+ Q [Equation (9)] is that it allows to take into account a heat measured by the calorimeter when a liquid is titrated with itself and the sign of this heat. Because it is necessary to apply work to the system in order to introduce an amount of liquid into the cell and push out an equivalent amount of liquid, this work must be positive. Since in this case Q

_{inj}= -W

_{inj}, the heat measured must to be negative. The heat shown in Figures 2-II agrees with this prediction.

_{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 [8] the following equation has been proposed that gives the temperature rise in a fluid from frictional flow in a tube:

^{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 Equation (88) ΔT depends on the nature of the fluid through its viscosity, density and heat capacity, on the geometry of the calorimetric system through the diameter and length of the needle and to the conditions of the experiment through the volume flow rate. When combining Equations (87) and (88) it results an expected influence of the volumetric flow rate (v’) in Q

_{inj}. This fact was shown experimentally in the Figure 2.7 of ref. [2].

^{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 [5], we demonstrated that if the plot of j

^{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}). Figure 6 shows an example when fraction F is composed of non-charged polymeric particles (component 2) and a cationic surfactant (component 3). The solvent in this case is water. From zero to t

_{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 [4–6] that concomitantly component 2 is in the presence of components 1 and 3 and component 3 is in the presence of components 1 and 2. Thus the notation j

^{o}

_{F;1}= x

_{f2}j

^{Δ}

_{2;1,3}+ x

_{f3}j

^{Δ}

_{3;1,2}indicates that, F is composed of components 2 and 3, which are interacting in a medium (component 1). 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.

^{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 [5,6], in a region of saturation of interactions, component 2 is interacting with a part of component 3 to form a fraction with constant composition. A fraction with constant composition is named a “pseudo-component [4–6].” This pseudo-component, composed of 2 and a part of 3, does not interact with the rest of component 3. A saturation of interactions is related to the formation of pseudo-components.

^{o}

_{F;1}in the region of saturation (Equation (90)) in the equation for calculating j

^{o}

_{3;1,2}from j

^{o}

_{F;1}(Equation (117)) and bearing in mind that:

_{3;1,2}/dns

_{3}. In the same way, the heat obtained when component 2 is the titrant can be written as:

_{3;1,2}/dns

_{3}in Equation (86) with respect to x

_{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 Equation (123) (in Appendix 1: “Limits at infinite dilution in multicomponent systems”) for enthalpies and volumes:

_{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 (Equation (86)) involves both a variation of enthalpy and a variation of volume. In general Δv

^{Δ}

_{3;1,2}is not zero. As example of this, Figure 7 shows the case of the interaction between non-charged polymeric particles and a surfactant. On the other hand, if there were no link between the variation of enthalpy and the heat of interaction measured by ITC this would affect the results of heats of interaction obtained with the technique, particularly in biophysical applications. This paradox can be solved as follows: models have been proposed [9–11] that indicate that the variation in volume for protein unfolding is very small. In addition it has been found experimentally that the variation in volume during the denaturation of lysozyme by a strong denaturant is very close to zero [12]. In our case, we have found that Δv

^{Δ}

_{F;1}can be neglected in the process of binding deciltrimethylammonium bromide to lysozyme [5] (see Figure 8). Supposing Δv

^{Δ}

_{F;1}≈ 0 in Equation (125) (in Appendix 1: “Limits at infinite dilution in multicomponent systems”), then:

_{3;1,2}/dns

_{3}(Equation (86)) yields for this type of processes:

_{1}h

_{1}Δv

^{Δ}

_{3;1,2}|≪ |Δh

^{Δ}

_{3;1,2}| holds.

## 5. Conclusions

_{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.

## Acknowledgments

## Appendix 1. Fraction of a System and Fraction Variables

_{1}, n

_{2}and n

_{3}are the number of moles of the components 1, 2 and 3, respectively. The Gibbs Equation [13] for J takes the form:

_{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.

_{F}= n

_{2}+ n

_{3}, and x

_{f3}= n

_{3}/(n

_{2}+ n

_{3}), which are related to the composition of F. The Gibbs equation for Equation (104) takes the form:

_{1;F}and j

_{F;1}are respectively:

_{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 [14] we can write the partial properties j

_{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:

_{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:

## Appendix 2. Limits at Infinite Dilution in Multicomponent Systems

_{F;1}at infinite dilution is defined as:

_{2;1,3}and j

_{3;1,2}are defined as:

_{f3}and combining the result with Equations (114) and (115) yields:

^{Δ}

_{2;1,3}and j

^{Δ}

_{3;1,2}in (114) we have:

^{∅︀}

_{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.

^{Δ}

_{2;1,3}and j

^{Δ}

_{3;12}in the Gibbs-Duhem type equation for the partial properties (Equation (115)) we have:

_{f3}and combining with Equations (121) and (123) yields:

## Appendix 3. The Region of High Dilution

_{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:

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

_{F}tending to zero in (131) and including (130) and (132) we obtain:

_{F}is to replace the partial properties as follows:

## Appendix 4. Basic Equations

_{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:

_{F}, x

_{f3}and V, the enthalpy takes the form:

_{1}, n

_{2}and n

_{3}and c

_{F}, x

_{f3}and V:

_{F}= x

_{F}ρ:

_{v}in the high dilution region can be obtained by substituting Equations (151) and (148) in (147):

_{v}can be written as:

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**Figure 1.**Typical performance of an isothermal titration calorimeter. The electronic details of the measurement of the calorimetric signal have been omitted for clarity.

**Figure 2.**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.

**Figure 3.**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}.

**Figure 4.**Variation in enthalpy between the different states of a differential concentration experiment of titration.

**Figure 5.**Calorimetric signal of the titration of toluene with toluene at 30 °C. The volume of titration was 200 μL.

**Figure 6.**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. [5]).

**Figure 7.**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. [5]).

**Figure 8.**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. [5]).

**Table 1.**Isothermal titration calorimeters that are currently manufactured and the method employed by each (full cell or half-full cell).

Calorimeter (Company) | Type of method: full cell or half-full cell |
---|---|

iTC_{200} (Microcal Inc.) | Full Cell (1) |

AUTO iTC_{200} (Microcal Inc.) | Full Cell (1) |

VP-ITC (Microcal Inc.) | Full Cell (1) |

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

TAM 2277 (TA Instruments) | Both, but the half-full cell method is most often used and is the strongly recommended method (2) |

TAM III ITC (TA Instruments) | Both, but the half-full cell method is most often used and is the strongly recommended method (2) |

^{(1)}Technical information supplied by MicroCal Inc.

^{(2)}Technical information supplied by TA Instruments.

© 2009 by the authors; licensee Molecular Diversity Preservation International, Basel, Switzerland. This article is an open-access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

## Share and Cite

**MDPI and ACS Style**

Grolier, J.-P.E.; Del Río, J.M. On the Physical Meaning of the Isothermal Titration Calorimetry Measurements in Calorimeters with Full Cells. *Int. J. Mol. Sci.* **2009**, *10*, 5296-5325.
https://doi.org/10.3390/ijms10125296

**AMA Style**

Grolier J-PE, Del Río JM. On the Physical Meaning of the Isothermal Titration Calorimetry Measurements in Calorimeters with Full Cells. *International Journal of Molecular Sciences*. 2009; 10(12):5296-5325.
https://doi.org/10.3390/ijms10125296

**Chicago/Turabian Style**

Grolier, Jean-Pierre E., and Jose Manuel Del Río. 2009. "On the Physical Meaning of the Isothermal Titration Calorimetry Measurements in Calorimeters with Full Cells" *International Journal of Molecular Sciences* 10, no. 12: 5296-5325.
https://doi.org/10.3390/ijms10125296