Comparison of Theory and Practice in the Framework for Quantifying Interaction Capacity through Six Interaction Parameters Using tert-Butanol as a Target Molecule.

Molecular interactions are important for various areas of research. Interactions between a target molecule and probe molecules having their own interaction capacity can be quantified via six interaction parameters. The theoretical interaction energy can be calculated from the interaction parameters, while that of experimental is measured using a calorimeter. These two methods are proposed in this work to calculate them. The first is based on an equation linking Hansen's and Drago's parameters. The second method is based on an experimental matrix formed by the interaction energies of tert-butanol with the probe molecules characterized by their six interaction parameters. Finally, the quality of the experiment matrix is checked for the effectiveness of the six experimental interaction parameters of the target molecule, which is tert-butanol. Then, these experimental values are compared with theoretical values from interaction parameters.

Introduction

Studying molecular interactions, which are also called noncovalent or intermolecular interactions, is essential for understanding biological structures and processes. They are important in various fields[1−13] such as drug design, material science, sensors, anotechnology, separation, and origins of life. Furthermore, complex solid and liquid molecules have their own interaction capacities, which can be quantified[14] using a set of six molecular interaction parameters ∂d, ∂p, Ea, Eb, Ca, and Cb.

Here, ∂d and ∂p (MPa1/2) are Hansen’s magnetic and electrical parameters,[15,16] and Ea, Eb, Ca, and Cb (kcal1/2 mol–1/2) are Drago’s interaction parameters[17] concerning chemical bonds having charge transfer and orbital overlap as processes. This indicates that there are three types of interactions, each of which has a well-determined origin.

The most important result of molecular interactions is the interaction energy of the interacting molecules. The latter can be determined experimentally via the mixing energy or via theoretical calculations based on interaction parameters.

To experimentally determine interaction energy, the use of probe molecules in addition to their six interaction parameters is required.

This study aims to establish the above-mentioned approach for determining both the interaction parameters and interaction energy. To this end, this work uses tert-butanol as a solute and diethyl ether, i-propylether, n-butylether, triethylalamine, diethylamide, pyridine, dimethylformamide, dimethylacetamide, and acetonitrile as solvents or probe molecules for tert-butanol.

Results

The two main results are presented in Tables and 2.

Table 1

Experimental Yexp (kcal/mol) and Calculated Interaction Energies Ycal (kcal/mol) and Relative Error (Yexp – Ycal / Ycal)100

solvents	Yexp	Ycal	relative error (%)
diethyl ether	14.20	13.67	3.73
i-propylether	13.48	12.86	4.59
n-butylether	14.93	14.27	4.42
triethylalamine	16.40	14.88	9.26
diethyl amine	16.28	14.93	8.29
pyridine	19.58	18.81	3.93
dimethylformamide	19.77	19.18	2.98
dimethylacetamide	19.08	18.42	3.45
acetonitrile	17.09	17.10	–0.05

Table 2

Estimated and Calculated Values of the Six Interaction Parameters ∂d, ∂p (cal1/2 cm–3/2), Eb, Cb, Ea, and Ca (kcal1/2 mol–1/2) of the Target Molecule tert-Butanol

interaction parameters of tert-butanol	∂d	∂p	Ea	Ca	Eb	Cb
estimated value (Table 7)	7.43	2.49	1.25	0.46	1.80	0.57
experimental value	6.72	2.41	2.58	0.20	1.74	0.80

The demonstration has been done to know that it is possible to calculate the interaction energies between molecules based on the knowledge of the interaction capacity defined by the six interaction parameters ∂d, ∂p, Ea, Eb, Ca, and Cb. The relative errors between the calculated and experimental values are low and acceptable.

Finally, the capacity of the interaction of the target molecule tert-butanol has been defined by the same six parameters. It is possible to explain the important deviation on Ea that can be seen on the second table.

Discussion

The objective of this work was to demonstrate that it is possible to define the interaction capacity of molecules using the mixing model proposed recently[14] by determining a group of six interaction parameters and then calculating the interaction energy of the same molecule in different solvents using these parameters. To this end, tert-butanol mixed with nine different solvents was used in this study.

The prediction of the solubility in different solvents was not considered in this work. In such a case, it would have been necessary to use an appropriate model taking into account of the enthalpic and entropic parts of the dissolution process.[25]

In this research paper, each molecule is characterized by its six interaction parameters and its molar volume. So from the point of view of the interaction, our paper works with a space of six dimensions and these six independent parameters (∂d, ∂p, Ea, Cb, Eb, and Cb) characterizing different types of interactions.

The reason for the six-dimensional space in this work is as follows. The energy of the cohesion or interaction, obviously, comes from three types of interactions: magnetic dipole–dipole (mobile charges), electric dipole–dipole (localized charges), and transfer processes of charges and orbital recovering.

For the hydrogen bonding interaction, it is necessary to mobilize the four parameters Ea, Ca, Eb, and Cb. In the case of a solute, moreover, our proposed model still requires the use of its molar volume V giving the notion of the molecular size.

For many authors having worked with the three Hansen’s parameters (∂d, ∂p, and ∂h), their space is a three-dimensional space. However, these three parameters are not homogeneous because the parameter ∂h is a function of four interaction components according to the equation

Therefore, different combinations of these four parameters Ea, Ca, Eb, and Cb having different values can give the same value of ∂h.

Consequently, the Hansen’s sphere is an apparent interaction sphere. Considering the hydrogen bonds, it is necessary to introduce several kinds of corrections, including thermodynamic corrections to get closer to reality.[26]

It is impossible to build any thermodynamic descriptive model with these three Hansen parameters, including the volume of the molecules as in our model. One of the important goals of scientific research is to be able to predict phenomena; we can only predict something on the basis of a descriptive model. Experience shows that the nature of the solvents plays an important role in chemical reactions. Between our method of quantification of interaction energy and the QSAR method,[27] the common point lies in obtaining an experiment matrix in which the parameters characterizing the solvents can be figured explicitly.

The cited work[27] used a 412 × 76 experiment matrix, while our work used a 9 × 6 experiment matrix. The difference between the two methods is in the choice of parameters for solvents. Our work uses six interaction parameters—two from Hansen’s type and four from the Drago type. All of which are extracted from the cohesion energy and the quality of the experiment matrix tested; in contrast, the work with the QSAR method used 24 parameters or descriptors for the solvents; these did not necessary arise from the energy of cohesion and there was no test conducted to ascertain the quality of the 412 × 76 experiment matrix used. This is a significant difference from the point of view of rigor. Ref (14) has been cited in our paper for comparison purposes.

The introduction of six interaction parameters and the construction of the interaction model were done gradually, and the first applications are in the choice of solvents to have the best performance for microencapsulation. The following Refs[3] and (28) have been cited in our paper for comparison purposes.

Although the nine solvents are not amphoteric,[24] it is an important condition for obtaining suitable values for V∂2h/n = EaEb + CaCb. They have been chosen because of their interaction parameters, which are partially published (Table ). In addition, their experimental mixing energies with tert-butanol have also been published,[18] and these are indispensable for the subsequent comparison of the theoretical and experimental interaction energies (Table ).

Table 3

Published Values of Molar Volumes V (cm3/mol) of Hansen’s Cohesive Parameters ∂d, ∂p, and ∂h (cal1/2 cm–3/2) and Drago’s Chemical Interaction Parameters Ea, Ca, Eb, and Cb (kcal1/2 mol–1/2)

solvent	V	∂d	∂p	∂h	Ea	Ca	Eb	Cb
tert-butanol	94.80	7.43	2.49	7.28	1.36[21]	0.51[21]
diethylether	104.89	7.03	1.41	2.49			1.80	1.63
i-propylether	142.20	6.69	1.02	1.19			1.95	1.66
n-butylether	170.36	7.13	2.10	2.20			1.89	1.67
triethylalamine	140.00	7.13	1.80	0.92			1.32	5.73
diethylamine	102.90	6.55	3.42	3.08			1.22	4.54
pyridine	80.87	9.28	4.30	2.88			1.78	3.54
dimethylformamide	77.40	8.50	6.69	5.52			2.19	1.31
dimethylacetamide	93.04	8.21	5.62	4.98			2.35	1.31
acetonitrile	52.86	7.47	8.79	2.98			1.64	0.71

Table 11

Presents a Comparison of Theoretical and Experimental Values for the Interaction Energiesa

solvent	Yexp	Ycal	relative error (%)
diethyl ether	14.20	13.67	3.73
i-propylether	13.48	12.86	4.59
n-butylether	14.93	14.27	4.42
triethylalamine	16.40	14.88	9.26
diethyl amine	16.28	14.93	8.29
pyridine	19.58	18.81	3.93
dimethylformamide	19.77	19.18	2.98
dimethylacetamide	19.08	18.42	3.45
acetonitrile	17.09	17.10	–0.05

The levels of the relative errors validate the proposed mixing model.[14] Comparison of the experimental Einterexp = Yexp (kcal/mol) and calculated interaction energies Eintertheo = Ycal (kcal/mol) for the selected case X1i = 9/10Ximax.

As can be seen from Table , it is impossible to directly determine the values of the six interaction parameters for tert-butanol and the nine solvents. Therefore, a method has been proposed herein for approaching them using the equation V∂2h/n = EaEb + CaCb. This equation has an infinite number of solutions even in a very limited range. However, there is only one solution that corresponds to the actual case. Various solutions were tested via an iterative method. The solution X1i = 9/10X1imax was retained as it was the most optimal. The complete list of the six parameters estimated for the ten molecules as a result of using the optimal solution is presented in Table .

Table 7

Molar Volume V (cm3/mol) of Hansen’s Cohesive Parameters ∂d, ∂p, and ∂h (cal1/2 cm–3/2), Drago’s Corrected Chemical Interaction Parameters Ea, Ca, Eb, and Cb (kcal1/2 mol–1/2), and V∂2h/n = EaEb + CaCb (kcal mol–1) for the Case where X1i = 9/10Ximax

solvent	V	∂d	∂p	∂h	V∂2h/2	Ea	Ca	Eb	Cb
tert-butanol	94.80	7.43	2.49	7.28	2.51	1.25	0.46	1.80	0.57
diethylether	104.89	7.03	1.41	2.49	0.33	0.18	0.02	1.66	1.50
i-propylether	142.20	6.69	1.02	1.19	0.10	0.05	0.007	1.80	1.53
n-butylether	170.36	7.13	2.10	2.20	0.41	0.21	0.03	1.74	1.54
triethylalamine	140.00	7.13	1.80	0.92	0.05	0.04	0.0003	1.21	5.27
diethylamine	102.90	6.55	3.42	3.08	0.50	0.40	0.11	1.12	4.18
pyridine	80.87	9.28	4.30	2.28	0.21	0.12	0.004	1.64	3.26
dimethylformamide	77.40	8.50	6.69	5.52	1.18	0.53	0.094	2.01	1.21
dimethylacetamide	93.04	8.21	5.62	4.98	1.15	0.48	0.093	2.16	1.21
acetonitrile	52.86	7.47	8.79	2.98	0.23	0.14	0.03	1.51	0.65

Furthermore, multilinear regression was applied to an experimental matrix; this allowed us to determine the values of the tert-butanol interaction parameters. Considering the part of the work as a control of the quality of the proposed interaction model, the chosen experiment matrix presented a result that is similar to the estimated values presented in Table .

However, Ea = 2.58 (kcal1/2 mol–1/2) was significantly higher. Using this value, the expression V∂2h/n = EaEb + CaCb gives (2.58 × 1.74 + 0.20 × 0.80) = 4.65 (kcal mol–1) compared to a Hansen’s value of 2.51 (kcal mol–1). This difference is due to the fact that the nine solvents are not amphoteric.[24] This means that their V∂2h/n value is close to zero, which must have had repercussions on the multilinear regression results.

Using them to calculate the interaction energies between tert-butanol and the nine solvents, results (Table ) that match well with the experimentally determined energies (Table ) were obtained with the mean relative error for the nine solvents being 4.51%.

Table 8

Various Contributions Made to the Theoretical Interaction Energy (kcal/mol) by tert-Butanol and the Nine-Selected Solvents

solvent	2V2∂d1∂d2 dispersive interaction	2V2∂p1∂p2 polar interaction	Ea1Eb2 + Ca1Cb2 chemical bond 1	Ea2Eb1+ Ca2Cb1 chemical bond 2	theoretical interaction energy
diethyl ether	9.88	0.65	0.34	2.8	13.67
i-propylether	9.36	0.47	0.09	2.94	12.86
n-butylether	10.03	0.96	0.40	2.88	14.27
triethylalamine	10.03	0.85	0.07	3.94	14.88
diethyl amine	9.21	1.62	0.78	3.32	14.93
pyridine	13.00	2.04	0.22	3.54	18.81
dimethylformamide	11.95	3.16	1.00	3.07	19.18
dimethylacetamide	11.60	2.64	0.92	3.26	18.42
acetonitrile	10.48	4.16	0.27	2.19	17.10

Table 10

Mixing Contributions ΔEmix (kcal/mol) (Ref (18)), Cavity Contributions ΔEcavity(i,j), (kcal/mol), and Vaporization Contributions ΔEvap (kcal/mol) of tert-Butanol to the Experimental Interaction Energy ΔEinterexp(i,j) (kcal/mol) between tert-Butanol and the Nine Solvents Obtained from Colorimetric Measurements

solvent	ΔEmix	ΔEcavity(i,j)	ΔEvap of tert-butanol	experimental interaction energies ΔEinterexp(i,j)
tert-butanol			10.72
diethylether	–1.67	5.15	10.72	14.2
i-propylether	–1.69	4.45	10.72	13.48
n-butylether	–1.19	5.40	10.72	14.93
triethylalamine	0.42	5.26	10.72	16.40
diethylamine	0.19	5.37	10.72	16.28
pyridine	–0.35	9.21	10.72	19.58
dimethylformamide	–1.08	10.13	10.72	19.77
dimethylacetamide	–0.73	9.09	10.72	19.08
acetonitrile	–2.63	9.00	10.72	17.09

Finally, by observing the different values of the interaction parameters presented in Table , we can determine whether the target molecules are basic or acidic. In the case of tert-butanol, the charge transfer (Ea, Eb) is substantially more important than the overlap orbital (Ca, Cb) process.

Table 13

Estimated Values (from Table ) and Calculated Values of the Six Interaction Parameters, ∂d, ∂p (cal1/2 cm–3/2), Eb, Cb, Ea, and Ca (kcal1/2 mol–1/2) of tert-Butanol for the Chosen Case where X1i = 9/10X1imax

interaction parameters of tert-butanol	∂d	∂p	Ea	Ca	Eb	Cb
estimated value (Table 7)	7.43	2.49	1.25	0.46	1.80	0.57
experimental value	6.72	2.41	2.58	0.20	1.74	0.80

Conclusions

The comparison of theoretically and experimentally determined values in the context of the interactions of tert-butanol with nine solvent molecules shows that the ability of a molecule to interact can be quantified using its six interaction parameters ∂d, ∂d, Ea, Ca, Eb, and Cb.

The bridging equation V∂2h/n = EaEb + CaCb is essential for elucidating the nature of ∂h.

The experimental energy of the interaction with different molecules can be determined using the expression

The components of the energy of the theoretical interaction can be calculated according to the following equation (Figure ).

Figure 1

Components of the energy of the theoretical interaction.

Calculation program used: Python program.

Theoretical Section and Calculation

To achieve the objectives of this work, it is necessary to have an innovative mixing model; this has been developedwhere i is the solute, j is the solvent, −ΔEmix(i,j) + ΔEvapi + Vi(∂2dj + (∂2pj/2)) + niVj∂2hj/nj – ΔVi(∂2dj + 3/2RT/Vj) is the experimental interaction energy, ΔEinterexp, ΔEmix(i,j) is the mixing energy of solute i into solvent j, ΔEvap(i) is the vaporization energy of solute i, Vi(∂2dj + (∂2pj/2)) + niVj∂2hj/nj – ΔVi(∂2dj + 3/2RT/Vj) is considered the cavity formation energy or the disturbance energy of the solvent bulk due to the presence of the solute, ΔEcav.

2Vi∂dj∂di + 2Vi∂pj∂pi + (EajEbi + CajCbi) + (EaiEbj + CaiCbj) is the theoretical interaction energy, ΔEintertheo, between solute i and solvent j, computable using the six interaction parameters.

With these definitions being recalled, eq becomes

tert-Butanol was used as a solute, and the following nine solvents were used: diethylether, i-propylether, n-butylether, triethylalamine, diethylamine, pyridine, dimethylformamide, dimethylacetamide, and acetonitrile.

The bridge equation linking the Hansen’s parameter ∂h to Drago’s four parameters Ea, Eb, Ca, and Cb (ref (14)) was used

From ref (18), the nine mixing energies of tert-butanol −ΔEmix(i,j) in the nine-selected solvents were obtained.

Hansen’s calculated vaporization energy of tert-butanol (ΔEvap), Hansen’s cohesion parameters ∂d, ∂p, and ∂h (ref (19)), and Drago’s chemical interaction parameters[17] for the nine-selected solvents were used.

Issue Regarding Drago’s Chemical Interaction Parameters

For the ten substances used in this study, the values of Drago’s parameters, Ea, Ca, Eb, and Cb, are incomplete.[17,20] A method for determining the missing values needs to be proposed.

Determination of the Missing Values of Drago’s Parameters

First, it needs to be highlighted that the experimental method proposed by Drago is not efficient enough to eliminate the polar contribution of the chemical interaction energy. According to Drago’s ECW model[17]

However, according to our model,[14] ΔH in fact must be

This is because the apolar solvent used in the mixing process cannot eliminate the polar contribution 2Vi∂pj∂pi from ΔH.

However, Drago gave the following expression for ΔH

Then,where W must be

In this study, the following equation will be considered

Therefore, the values of Drago’s parameters Ea, Eb, Ca, and Cb are over-estimated, and they need to be corrected so that they can be reverted to the Hansen’s scale.

The proposed correction method is the following. From the Drago–Wayland parameters,[21] methanol and ethanol were chosen because the values of their four parameters are exceptionally published (Table ).

Table 4

Published Molar Volumes V (cm3/mol), Hansen’s Cohesive Parameters ∂h (cal1/2 cm–3/2) and Drago’s Chemical Interaction Parameters Ea, Ca, Eb, and Cb (kcal1/2 mol–1/2) for Methanol and Ethanol

solvent	V	∂h	V∂2h/2 (Hansen)	Ea	Ca	Eb	Cb	EaEb + CaCb (Drago)
methanol	40.7	10.93	2.43	1.25	0.75	1.80	0.70	2.78
ethanol	58.5	9.51	2.65	1.34	0.69	1.85	1.10	3.24

For methanol, (Hansen)/(Drago) = 2.43/2.78 = 0.87, the over estimation is approximately 14%. For ethanol, (Hansen)/(Drago) = 2.65/3.24 = 0.82, the over estimation is approximately 22%.

Given that the errors for methanol and ethanol are 0.87 and 0.82, respectively, the value of the correction factor will be the mean value, 0.85.

Finally, we obtained

Thus, using 0.92 as a correction factor, it is possible to obtain the values of Drago’s parameters adapted to the Hansen’s scale (Table ).

Table 5

Molar Volumes V (cm3/mol), Hansen’s Cohesive Parameters ∂d, ∂p, and ∂h (cal1/2 cm–3/2), and Drago’s Corrected Chemical Interaction Parameters Ea, Ca, Eb, and Cb (kcal1/2 mol–1/2)

solvent	V	∂d	∂p	∂h	Ea	Ca	Eb	Cb
tert-butanol	94.80	7.43	2.49	7.28	1.25	0.46
diethylether	104.89	7.03	1.41	2.49			1.66	1.50
i-propylether	142.20	6.69	1.02	1.19			1.80	1.53
n-butylether	170.36	7.13	2.10	2.20			1.74	1.54
triethylalamine	140.00	7.13	1.80	0.92			1.21	5.27
diethylamine	102.90	6.55	3.42	3.08			1.12	4.18
pyridine	80.87	9.28	4.30	2.88			1.64	3.26
dimethylformamide	77.40	8.50	6.69	5.52			2.01	1.21
dimethylacetamide	93.04	8.21	5.62	4.98			2.16	1.21
acetonitrile	52.86	7.47	8.79	2.98			1.51	0.65

Now, the equation V∂2h/n = (EaEb + CaCb) (kcal mol–1), with the values given in Table , is used to construct ten equations, which are presented in Table .

Table 6

Molar Volumes V (cm3/mol), Hansen’s Cohesive Parameters ∂h (cal1/2 cm–3/2), V∂2h/n (kcal mol–1), and the Ten Equations of the Formed V∂2h/n = EaEb + CaCb (kcal mol–1) Corresponding to the Solute tert-Butanol and the Nine Solvents

solvent	V	∂h	n	V∂2h/n	V∂2h/n = EaEb + CaCb
tert-butanol	94.80	7.28	2	2.51	2.51 = 1.25Eb + 0.46Cb
diethylether	104.89	2.49	2	0.33	0.33 = 1.66Ea + 1.50Ca
i-propylether	142.20	1.19	2	0.10	0.10 = 1.80Ea + 1.53Ca
n-butylether	170.36	2.20	2	0.41	0.41 = 1.74Ea + 1.54Ca
triethylalamine	140.00	0.92	2	0.05	0.05 = 1.21Ea + 5.27Ca
diethylamine	102.90	3.08	2	0.50	0.50 = 1.12Ea + 4.18Ca
pyridine	80.87	2.88	2	0.21	0.21 = 1.64Ea + 3.26Ca
dimethylformamide	77.40	5.52	2	1.18	1.18 = 2.01Ea + 1.21Ca
dimethylacetamide	93.04	4.98	2	1.15	1.15 = 2.16Ea + 1.21Ca
acetonitrile	52.86	2.98	2	0.23	0.23 = 1.51Ea + 0.65Ca

Each of these equations has two unknowns, and they are Eb and Cb for tert-butanol and Ea and Ca for the nine solvents.

Substitute X1i = Ea for each of the nine chosen solvents and X1i = Eb for tert-butanol. In addition, substitute X2i = Ca for each of the nine chosen solvents and X2i = Cb for tert-butanol in the ten equations.

Then, all of the ten equations shown in Table having the two unknowns X1i and X2i can be represented in the following form

A graph of X1i as a function of X2i is given in Figure .

Figure 2

Graph of X1i as a function of X2i.

Any point on the straight line plotted in the above figure is a solution of the equationX1i = X1imax when X2i = 0, and X2i = X2imax when X1i = 0.

For tert-butanol, which is our target molecule, the values of Eb and Cb must be in the following limits according to the equation 2.51 = 1.25Eb + 0.45Cb

The most optimal case X1i = (9/10)X1imax has been chosen after having tested many options to obtain solutions that respect these limits. Using X1i = (9/10)X1imax, it is possible to calculate all values of the variable X2i (Table ).

The expression of the theoretical interaction energy between tert-butanol and the nine solvents has the following form

The chemical bonding interaction has two parts

Bonding between a solvent molecule and a solute molecule involves eight parameters.

This combination of two molecules in turn gives two bonds, which are given by eqs and 10.

Using the values shown in Tables –7, it is possible to calculate the different contributions of tert-butanol and the nine-selected solvents to the theoretical interaction energy.

Determination of the Experimental Interaction Energy

The expression for the experimental interaction energy isin a simpler form

Table gives an overview of different contributions to cavity formation energy.

Table 9

Dispersive Contribution V2∂2d1, Polar Contribution V2∂2p1/2, Chemical Contribution V∂2h/2, and Mechanical Contribution ΔVi(∂2dj + 3/2RT/Vj) (Ref (18)) to the Cavity Formation Energy ΔEcav(i,j)

solvent	V2∂2d1 (kcal/mol)	V2∂2p1/2 (kcal/mol)	V∂2h/2 (kcal/mol)	ΔVi(∂2dj + 3/2RT/Vj)a	ΔEcavité(i,j)(kcal/mol)
diethylether	4.69	0.09	0.33	0.04	5.15
i-propylether	4.24	0.09	0.10	0.02	4.45
n-butylether	4.82	0.20	0.41	–0.03	5.40
triethylalamine	4.82	0.15	0.05	0.24	5.26
diethylamine	4.07	0.55	0.50	0.25	5.37
pyridine	8.16	0.87	0.21	–0.03	9.21
dimethylformamide	6.84	2.12	1.18	–0.01	10.13
dimethylacetamide	6.38	1.50	1.15	0.06	9.09
acetonitrile	5.29	3.66	0.23	–0.18	9.00

Vi = 94.8 cm3 mol–1.

The addition of the published mixing energies and the vaporization energy of the solute with the cavity formation energies allows us to obtain the nine experimental energies of interaction between tert-butanol and the solvents (Table ).

Overview of the Experiment Matrix to Be Used for Calculating the Interaction Parameters of tert-Butanol

From Tables and 10, the experiment matrix can be written as follows:where (Y) represents the matrix column of experimental interaction energies, (X) represents the experiment matrix, and (b) is the column matrix of coefficients to be calculated.

Ideally, the experiment matrix (X) must be orthogonal so that the coefficients bi are independent.

However, in the case where the inflation factor (Table ), F(bi), of each coefficient, bi is in the range of 1–10, (X) can be used. Outside of this range, the coefficients bi are biased.[22,23] The inflation factor, F(bi), of each coefficient, bi can be calculated according the following equationswhereand

Table 12

Inflation Factors F(bi) of the Experiment Matrix for the Case X1i = 9/10X1imax

F(b1)	F(b2)	F(b3)	F(b4)	F(b5)	F(b6)
5.20	5.04	2.03	1.90	8.63	5.57

Calculation of the Six Interaction Parameters of the Target tert-Butanol

The multilinear regression when applied to the proposed experiment matrix gives the experimental values for the six interaction parameters. We can now compare these experimental values (Table ) with the estimated values presented in Table .