Get 20M+ Full-Text Papers For Less Than $1.50/day. Start a 14-Day Trial for You or Your Team.

Learn More →

Morse potential specific heat with applications: an integral equations theory based

Morse potential specific heat with applications: an integral equations theory based The specific heat in its molar form or mass form is a significant thermal property in the study of the thermal capac‑ ity of the described system. There are two basic methods for the determination of the molar specific heat capacity, one of them is the experimental procedure and the other is the theoretical procedure. The present study deals with finding a formula of the molar specific heat capacity using the theory of the integral equations for Morse interaction which is a very important potential for the study of the general oscillations in the quantum mechanics. We use the approximation (Mean‑Spherical) for finding the total energy of the compositions described by Morse interaction. We find two formulas of the heat capacity, one at a constant pressure and the other at a constant volume. We conclude that the Morse molar specific heat is temperature dependent via the inverse square low with respect to temperature. Besides, we find that the Morse molar specific heat is proportional to the square of the Morse interaction well depth. Also, we find that the Morse molar specific heat depends on the particles’ diameter, the bond distance of Morse inter ‑ action, the width parameter of Morse interaction, and the volumetric density of the system. We apply the formula of the specific heat for finding the specific heat of the vibrational part for two dimer which are the lithium and caesium dimers and for the hydrogen fluoride, hydrogen chloride, nitrogen, and hydrogen molecules. Keywords: Specific heat, Morse potential, Quantum oscillator, Fractional volume, Integral equations Introduction interaction and from this formula, we find a formula for One of the most significant thermal properties is the the molar specific-heat capacity of the Morse interaction. molar specific-heat, which is considered the main prop - The volume-heat capacity is found from the equation: erties in the calorimeters measurements. The latter C (T ) = E (1) measurements is the basic experimental procedures for finding the values of the molar specific-heat for a specific The right-side of the previous equation represents system, and also, the molar specific-heat capacity can be the first derivative of the energy with respect to the determined through a theoretical procedures derived absolute temperature. We use the previous main equa from the principles of the thermodynamics or derived tion for finding the Molar specific-heat for the Morse from the principles of the statistical mechanics which potential which is an important potential for describing we focus on. We employ the theory of the closed integral the vibrational cases, especially, in quantum mechanics equations of the statistical-mechanics for finding a for - [1]. In present work, we employ the integral equations mula of the full energy of a system described by a Morse theory solutions for the low density systems for deriving an equation of the molar specific-heat capacity for the Morse interaction where the theory of integral equations *Correspondence: mhdm‑ra@scs‑net.org; mn41@live.com Faculty of Sciences, Damascus University, Damascus, Syrian Arab Republic has multiple applications in lots of properties of the © The Author(s) 2022. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http:// creat iveco mmons. org/ licen ses/ by/4. 0/. The Creative Commons Public Domain Dedication waiver (http:// creat iveco mmons. org/ publi cdoma in/ zero/1. 0/) applies to the data made available in this article, unless otherwise stated in a credit line to the data. Al‑Raeei BMC Chemistry (2022) 16:22 Page 2 of 6 physical systems. For instance, the theory was applied where r is the distance of the particles, n is number of the * t for discussing: some polymeric systems with experimen- density, c (r) is the correlation function-direct while h (r) ** tal data [2], some diatomic molecules [3], the nematic is the total correlation function and c (r) is the correla- case [4], some electrical properties in the liquids [5], the tion function-indirect which is: hard sphere as an analytical study [6], deriving equa- ∗∗ ∗ ′ t ′ ′ tions of state for a specific type of dispersion [7 ], multi- c (r) = n c �r −�r h r d�r (4) atomic systems of the fluids [8 ], the structure-factor of a charged system [9]. In this study, we apply the mean- As we see from the two equations-3 and 4, the solu- spherical-approximation for deriving a specific heat tions of the Ornstein_Zernike equation can be found if formula of Morse potential which can be derived in mul- we employ another equation which can be resulted from tiple form such as: many approximations formulas included in the simple fluids theory such as the mean spherical approximation U (r) = K [exp (2a(r − r) − 2 exp (a(r − r)))] Morse 0 0 0 MSA and other approximations. In this work, we use (2) MS-approximation for deriving the molar specific-heat where Κ represents the energy (equilibrium) of the capacity of the Morse potential. We start from the gen- interaction potential, a is the width of well parameter eral relationship of the full energy: and r is the equilibrium bond distance. The Morse potential has multiple applications in lots of chemi- 4nπN 3Tk N E = r g(r)U (r)dr + Morse (5) cal physics subjects, for instance: the discussing of the 2 2 full thermal properties of a specific system such as the dia - mond class materials and finding the constants of the N represent a number of particles in the described sys- vibrational force and the elastic properties [10–12], the tem and the constant in the last term of the equation is discussing of the correlations in alloys phases [13], the Boltzmann constant. Now, if we use the MS-approxima- discussing of the spectral analysis [14], in the study of tion and the formula of the Morse potential in the full some quantum effects [15, 16], discussing the energy energy formula, we find: vibrational states [17, 18], discussing the alpha decay 3Tk N 4nπK N B 0 [19], the study of the structures with other potentials E = + G + G (6) 1 2 2 2 [20–22], the study of the some dimers where this poten- tial has wide applications [22, 23]. In the section-2 of this where: article, we illustrate the method of deriving of the spe- cific heat equation, and in the section-3, we discussed some aspects of the equation which we derive in addi- G = r drg (r)U (r) (7) 1 0 Morse tion to the applications of it. While in the last section of the article, we inserted some conclusions points. K K 0 0 2 2 G = [exp (2a(r − r)) − 2 exp (a(r − r))] r − r exp (2a(r − r)) + 2 r exp (a(r − r)) dr (8) 2 0 0 0 0 Tk Tk B B The heat volumetric capacity for Morse potential Or in another form: One of the most significant equations in the integral 3Tk N 4nπK N B 0 equation theory is the Ornstein_Zernike (O_Z) equation E = + G + G (9) 3 4 2 2 which describes the correlation between the particles in the system as direct correlation and indirect correlation where: and this equation is given as follows: t ∗∗ ∗ 2 2 h (r) ≡ c (r) + c (r) (3) G = r − 4 r exp (2a(r − r)) dr 3 0 (10) Tk K K 0 0 2 2 2 (11) G = r exp (4a(r − r)) + 4 r exp (3a(r − r)) − 2r exp (a(r − r)) dr 4 0 0 0 Tk Tk B B d A l‑Raeei BMC Chemistry (2022) 16:22 Page 3 of 6 Or in simpler form: And we used the following reduced Morse interaction parameters: 3Tk N 4nπK N B 0 E = + ∗∗ ∗ ay = y 2 2 (17) (12) K K K 0 0 0 1 − 4 I + 4 I − 2I − I 2 3 1 4 ∗∗ Tk Tk Tk B B B r = y + r 0 (18) where: ∗∗ ∗∗∗ y a = y (19) The integral by partition method can be applied to the ∗2 ∗ ∗ integrals in the four Eqs.  (13–16) and using the reduced I = η (r , a) dy exp −y 1 1 0 parameters we find that the full energy of the system (13) written as follows:   � � � � 1 K d d 1 1 − 4 exp (2a(r − d)) + +   2 3 2 Tk a a 2a     � �   3Tk N 4nπK N K d 2d 2 B 0   (20) E = + + 4 exp (3a(r − d)) + +   2 3 2 2 3Tk a 3a 9a  B   � � � � 2 2   d 2d 2 K d d 1 − 2 exp (a(r − d)) + + − exp (4a(r − d)) + + 0 0 2 3 2 3 a a a 4Tk a 2a 8a If we apply the first equation in the work on the equa ∗2 ∗ ∗ I = η (r , a) dy exp −2y 2 2 0 tion-20, we find that the heat volumetric capacity is given (14) as:   � � d d 1 2 exp (2a(r − d)) + +   2 3 3k N 2nπK N a a 2a B   V 0 (21) C (T ) = +   � � � � 2 2 2 k T   exp 4a r − d d d 1 4 d 2d 2 B ( ( )) + + + − exp (3a(r − d)) + + 2 3 2 3 4 a 2a 8a 3 a 3a 9a Results and discussion The Morse and the kinetic parts of the specific heat The formula (21) is the main relationship which we found ∗2 ∗ ∗ in the present work which represent the heat volumetric I = η (a, r ) dy exp −3y 3 3 0 (15) capacity. First, we can find the heat capacity at constant pressure from the relationship which relates with the heat volumetric capacity as follows: ∗4 ∗ ∗ p V 2 I = C (a, r ) dy exp −4y C (T ) − C (T ) = V χ(T )α T (22) 4 2 0 (16) Which gives us:   � � � � 2 2 d d 1 exp (4a(r − d)) d d 1 2 exp (2a(r − d)) + + + + +   2 3 2 3 3k N 2nπK N a a 2a 4 a 2a 8a B   p 0 C (T) = + (23)   � � 2 2 2 k T   B 4 d 2d 2 − exp (3a(r − d)) + + 2 3 3 a 3a 9a + V α χ(T)T Al‑Raeei BMC Chemistry (2022) 16:22 Page 4 of 6 Table 1 The specific heat of the hydroen fluoride at the room Table 3 The specific heat of the caesium dimer at the room temperature temperature. O V O V −23 t(C ) M (g/mol) ρ (g/cc) C (J/gK) t(C ) M (g/mol) C (J/gK ) × 10 25 20.0100 0.00115 0.6230 25 265.8109 9.2434 Table 2 The specific heat of the hydrogen chloride at the room Table 4 The specific heat of the lithium dimer at the room temperature. temperature. O V O V −23 t(C ) M (g/mol) ρ(g/cc) C (J/gK) t(C ) M (g/mol) C (J/gK ) × 10 25 36.46 0.00149 0.3419 25 13.8800 3.4208 low temperature range. In addition, we see that the Morse where χ(Τ) is the thermal-compressibility, while α is the part of the molar specific heat is depends on the well depth thermal expansion coefficient of the described system. of Morse potential and is proportional to the square of the However, we have two formulas of the heat capacity well depth. Also, the Morse part of the Molar specific heat (pressure and volume), the two are semi-equal for the depends on the diameter of the particles composing the compositions which had a very small value of the thermal described system, and the parameter which determine the expansion parameter, for instance, soft materials. width of the Morse potential, and the bond distance, and Besides, we find the molar formalism of the volumetric the volumetric density of the system. heat capacity as follows:   � � � � 2 2 d d 1 exp (4a(r − d)) d d 1 2 exp (2a(r − d)) + + + + +   2 3 2 3 3R 2nπRK a a 2a 4 a 2a 8a   V 0 (24) C (T ) = +  � �  2 2 2   k T 4 d 2d 2 − exp (3a(r − d)) + + 2 3 3 a 3a 9a Which represents the specific heat for a one mole of the Applications of the Morse specific heat formula described composition. As we see from the equation-24, We use the formula of the Morse specific heat which we the molar specific heat can be written as: derived for some application where we start from the hydrogen fluoride (HF) where we calculated the spe - V V V C (T ) = C (T ) + C (T ) (25) Id cific heat at the room temperature and we illustrated the result of this application in Table  1 which contains the With: molar mass of the hydrogen fluoride molecule, the den - 3R sity (volume) of the hydrogen fluoride, and the specific C (T ) ≡ (26) Id heat of the hydrogen fluoride molecule.   � � d d 1 2 exp (2a(r − d)) + +   2 3 2nπRK a a 2a   C (T ) ≡   � � � � M (27) 2 2 2   k T exp (4a(r − d)) d d 1 4 d 2d 2 + + + − exp (3a(r − d)) + + 2 3 2 3 4 a 2a 8a 3 a 3a 9a where the first part of the formula (25) represents the Besides, we use the formula for finding the specific heat kinetic molar specific heat (equation-26) and this part is of the hydrogen chloride (HCl) at the temperature and independent of the absolute temperature. Besides, the sec- we illustrated the result in the Table 2 which contains the ond part represents the molar specific heat of the Morse molar mass of the hydrogen chloride molecule, the den- interaction (equation-27), and as we see, the Morse part is sity (volume) of the hydrogen chloride, and the specific depends on the absolute temperature and this part is pro- heat of the hydrogen chloride molecule. portional to inverse of the square of the absolute tempera- The two previous applications of the formula which we ture which means that the Morse part is more effective in derived are about the diatomic and linear molecules and A l‑Raeei BMC Chemistry (2022) 16:22 Page 5 of 6 are composed of two type of atoms, in addition, we use Table 5 The specific heat of the hydrogen and nitrogen at the room temperature. the formula of the specific heat of the Morse potential part specific heat for another part of the molecules which O V t (C ) M (g/mol) ρ (g/cc) C (J/gK) is Cs2 [23] (Caesium dimer) which is composed of one Hydrogen 25 2.0160 0.0009 6.1834 type of atoms. We calculated the Morse part specific heat Nitrogen 25 28.0140 0.0013 0.4450 for this dimer and the result of this calculation contained in Table 3 which includes the molar mass and the specific heat of this dimer. The result of the specific heat of the caesium dimer can Conclusions be compared of the value resulted from the study [23] We derived a formula of the Morse potential specific heat where we see that the two vibrational part of the specific using the theory of the integral equations. We employed heat are near to each other. Also, we use the formula of the mean-spherical-approximation for that purpose. the specific heat of Morse potential for the lithium dimer First, we found a formula for the total energy of the com- for finding the vibrational part of the specific heat of this position described with Morse interaction, and based dimer at the room temperature and we illustrated the on the energy formula, we derived the required formula result of this calculation in Table  4 which includes the of the molar specific heat for the Morse interaction. We molar mass and the specific heat of this dimer. derived two formulas of the heat capacity for the Morse Finally, we use the formula of the specific heat for interaction, one for the constant pressure and the other finding the specific heat at the room temperature of the for the constant volume. hydrogen [25] and nitrogen [24] molecules. We found that, the molar specific heat capacity of the As we see from the last table (Table  5), the specific Morse potential is depends on the absolute temperature heat of the hydrogen molecule and nitrogen molecule at of the systems via the inverse-square low of the absolute the room temperature have the same rank of the values temperature. Also, we found that the specific molar heat found in the literatures. of the Morse interaction is function to the particles’ diam- eter, the bond distance of the Morse interaction, the width The ratio γ of the well parameter, the volumetric density of the sys- Additional significant property of the material which tem, and the depth of the Morse well. We found that the is the ratio γ can be found from the formula which we Morse molar specific heat is proportional to the square of derived in this work where this property represents the the depth well of the Morse interaction. We applied the ratio between the heat capacity at constant V and at con- formula for six different molecules which are the lithium stant P and defined as: and caesium dimers, the hydrogen fluoride, hydrogen chloride, nitrogen, and hydrogen molecules and we found C (T ) γ = (28) that the values are near the values in other literatures. C (T ) The derived formula in the present work is applied which becomes given by the following formula using the for finding the specific heat capacity for the oscilla - two equations-21 and 23: tions part as a general case which is represented via the V α χ(T)T γ = 1+  � � 2 2 3k N 2nπK N d d 1 + (2 exp (2a(r − d))) + +   2 2 3 (29) 2 k T a a 2a  B    � � � � 2 2   exp (4a(r − d)) d d 1 4 d 2d 2 + + + − exp (3a(r − d)) + + 2 3 2 3 4 a 2a 8a 3 a 3a 9a As we see from the ratio γ of the Morse potential the Morse potential, for instance, the diatomic molecules as ratio depends on the Morse parameters and absolute the hydrogen chloride molecule and hydrogen fluoride temperature of the system in addition to the Bulk modu- molecule. lus of the system at this temperature. Al‑Raeei BMC Chemistry (2022) 16:22 Page 6 of 6 Acknowledgements 12. Nguyen DB, Trinh HP. High‑pressure study of thermodynamic parameters Not applicable. of diamond‑type structured crystals using interatomic morse potentials. J Eng Appl Sci. 2021. https:// doi. org/ 10. 1186/ s44147‑ 021‑ 00015‑x. Authors’ contributions 13. Duc NB. Dependence on temperature and pressure of second cumulant MA contributed to the conceptualization, methodology and derivation of the and correlation effects during EXAFS in intermetallic alloys. Heliyon. 2021. method, software, visualization, writing and editing of the article. All authors https:// doi. org/ 10. 1016/j. heliy on. 2021. e08157. read and approved the final manuscript. 14. Awad L, ElK ‑ ork N, Chamieh G, Korek M. Theoretical electronic structure with rovibrational studies of the molecules YP, YP+ and YP. Spec ‑ trochim ActaA: ‑ Funding Mol Biomol Spectrosc. 2022. https:// doi. org/ 10. 1016/j. saa. 2021. 120544. Not applicable. 15. Abebe OJ, Obeten OP, Okorie US, Ikot AN. Spin and pseudospin sym‑ metries of the dirac equation for the generalised morse potential and a Availability of data and materials class of Yukawa potential. Pramana–J Phys. 2021. https:// doi. org/ 10. 1007/ All data generated or analysed during this study are included in this published s12043‑ 021‑ 02131‑y. article. 16. Singh CA, Devi OB. Ladder operators for the kratzer oscillator and the morse potential. Int J Quantum Chem. 2006;106(2):415–25. https:// doi. org/ 10. 1002/ qua. 20775. Declarations 17. Rasheed NK, Ayaash AN. A comparative study of potential energy curves ana‑ lytical representations for co, n2, p2, and scf in their ground electronic states. Ethics approval and consent to participate Iraqi J Sci. 2021;62(8):2536–42. https:// doi. org/ 10. 24996/ ijs. 2021. 62.8.6. Not applicable. 18. Belfakir A, Hassouni Y, Curado EMF. Construction of coherent states for morse potential: a su(2)‑like approach. Phys Lett A: Gen At Solid State Consent for publication Phys. 2020. https:// doi. org/ 10. 1016/j. physl eta. 2020. 126553. Not applicable. 19. Koyuncu F. A new potential model for alpha decay calculations. Nucl Phys A. 2021. https:// doi. org/ 10. 1016/j. nuclp hysa. 2021. 122211. Competing interests 20. Jacobson DW, Thompson GB. Revisting lennard jones, morse, and N‑M Competing interest is not found. potentials for metals. Comput Mater Sci. 2022. https:// doi. org/ 10. 1016/j. comma tsci. 2022. 111206. Received: 3 January 2022 Accepted: 15 March 2022 21. Al‑Raeei M. Applying fractional quantum mechanics to systems with electrical screening effects. Chaos, Solitons Fractals. 2021. https:// doi. org/ 10. 1016/j. chaos. 2021. 111209. 22. Pingak RK, Johannes AZ, Ngara ZS, Bukit M, Nitti F, Tambaru D, Ndii MZ. Accuracy of morse and morse‑like oscillators for diatomic molecular References interaction: a comparative study. Results Chem. 2021. https:// doi. org/ 10. 1. Al‑Raeei M. The use of classical and quantum theoretical Physics methods 1016/j. rechem. 2021. 100204. in the study of complex systems and their applications in condensed 23. Horchani R, Jelassi H. Eec ff t of quantum corrections on thermodynamic matter Physics and quantum information theory. Damascus, the Syrian properties for dimers. Chem Phys. 2020. https:// doi. org/ 10. 1016/j. chemp Arab Republic, PhD thesis. 2021. hys. 2020. 110692. 2. Benoît H, Benmouna M, Wu W. Static scattering from multicomponent 24. Gyuk NJ, Abubakar MS, Abdu SG. Using asymptotic iteration method polymer and copolymer systems. Macromolecules. 1990;23(5):1511–7. (aim) for solving differential equations: case study of vibrational states of https:// doi. org/ 10. 1021/ ma002 07a045. diatomic molecule. Sci World J. 2019;14:3. 3. D’yakonov SG, D’yazaveev KN, D’yakonov GS. Calculation of the ther‑ 25. Horchani R, Jelassi H. The rotating Morse potential model for diatomic modynamic properties of diatomic substances based on the ornstein‑ molecules in the tridiagonal J‑matrix representation: I Bound states. J zernike equation. Teplofizika Vysokikh Temperatur. 2004;42(2):221–7. Phys B: At Mol Opt Phys. 2007;40:4245–57. https:// doi. org/ 10. 1088/ 0953‑ 4. Holovko MF, Sokolovska TG. Analytical solution of the ornstein‑zernike 4075/ 40/ 21/ 011. equation with the mean spherical closure for a nematic phase. J Mol Liq. 1999;82(3):161–81. https:// doi. org/ 10. 1016/ s0167‑ 7322(99) 00098‑7. Publisher’s Note 5. Krienke H. Thermodynamical, structural, and dielectric properties of Springer Nature remains neutral with regard to jurisdictional claims in pub‑ molecular liquids from integral equation theories and from simulations. lished maps and institutional affiliations. Pure Appl Chem. 2004;76(1):63–70. https:// doi. org/ 10. 1351/ pac20 04760 6. Mier Y, Teran L, Corvera E, Gonzalez AE. Analytical solution of the mean‑ spherical approximation for a system of hard spheres with a surface adhesion. Phys Rev A. 1989;39(1):371–3. https:// doi. org/ 10. 1103/ PhysR evA. 39. 371. 7. Al‑Raeei M. An equation of state for london dispersion interaction with thermodynamic inconsistent terms. Results Chem. 2022. https:// doi. org/ 10. 1016/j. rechem. 2022. 100296. Re Read ady y to to submit y submit your our re researc search h ? Choose BMC and benefit fr ? Choose BMC and benefit from om: : 8. Ohba M, Arakawa K. The asymptotic behavior of the site‑site ornstein‑ zernike equation and RISM‑2 approximations for polyatomic molecular fast, convenient online submission fluids. J Phys Soc Jpn. 1986;55(9):2955–62. https:// doi. org/ 10. 1143/ JPSJ. 55. 2955. thorough peer review by experienced researchers in your field 9. Wu C, Chan DYC, Tabor RF. A simple and accurate method for calculation rapid publication on acceptance of the structure factor of interacting charged spheres. J Colloid Interface support for research data, including large and complex data types Sci. 2014;426:80–2. https:// doi. org/ 10. 1016/j. jcis. 2014. 03. 023. 10. Tarabichi S, Achkar B, El‑Daher MS. Physics for the prearatory year of • gold Open Access which fosters wider collaboration and increased citations medical colleges. the Syrian Arab Republic: Ministry of Higher Eduaction maximum visibility for your research: over 100M website views per year Publishing; 2016. 11. Al‑Raeei M. Bulk modulus for Morse potential interaction with the dis‑ At BMC, research is always in progress. tribution function based. Chem Thermodyn Therm Anal. 2022;6:100046. https:// doi. org/ 10. 1016/j. ctta. 2022. 100046. Learn more biomedcentral.com/submissions http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png BMC Chemistry Springer Journals

Morse potential specific heat with applications: an integral equations theory based

BMC Chemistry , Volume 16 (1) – Mar 27, 2022

Loading next page...
 
/lp/springer-journals/morse-potential-specific-heat-with-applications-an-integral-equations-U6GWfNg1of

References (29)

Publisher
Springer Journals
Copyright
Copyright © The Author(s) 2022
eISSN
2661-801X
DOI
10.1186/s13065-022-00811-3
Publisher site
See Article on Publisher Site

Abstract

The specific heat in its molar form or mass form is a significant thermal property in the study of the thermal capac‑ ity of the described system. There are two basic methods for the determination of the molar specific heat capacity, one of them is the experimental procedure and the other is the theoretical procedure. The present study deals with finding a formula of the molar specific heat capacity using the theory of the integral equations for Morse interaction which is a very important potential for the study of the general oscillations in the quantum mechanics. We use the approximation (Mean‑Spherical) for finding the total energy of the compositions described by Morse interaction. We find two formulas of the heat capacity, one at a constant pressure and the other at a constant volume. We conclude that the Morse molar specific heat is temperature dependent via the inverse square low with respect to temperature. Besides, we find that the Morse molar specific heat is proportional to the square of the Morse interaction well depth. Also, we find that the Morse molar specific heat depends on the particles’ diameter, the bond distance of Morse inter ‑ action, the width parameter of Morse interaction, and the volumetric density of the system. We apply the formula of the specific heat for finding the specific heat of the vibrational part for two dimer which are the lithium and caesium dimers and for the hydrogen fluoride, hydrogen chloride, nitrogen, and hydrogen molecules. Keywords: Specific heat, Morse potential, Quantum oscillator, Fractional volume, Integral equations Introduction interaction and from this formula, we find a formula for One of the most significant thermal properties is the the molar specific-heat capacity of the Morse interaction. molar specific-heat, which is considered the main prop - The volume-heat capacity is found from the equation: erties in the calorimeters measurements. The latter C (T ) = E (1) measurements is the basic experimental procedures for finding the values of the molar specific-heat for a specific The right-side of the previous equation represents system, and also, the molar specific-heat capacity can be the first derivative of the energy with respect to the determined through a theoretical procedures derived absolute temperature. We use the previous main equa from the principles of the thermodynamics or derived tion for finding the Molar specific-heat for the Morse from the principles of the statistical mechanics which potential which is an important potential for describing we focus on. We employ the theory of the closed integral the vibrational cases, especially, in quantum mechanics equations of the statistical-mechanics for finding a for - [1]. In present work, we employ the integral equations mula of the full energy of a system described by a Morse theory solutions for the low density systems for deriving an equation of the molar specific-heat capacity for the Morse interaction where the theory of integral equations *Correspondence: mhdm‑ra@scs‑net.org; mn41@live.com Faculty of Sciences, Damascus University, Damascus, Syrian Arab Republic has multiple applications in lots of properties of the © The Author(s) 2022. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http:// creat iveco mmons. org/ licen ses/ by/4. 0/. The Creative Commons Public Domain Dedication waiver (http:// creat iveco mmons. org/ publi cdoma in/ zero/1. 0/) applies to the data made available in this article, unless otherwise stated in a credit line to the data. Al‑Raeei BMC Chemistry (2022) 16:22 Page 2 of 6 physical systems. For instance, the theory was applied where r is the distance of the particles, n is number of the * t for discussing: some polymeric systems with experimen- density, c (r) is the correlation function-direct while h (r) ** tal data [2], some diatomic molecules [3], the nematic is the total correlation function and c (r) is the correla- case [4], some electrical properties in the liquids [5], the tion function-indirect which is: hard sphere as an analytical study [6], deriving equa- ∗∗ ∗ ′ t ′ ′ tions of state for a specific type of dispersion [7 ], multi- c (r) = n c �r −�r h r d�r (4) atomic systems of the fluids [8 ], the structure-factor of a charged system [9]. In this study, we apply the mean- As we see from the two equations-3 and 4, the solu- spherical-approximation for deriving a specific heat tions of the Ornstein_Zernike equation can be found if formula of Morse potential which can be derived in mul- we employ another equation which can be resulted from tiple form such as: many approximations formulas included in the simple fluids theory such as the mean spherical approximation U (r) = K [exp (2a(r − r) − 2 exp (a(r − r)))] Morse 0 0 0 MSA and other approximations. In this work, we use (2) MS-approximation for deriving the molar specific-heat where Κ represents the energy (equilibrium) of the capacity of the Morse potential. We start from the gen- interaction potential, a is the width of well parameter eral relationship of the full energy: and r is the equilibrium bond distance. The Morse potential has multiple applications in lots of chemi- 4nπN 3Tk N E = r g(r)U (r)dr + Morse (5) cal physics subjects, for instance: the discussing of the 2 2 full thermal properties of a specific system such as the dia - mond class materials and finding the constants of the N represent a number of particles in the described sys- vibrational force and the elastic properties [10–12], the tem and the constant in the last term of the equation is discussing of the correlations in alloys phases [13], the Boltzmann constant. Now, if we use the MS-approxima- discussing of the spectral analysis [14], in the study of tion and the formula of the Morse potential in the full some quantum effects [15, 16], discussing the energy energy formula, we find: vibrational states [17, 18], discussing the alpha decay 3Tk N 4nπK N B 0 [19], the study of the structures with other potentials E = + G + G (6) 1 2 2 2 [20–22], the study of the some dimers where this poten- tial has wide applications [22, 23]. In the section-2 of this where: article, we illustrate the method of deriving of the spe- cific heat equation, and in the section-3, we discussed some aspects of the equation which we derive in addi- G = r drg (r)U (r) (7) 1 0 Morse tion to the applications of it. While in the last section of the article, we inserted some conclusions points. K K 0 0 2 2 G = [exp (2a(r − r)) − 2 exp (a(r − r))] r − r exp (2a(r − r)) + 2 r exp (a(r − r)) dr (8) 2 0 0 0 0 Tk Tk B B The heat volumetric capacity for Morse potential Or in another form: One of the most significant equations in the integral 3Tk N 4nπK N B 0 equation theory is the Ornstein_Zernike (O_Z) equation E = + G + G (9) 3 4 2 2 which describes the correlation between the particles in the system as direct correlation and indirect correlation where: and this equation is given as follows: t ∗∗ ∗ 2 2 h (r) ≡ c (r) + c (r) (3) G = r − 4 r exp (2a(r − r)) dr 3 0 (10) Tk K K 0 0 2 2 2 (11) G = r exp (4a(r − r)) + 4 r exp (3a(r − r)) − 2r exp (a(r − r)) dr 4 0 0 0 Tk Tk B B d A l‑Raeei BMC Chemistry (2022) 16:22 Page 3 of 6 Or in simpler form: And we used the following reduced Morse interaction parameters: 3Tk N 4nπK N B 0 E = + ∗∗ ∗ ay = y 2 2 (17) (12) K K K 0 0 0 1 − 4 I + 4 I − 2I − I 2 3 1 4 ∗∗ Tk Tk Tk B B B r = y + r 0 (18) where: ∗∗ ∗∗∗ y a = y (19) The integral by partition method can be applied to the ∗2 ∗ ∗ integrals in the four Eqs.  (13–16) and using the reduced I = η (r , a) dy exp −y 1 1 0 parameters we find that the full energy of the system (13) written as follows:   � � � � 1 K d d 1 1 − 4 exp (2a(r − d)) + +   2 3 2 Tk a a 2a     � �   3Tk N 4nπK N K d 2d 2 B 0   (20) E = + + 4 exp (3a(r − d)) + +   2 3 2 2 3Tk a 3a 9a  B   � � � � 2 2   d 2d 2 K d d 1 − 2 exp (a(r − d)) + + − exp (4a(r − d)) + + 0 0 2 3 2 3 a a a 4Tk a 2a 8a If we apply the first equation in the work on the equa ∗2 ∗ ∗ I = η (r , a) dy exp −2y 2 2 0 tion-20, we find that the heat volumetric capacity is given (14) as:   � � d d 1 2 exp (2a(r − d)) + +   2 3 3k N 2nπK N a a 2a B   V 0 (21) C (T ) = +   � � � � 2 2 2 k T   exp 4a r − d d d 1 4 d 2d 2 B ( ( )) + + + − exp (3a(r − d)) + + 2 3 2 3 4 a 2a 8a 3 a 3a 9a Results and discussion The Morse and the kinetic parts of the specific heat The formula (21) is the main relationship which we found ∗2 ∗ ∗ in the present work which represent the heat volumetric I = η (a, r ) dy exp −3y 3 3 0 (15) capacity. First, we can find the heat capacity at constant pressure from the relationship which relates with the heat volumetric capacity as follows: ∗4 ∗ ∗ p V 2 I = C (a, r ) dy exp −4y C (T ) − C (T ) = V χ(T )α T (22) 4 2 0 (16) Which gives us:   � � � � 2 2 d d 1 exp (4a(r − d)) d d 1 2 exp (2a(r − d)) + + + + +   2 3 2 3 3k N 2nπK N a a 2a 4 a 2a 8a B   p 0 C (T) = + (23)   � � 2 2 2 k T   B 4 d 2d 2 − exp (3a(r − d)) + + 2 3 3 a 3a 9a + V α χ(T)T Al‑Raeei BMC Chemistry (2022) 16:22 Page 4 of 6 Table 1 The specific heat of the hydroen fluoride at the room Table 3 The specific heat of the caesium dimer at the room temperature temperature. O V O V −23 t(C ) M (g/mol) ρ (g/cc) C (J/gK) t(C ) M (g/mol) C (J/gK ) × 10 25 20.0100 0.00115 0.6230 25 265.8109 9.2434 Table 2 The specific heat of the hydrogen chloride at the room Table 4 The specific heat of the lithium dimer at the room temperature. temperature. O V O V −23 t(C ) M (g/mol) ρ(g/cc) C (J/gK) t(C ) M (g/mol) C (J/gK ) × 10 25 36.46 0.00149 0.3419 25 13.8800 3.4208 low temperature range. In addition, we see that the Morse where χ(Τ) is the thermal-compressibility, while α is the part of the molar specific heat is depends on the well depth thermal expansion coefficient of the described system. of Morse potential and is proportional to the square of the However, we have two formulas of the heat capacity well depth. Also, the Morse part of the Molar specific heat (pressure and volume), the two are semi-equal for the depends on the diameter of the particles composing the compositions which had a very small value of the thermal described system, and the parameter which determine the expansion parameter, for instance, soft materials. width of the Morse potential, and the bond distance, and Besides, we find the molar formalism of the volumetric the volumetric density of the system. heat capacity as follows:   � � � � 2 2 d d 1 exp (4a(r − d)) d d 1 2 exp (2a(r − d)) + + + + +   2 3 2 3 3R 2nπRK a a 2a 4 a 2a 8a   V 0 (24) C (T ) = +  � �  2 2 2   k T 4 d 2d 2 − exp (3a(r − d)) + + 2 3 3 a 3a 9a Which represents the specific heat for a one mole of the Applications of the Morse specific heat formula described composition. As we see from the equation-24, We use the formula of the Morse specific heat which we the molar specific heat can be written as: derived for some application where we start from the hydrogen fluoride (HF) where we calculated the spe - V V V C (T ) = C (T ) + C (T ) (25) Id cific heat at the room temperature and we illustrated the result of this application in Table  1 which contains the With: molar mass of the hydrogen fluoride molecule, the den - 3R sity (volume) of the hydrogen fluoride, and the specific C (T ) ≡ (26) Id heat of the hydrogen fluoride molecule.   � � d d 1 2 exp (2a(r − d)) + +   2 3 2nπRK a a 2a   C (T ) ≡   � � � � M (27) 2 2 2   k T exp (4a(r − d)) d d 1 4 d 2d 2 + + + − exp (3a(r − d)) + + 2 3 2 3 4 a 2a 8a 3 a 3a 9a where the first part of the formula (25) represents the Besides, we use the formula for finding the specific heat kinetic molar specific heat (equation-26) and this part is of the hydrogen chloride (HCl) at the temperature and independent of the absolute temperature. Besides, the sec- we illustrated the result in the Table 2 which contains the ond part represents the molar specific heat of the Morse molar mass of the hydrogen chloride molecule, the den- interaction (equation-27), and as we see, the Morse part is sity (volume) of the hydrogen chloride, and the specific depends on the absolute temperature and this part is pro- heat of the hydrogen chloride molecule. portional to inverse of the square of the absolute tempera- The two previous applications of the formula which we ture which means that the Morse part is more effective in derived are about the diatomic and linear molecules and A l‑Raeei BMC Chemistry (2022) 16:22 Page 5 of 6 are composed of two type of atoms, in addition, we use Table 5 The specific heat of the hydrogen and nitrogen at the room temperature. the formula of the specific heat of the Morse potential part specific heat for another part of the molecules which O V t (C ) M (g/mol) ρ (g/cc) C (J/gK) is Cs2 [23] (Caesium dimer) which is composed of one Hydrogen 25 2.0160 0.0009 6.1834 type of atoms. We calculated the Morse part specific heat Nitrogen 25 28.0140 0.0013 0.4450 for this dimer and the result of this calculation contained in Table 3 which includes the molar mass and the specific heat of this dimer. The result of the specific heat of the caesium dimer can Conclusions be compared of the value resulted from the study [23] We derived a formula of the Morse potential specific heat where we see that the two vibrational part of the specific using the theory of the integral equations. We employed heat are near to each other. Also, we use the formula of the mean-spherical-approximation for that purpose. the specific heat of Morse potential for the lithium dimer First, we found a formula for the total energy of the com- for finding the vibrational part of the specific heat of this position described with Morse interaction, and based dimer at the room temperature and we illustrated the on the energy formula, we derived the required formula result of this calculation in Table  4 which includes the of the molar specific heat for the Morse interaction. We molar mass and the specific heat of this dimer. derived two formulas of the heat capacity for the Morse Finally, we use the formula of the specific heat for interaction, one for the constant pressure and the other finding the specific heat at the room temperature of the for the constant volume. hydrogen [25] and nitrogen [24] molecules. We found that, the molar specific heat capacity of the As we see from the last table (Table  5), the specific Morse potential is depends on the absolute temperature heat of the hydrogen molecule and nitrogen molecule at of the systems via the inverse-square low of the absolute the room temperature have the same rank of the values temperature. Also, we found that the specific molar heat found in the literatures. of the Morse interaction is function to the particles’ diam- eter, the bond distance of the Morse interaction, the width The ratio γ of the well parameter, the volumetric density of the sys- Additional significant property of the material which tem, and the depth of the Morse well. We found that the is the ratio γ can be found from the formula which we Morse molar specific heat is proportional to the square of derived in this work where this property represents the the depth well of the Morse interaction. We applied the ratio between the heat capacity at constant V and at con- formula for six different molecules which are the lithium stant P and defined as: and caesium dimers, the hydrogen fluoride, hydrogen chloride, nitrogen, and hydrogen molecules and we found C (T ) γ = (28) that the values are near the values in other literatures. C (T ) The derived formula in the present work is applied which becomes given by the following formula using the for finding the specific heat capacity for the oscilla - two equations-21 and 23: tions part as a general case which is represented via the V α χ(T)T γ = 1+  � � 2 2 3k N 2nπK N d d 1 + (2 exp (2a(r − d))) + +   2 2 3 (29) 2 k T a a 2a  B    � � � � 2 2   exp (4a(r − d)) d d 1 4 d 2d 2 + + + − exp (3a(r − d)) + + 2 3 2 3 4 a 2a 8a 3 a 3a 9a As we see from the ratio γ of the Morse potential the Morse potential, for instance, the diatomic molecules as ratio depends on the Morse parameters and absolute the hydrogen chloride molecule and hydrogen fluoride temperature of the system in addition to the Bulk modu- molecule. lus of the system at this temperature. Al‑Raeei BMC Chemistry (2022) 16:22 Page 6 of 6 Acknowledgements 12. Nguyen DB, Trinh HP. High‑pressure study of thermodynamic parameters Not applicable. of diamond‑type structured crystals using interatomic morse potentials. J Eng Appl Sci. 2021. https:// doi. org/ 10. 1186/ s44147‑ 021‑ 00015‑x. Authors’ contributions 13. Duc NB. Dependence on temperature and pressure of second cumulant MA contributed to the conceptualization, methodology and derivation of the and correlation effects during EXAFS in intermetallic alloys. Heliyon. 2021. method, software, visualization, writing and editing of the article. All authors https:// doi. org/ 10. 1016/j. heliy on. 2021. e08157. read and approved the final manuscript. 14. Awad L, ElK ‑ ork N, Chamieh G, Korek M. Theoretical electronic structure with rovibrational studies of the molecules YP, YP+ and YP. Spec ‑ trochim ActaA: ‑ Funding Mol Biomol Spectrosc. 2022. https:// doi. org/ 10. 1016/j. saa. 2021. 120544. Not applicable. 15. Abebe OJ, Obeten OP, Okorie US, Ikot AN. Spin and pseudospin sym‑ metries of the dirac equation for the generalised morse potential and a Availability of data and materials class of Yukawa potential. Pramana–J Phys. 2021. https:// doi. org/ 10. 1007/ All data generated or analysed during this study are included in this published s12043‑ 021‑ 02131‑y. article. 16. Singh CA, Devi OB. Ladder operators for the kratzer oscillator and the morse potential. Int J Quantum Chem. 2006;106(2):415–25. https:// doi. org/ 10. 1002/ qua. 20775. Declarations 17. Rasheed NK, Ayaash AN. A comparative study of potential energy curves ana‑ lytical representations for co, n2, p2, and scf in their ground electronic states. Ethics approval and consent to participate Iraqi J Sci. 2021;62(8):2536–42. https:// doi. org/ 10. 24996/ ijs. 2021. 62.8.6. Not applicable. 18. Belfakir A, Hassouni Y, Curado EMF. Construction of coherent states for morse potential: a su(2)‑like approach. Phys Lett A: Gen At Solid State Consent for publication Phys. 2020. https:// doi. org/ 10. 1016/j. physl eta. 2020. 126553. Not applicable. 19. Koyuncu F. A new potential model for alpha decay calculations. Nucl Phys A. 2021. https:// doi. org/ 10. 1016/j. nuclp hysa. 2021. 122211. Competing interests 20. Jacobson DW, Thompson GB. Revisting lennard jones, morse, and N‑M Competing interest is not found. potentials for metals. Comput Mater Sci. 2022. https:// doi. org/ 10. 1016/j. comma tsci. 2022. 111206. Received: 3 January 2022 Accepted: 15 March 2022 21. Al‑Raeei M. Applying fractional quantum mechanics to systems with electrical screening effects. Chaos, Solitons Fractals. 2021. https:// doi. org/ 10. 1016/j. chaos. 2021. 111209. 22. Pingak RK, Johannes AZ, Ngara ZS, Bukit M, Nitti F, Tambaru D, Ndii MZ. Accuracy of morse and morse‑like oscillators for diatomic molecular References interaction: a comparative study. Results Chem. 2021. https:// doi. org/ 10. 1. Al‑Raeei M. The use of classical and quantum theoretical Physics methods 1016/j. rechem. 2021. 100204. in the study of complex systems and their applications in condensed 23. Horchani R, Jelassi H. Eec ff t of quantum corrections on thermodynamic matter Physics and quantum information theory. Damascus, the Syrian properties for dimers. Chem Phys. 2020. https:// doi. org/ 10. 1016/j. chemp Arab Republic, PhD thesis. 2021. hys. 2020. 110692. 2. Benoît H, Benmouna M, Wu W. Static scattering from multicomponent 24. Gyuk NJ, Abubakar MS, Abdu SG. Using asymptotic iteration method polymer and copolymer systems. Macromolecules. 1990;23(5):1511–7. (aim) for solving differential equations: case study of vibrational states of https:// doi. org/ 10. 1021/ ma002 07a045. diatomic molecule. Sci World J. 2019;14:3. 3. D’yakonov SG, D’yazaveev KN, D’yakonov GS. Calculation of the ther‑ 25. Horchani R, Jelassi H. The rotating Morse potential model for diatomic modynamic properties of diatomic substances based on the ornstein‑ molecules in the tridiagonal J‑matrix representation: I Bound states. J zernike equation. Teplofizika Vysokikh Temperatur. 2004;42(2):221–7. Phys B: At Mol Opt Phys. 2007;40:4245–57. https:// doi. org/ 10. 1088/ 0953‑ 4. Holovko MF, Sokolovska TG. Analytical solution of the ornstein‑zernike 4075/ 40/ 21/ 011. equation with the mean spherical closure for a nematic phase. J Mol Liq. 1999;82(3):161–81. https:// doi. org/ 10. 1016/ s0167‑ 7322(99) 00098‑7. Publisher’s Note 5. Krienke H. Thermodynamical, structural, and dielectric properties of Springer Nature remains neutral with regard to jurisdictional claims in pub‑ molecular liquids from integral equation theories and from simulations. lished maps and institutional affiliations. Pure Appl Chem. 2004;76(1):63–70. https:// doi. org/ 10. 1351/ pac20 04760 6. Mier Y, Teran L, Corvera E, Gonzalez AE. Analytical solution of the mean‑ spherical approximation for a system of hard spheres with a surface adhesion. Phys Rev A. 1989;39(1):371–3. https:// doi. org/ 10. 1103/ PhysR evA. 39. 371. 7. Al‑Raeei M. An equation of state for london dispersion interaction with thermodynamic inconsistent terms. Results Chem. 2022. https:// doi. org/ 10. 1016/j. rechem. 2022. 100296. Re Read ady y to to submit y submit your our re researc search h ? Choose BMC and benefit fr ? Choose BMC and benefit from om: : 8. Ohba M, Arakawa K. The asymptotic behavior of the site‑site ornstein‑ zernike equation and RISM‑2 approximations for polyatomic molecular fast, convenient online submission fluids. J Phys Soc Jpn. 1986;55(9):2955–62. https:// doi. org/ 10. 1143/ JPSJ. 55. 2955. thorough peer review by experienced researchers in your field 9. Wu C, Chan DYC, Tabor RF. A simple and accurate method for calculation rapid publication on acceptance of the structure factor of interacting charged spheres. J Colloid Interface support for research data, including large and complex data types Sci. 2014;426:80–2. https:// doi. org/ 10. 1016/j. jcis. 2014. 03. 023. 10. Tarabichi S, Achkar B, El‑Daher MS. Physics for the prearatory year of • gold Open Access which fosters wider collaboration and increased citations medical colleges. the Syrian Arab Republic: Ministry of Higher Eduaction maximum visibility for your research: over 100M website views per year Publishing; 2016. 11. Al‑Raeei M. Bulk modulus for Morse potential interaction with the dis‑ At BMC, research is always in progress. tribution function based. Chem Thermodyn Therm Anal. 2022;6:100046. https:// doi. org/ 10. 1016/j. ctta. 2022. 100046. Learn more biomedcentral.com/submissions

Journal

BMC ChemistrySpringer Journals

Published: Mar 27, 2022

Keywords: Specific heat; Morse potential; Quantum oscillator; Fractional volume; Integral equations

There are no references for this article.