Computational Investigation of Heat-Induced Thermodynamic Properties in Benzimidazole and Its Derivatives

Introduction: Benzimidazole is a fragrant heterocyclic composite. The manufacturing of benzimidazole primarily depends on poly heterocycles; took the attention of chemists from previous few decades because of its functions as a pharmacophore in healthcare pharmacy, and medical specialty[1].

This compound possesses several medicine’s properties until currently, the foremost outstanding benzimidazole moiety is N- ribosyl dimethyl benzimidazole, which can be created in nature, and it is the centred matter for atomic number 27 in cobalamin[2].

Fig. 1. Structure of benzimidazole: (a) chemical and (b) Gaussian view[3].

Benzimidazole exhibits a wide range of biological activities, including antimicrobial, antifungal, antihistaminic, antiviral, antioxidant, anticancer, and anti-ulcer properties [4]. Moreover, when a benzimidazole is substituted with proline bisamide, it has been found to aid in treating insomnia and promoting sleep, acting as a potent orexin inhibitor identified through high-throughput screening [4]. For these reasons, benzimidazole derivatives are considered significant molecules of interest in medicinal chemistry. Orexin itself plays diverse roles in the central nervous system, including regulation of the normal sleep–wake cycle and eating behavior. Dysfunction of neurons responsible for orexin production is associated with conditions such as narcolepsy [5].

The biological relevance of many heterocyclic building blocks is reflected in their chemical structures, particularly due to the structural similarity of purine nucleobases with benzimidazole derivatives [5]. These features contribute to the broad therapeutic applications of benzimidazole-based drugs, although the method of preparation—whether in vitro or in vivo—can influence their activity. Substituted benzimidazoles, in particular, serve as potent inhibitors of parietal cell (H⁺/K⁺ ATPase) proton pumps [6,7].

Additionally, the heat capacity of an object can be defined as the energy transferred to it per unit change in temperature, as discussed in [8,9,10].

heat capacities are like the mass, where their value depends on the material and the amount of that material,  the specific heat can be used only for calculating the properties of actual materials so these specific heats are like density, that’s why they depend only on materials while specific heats are very organized,  as well as the organizing of specific heats at constant pressure and constant volume, specific heats are given as heat capacity per unit mass, heat capacity per mole, or heat capacity per particle, that’s mean, c = C=m, c = C=n; or c = C=N: In elementary physics mass specific heats are common wherein chemistry specific heats are common[10,11].

If we add the first law of thermodynamics with the definition of heat capacity, we can develop general expressions for heat capacities at constant volume and constant pressure[11]. Writing the first law in the form q = 4E + p4V, and inserting this into:

the formula will be as[11]:

Thermodynamics can be considered as the study of the relationship between heat (or energy) and work. Enthalpy can be considered as a central factor in thermodynamics this is the heat content of a system[10].

The heat that passes into or out of the system during a reaction is the enthalpy change, whether the enthalpy of the system increases (i.e. when energy is added) or decreases (because energy is given off) is a crucial factor that determines whether a reaction can happen, on the other hand, the energy of the molecules undergoing change can be called as the “internal enthalpy” or “enthalpy of the system”, These two expressions refer to the same thing. Similarly, the energy of the molecules that do not take part in the reaction is called the “external enthalpy” or the “enthalpy of the surroundings”[12].

In thermodynamics, entropy (S) can be defined as the measure of the number of specific ways in which a thermodynamic system may be arranged, generally, entropy can be understood as a measure of mess.

According to the second law of thermodynamics: the entropy of an isolated system never decreases, it will naturally proceed towards thermodynamic equilibrium of the configuration with maximum entropy.

Systems that are not isolated properly may decrease in entropy provided they increase the entropy of their environment by at least that same amount.  Since entropy is a state function, the change in the entropy of a system is the same for any process that goes from a given initial state to a given final state, whether the process is reversible or irreversible.  where irreversible processes increase the combined entropy of the system and its environment. The two definitions of entropy are the thermodynamic definition and the statistical mechanic’s definition[13].

Historically, the classical thermodynamics definition expanded first. In the classical thermodynamics point of view the system is consist of very large numbers of constituents (atoms, molecules), and the condition of the system is described by the average thermodynamic properties of those constituents these details of the system’s constituents are not directly considered but their behavior is described by macroscopically averaged properties of the early classical definition of the properties of the system supposed to be balanced[14].

The classical thermodynamic definition of entropy has more recently been extended into the area of non-balanced thermodynamics. Later, the thermodynamic properties, including entropy, were given an alternative definition in terms of the statistics of the motions of the microscopic constituents of a system — modeled at first classically. The statistical mechanic’s description of the behavior of a system is necessary as the definition of the properties of a system using regular thermodynamics become an increasingly unreliable method of predicting the final state of a system that is subject to some process[14].

2. Computational Method

Theoretical quantum chemical density functional computations were carried out at Becke 3-Lee-Yang-Parr (B3LYP) [15,16] level with 6-311++G(d,p) basis set by using Gaussian 09W program package [11]. The visualizations of the computed structural, spectroscopic and electronic properties were carried out with GaussView5 program package [17]. The initial geometry for the molecular optimization was drown using GaussView.

In this article, Benzimidazole and some of its derivatives were used to study the effect of changing the temperature on the thermodynamic properties; all compounds were in the gas stage[17].

The hardness is a half of the energy gap between HOMO and LUMO that

 The softness can be calculated from hardness that:

              (Hard  Soft Acid Bas (HSAB) principle 

Electronegativity can be calculated from E_HOMO and E_LUMO using the following equation:

                             E_HOMO + E_LUMO)

From electronegativity, chemical potential can estimated that μ = -χ

                             E_HOMO + E_LUMO)

Electrophilicity index (w) and nucleofugality (DEn) and electrofugality (DEe) can be taken from the chemical potential μ, and hardness η by the following equations respectively[18].

Nucleofugality (DEn)               (( μ +η)²)/2η

Electrophilicity index (ω)   

Electrofugality (DEe)              

3. Results and discussion

3.1. Molecular geometric structure

The  derivatives of  benzimidazole moiety will show very strong antioxidant properties when Benzimidazoles was substituted with phenyl, p-chlorophenyl, p-methoxyphenyl, and pyridinyl rings. A diagram of some benzimidazoles derivarives is given in Fig.2. Benzimidazole and some of its derivatives are shown in (Table1): The unsubstituted derivative 1 (R1, R2 = H) was less active. It could also be demonstrated that moving the methoxy residue from the 4 to the 7 position (2) is tolerated.[13]

Fig. 2. A diagram of compounds in vitro activity.

Replacement of the 7-OMe residue with a methyl-group was also tolerated (3). Introduction of an additional methyl residue at position 7 of 9 leading to (4) did not have much impact on potency of the resulting compound. Surprisingly, however, was the 10-fold increase in potency of (5) where the positions of the 4- and the 7-residues have been switched. Further increases in activity could be achieved with the introduction of halogens or trifluoromethyl residues at position 4, with bromo- and iodo-residues being most preferred (6, 7and 8).

The 4,7-dimethoxy- and the 4-methoxy- 7-chloro-derivatives (9) and (10) proved to be significantly less potent than the 4-chloro-7-methoxy derivative (6). It was observed, that fairly lipophilic R1 residues were needed for good potency[13].

Table1. Benzimidazole and some of its derivatives.

Compd.	R1	R2	R3
1	Methoxy-ethyl	H	H
2	Methoxy-ethyl	MeO	H
3	Methoxy-ethyl	Me	H
4	Methoxy-ethyl	Me	MeO
5	Methoxy-ethyl	MeO	Me
6	Methoxy-ethyl	MeO	Cl
7	Methoxy-ethyl	MeO	Br
8	Methoxy-ethyl	MeO	Cf3
9	Methoxy-ethyl	MeO	MeO
10	Methoxy-ethyl	Cl	MeO
11	Methoxy-ethyl	Br	Br

The optimized (computed = ideal form) molecular structures (Fig. 1) along with

the numbering scheme for compounds were shown in Fig. 3.

Fig. 3. The optimized structure of the compound 1.

3.2. Thermodynamic properties

The sum of translation, rotational, vibrational, and electronic energies gives the total energy of the molecule where the calculated minimum energy (SCF energy) agree with the optimized structure of the molecule[19].

Zero-point Vibrational energy is the lowest acceptable energy of the quantum mechanical system, and the amount of heat desired to raise the temperature of a substance by one degree is the heat capacity as The quantitative measure of randomness in the system is entropy[9].

Using B3LYP/6-311++G(d,p) basis set, the standard statistical thermodynamic parameters such as enthalpy (H), heat capacity (C) and entropy (S), for the title molecule were obtained from the theoretical harmonic frequencies in the range of temperature 200-1000 K. A scale factor for frequencies (0.96) was used in the calculations[19,20].

Following with the equipartition theorem, the translational energy, rotational energy, and molecular vibrational intensities increase with temperature. As a result, the thermodynamic functions increase with increasing temperature ranging from 200 to 1000 K as shown in tables[21].

The linking equations between enthalpy, heat capacity, entropy, and temperatures were done by linear and quadratic formulas and the corresponding fitting factors (R²) for these thermodynamic properties are shown for (1)-(11) compound, respectively. By using these equations, we can say that the attributes at any point of temperature without further computational procedures. It is inferred from the results that the thermodynamic parameters linearly depend on the temperature value[21,22].

The correlations between the thermodynamic properties and temperatures T values are shown in Fig. 5-15. The corresponding fitting equations are as follows for 1-11 compounds, respectively[23].

Fig. 5. Correlation graphics of thermodynamic properties and temperatures for (1) compound.

The correlation equations are as follows:

H= -5.96133+0.04249T +8.48813×10^-5T² ; (R² = 0.9997)

S= 66.00826+ 0.32448T  -6.35905×10^-5T² ; (R² = 0.99999)

C= -6.71942+ 0.34355T  -1.44114×10^-4T² ; (R² = 0.99971)

Fig. 6. Correlation graphics of thermodynamic properties and temperatures for (2) compound.

The correlation equations are as follows:

H= -6.68091+0.04827T +9.16594×10^-5T² ; (R² = 0.99972)

S= 66.6354+ 0.36041T  -7.32468×10^-5T² ; (R² = 0.99999)

C= -3.79275+ 0.36771T  -1.5298×10^-4T² ; (R² = 0.99974)

Fig. 7. Correlation graphics of thermodynamic properties and temperatures for (3) compound.

The correlation equations are as follows:

H= -6.39474+0.0456T +9.01951×10^-5T² ; (R² = 0.99972)

S= 65.66912+ 0.347T  -6.86782×10^-5T² ; (R² = 0.99999)

C= -5.5146+ 0.36129T  -1.50086×10^-4T² ; (R² = 0.99974)

Fig. 8. Correlation graphics of thermodynamic properties and temperatures for (4) compound.

The correlation equations are as follows:

H= -6.99359+0.05146T +9.69257×10^-5T² ; (R² = 0.99974)

S= 68.68421+ 0.38328T  -7.85662×10^-5T² ; (R² = 1)

C= -2.17709+ 0.38424T  -1.58077×10^-4T² ; (R² = 0.99976)

Fig. 9. Correlation graphics of thermodynamic properties and temperatures for (5) compound.

The correlation equations are as follows:

H= -6.89248+0.0517T +9.67653×10^-5T² ; (R² = 0.99974)

S= 70.09878+ 0.38406T  -7.9094×10^-5T² ; (R² = 1)

C= -1.54613+ 0.38257T  -1.57022×10^-4T² ; (R² = 0.99976)

Fig. 10. Correlation graphics of thermodynamic properties and temperatures for (6) compound.

The correlation equations are as follows:

H= -7.01425+0.05263T +9.02454×10^-5T² ; (R² = 0.99973)

S= 70.76704+ 0.37648T  -8.16797×10^-5T² ; (R² = 1)

C= 0.87428+ 0.36472T  -1.52744×10^-4T² ; (R² = 0.99974)

Fig. 11. Correlation graphics of thermodynamic properties and temperatures for (7) compound.

The correlation equations are as follows:

H= -7.29712+0.05461T +8.91579×10^-5T² ; (R² = 0.99977)

S= 75.02263+ 0.37823T  -8.27335×10^-5T² ; (R² = 1)

C= -1.77297+ 0.3629T  -1.51719×10^-4T² ; (R² = 0.99974)

Fig. 12. Correlation graphics of thermodynamic properties and temperatures for (8) compound.

The correlation equations are as follows:

H= -7.92745+0.05984T +9.56366×10^-5T² ; (R² = 0.99973)

S= 74.98362+ 0.41332T  -9.26938×10^-5T² ; (R² = 1)

C= 2.37237+ 0.39502T  -1.68661×10^-4T² ; (R² = 0.9997)

Fig. 13. Correlation graphics of thermodynamic properties and temperatures for (9) compound.

The correlation equations are as follows:

H= -8.52756+0.05378T +1.00677×10^-4T² ; (R² = 0.9975)

S= 70.67426+ 0.39726T  -8.34602×10^-5T² ; (R² = 1)

C= -0.1331+ 0.38997T  -1.60547×10^-4T² ; (R² = 0.99977)

Fig. 14. Correlation graphics of thermodynamic properties and temperatures for (10) compound.

The correlation equations are as follows:

H= -7.06272+0.05297T +9.02408×10^-5T² ; (R² = 0.99973)

S= 71.75987+ 0.3766T  -8.17244×10^-5T² ; (R² = 1)

C= 0.76259+ 0.36522T  -1.5313×10^-4T² ; (R² = 0.99974)

Fig. 15. Correlation graphics of thermodynamic properties and temperatures for (11) compound.

The correlation equations are as follows:

H= -6.71624+0.05271T +8.16403×10^-5T² ; (R² = 0.99973)

S= 76.49348+ 0.35947T  -8.21179×10^-5T² ; (R² = 1)

C= 3.65236+ 0.33617T  -1.43142×10^-4T² ; (R² = 0.9997)

The figures bellow illustrate the thermodynamic properties change due to the temp. changes with the correlation values for all compounds.

HOMO and LUMO Calculations:

High Occupied Molecular Orbitals HOMO of tittle compounds using B3LYP/6-311g(d,p) were reported in Table below (Gas phase)[24].

	energy gap	Homo	Lumo	X (electro negativity)	hardness	S (softness)	µ (chemical potential)	⍵ (electrophilicity index)	△En	△Ee
1	4.82	-5.86	-1.04	3.45	2.41	0.20746888	-3.45	2.46939834	0.22439834	7.12439834
2	4.8	-5.89	-1.09	3.49	2.4	0.20833333	-3.49	2.53752083	0.24752083	7.22752083
3	4.45	-5.40	-0.95	3.175	2.225	0.2247191	-3.175	2.26530899	0.20280899	6.55280899
4	4.55	-5.51	-0.96	3.235	2.275	0.21978022	-3.235	2.30004945	0.20254945	6.67254945
5	4.61	-5.76	-1.15	3.455	2.305	0.21691974	-3.455	2.58937636	0.28687636	7.19687636
6	4.57	-5.73	-1.16	3.445	2.285	0.21881838	-3.445	2.59694201	0.29444201	7.18444201
7	4.87	-6.15	-1.28	3.715	2.435	0.20533881	-3.715	2.8339271	0.3364271	7.7664271
8	4.33	-5.20	-0.87	3.035	2.165	0.23094688	-3.035	2.1273037	0.1748037	6.2448037
9	4.44	-5.64	-1.20	3.42	2.22	0.22522523	-3.42	2.63432432	0.32432432	7.16432432
10	4.69	-6.18	-1.49	3.835	2.345	0.21321962	-3.835	3.13586887	0.47336887	8.14336887
11	4.49	-5.69	-1.20	3.445	2.245	0.22271715	-3.445	2.64321269	0.32071269	7.21071269

Figure 16: The energy of HOMO and LUMO B3LYP in gas phase

Conclusions

Density functional theory B3LYP/6-311G (d,p) calculations were carried out for further study on the thermodynamic properties for the molecule. The thermodynamical parameters like heat capacity, entropy and enthalpy are found increasing with the increase of the temperature (from 200 K to 1000K).

Through The optimization of the title compounds by using Lee-Yang-Parr correlation functional (B3LYP) and , our  calculations indicate that the energy gaps values between HOMO and LUMO in the tittle compounds  increased by substitution and were effected by presents of benzene ring in case of the second compound. These changes led to changes in the other chemical calculation that dependent on the energy gap values such as electrofugality ∆Ee and nucleofugality ∆En, chemical potential, chemical hardness, electrophilicity index and electronegativity.

As in future scope, the same studies can be made for the same substances or compounds but in other phases like water or solid phase, and make a comparison between them to find the most suitable situation for the compound to make the best drug delivery system for them.

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