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Estimation of Thermodynamic Properties of Binary Liquid Mixtures on the Basis of Statistical Mechanical Theories

Year 2013, Volume: 16 Issue: 1, 10 - 19, 14.12.2012

J. D. Pandey Prakash Chandra Rupali Sethi Vinay Sanguri

Abstract

Thermodynamic properties of liquids and liquid mixtures play very important role in understanding the nature of molecular interactions occurring in the system. In the present work different thermodynamic properties of 15 pure liquids and 34 equimolar binary liquid mixtures of benzene, toluene, p-xylene, chlorobenzene and 1-chloronaphthalene with linear and branched alkanes have been computed with the help of Flory’s statistical theory (FST), Hard sphere equation of state (HSE) and Hole theory (HT) simultaneously. The calculated values are compared with the experimental findings collected from literature and quite satisfactory results are obtained.

Keywords

Flory statistical theory; hard sphere equation of state; hole theory; binary liquid mixtures

References

  • Abe, A. & Flory, P. J. (1965). The Thermodynamic Properties of Mixtures of Small, Nonpolar Molecules. J. Am. Chem. Soc., 87, 1838-1846.
  • Acosta, J., Acre, A., Rodil, E., & Soto, A.(2001). Densities, Speeds of Sound, Refractive indices and the corresponding changes of Mixing at 25 atmospheric pressure for systems composed by Ethyl acetate, Hexane and Acetone. J. Chem. Eng. Data, 46(5), 1176-1180. C and
  • Acree (Jr), W. E. (1984). Thermodynamic Properties of Nonelectrolyte Solutions, I N C London Ltd: Academic Press.
  • Ali, A. & Nain, A. K. (2000). Study of molecular interactions in non-aqueous binary liquid mixtures through ultrasonic measurements. J. Pure Appl. Ultrason., 22, 10-15.
  • Ali, A., & Nain, A.K. (2002).Ultrsonic & Volumetric Study of Binary Mixtures of Benzyl Alcohol with Amides. Bull.Chem. Soc.Japan, 75(4), 681-687.
  • Ali, A. & Nain, A. K. (2002). Ultrasonic study of molecular interaction in binary liquid mixtures at 30oC. Pramana 58(4), 695-701.
  • Ali, A., Nain, A. K., Sharma, V. K. & Ahmad, S. (2000). Acoustical studies of molecular interaction in binary liquid mixtures at 35o C. J. Acoust. Soc. India, 28, 283- 288.
  • Ali, A., Tiwari, K., Nain, A.K., & Chakravortty, V. (2000). Ultrasonic study of molecular interaction in ternary mixtures of dimethyl sulphoxides (1) + carbon tetrachloride (2) + aromatic hydrocarbons (3) at 308.15K. Indian J. Phys., 74B, 351-355.
  • rce. ., rce, . r., geitos, . ., odil, E., odr gue , ., Soto, A. (2000). . Physical and equilibrium properties of diisopropyl ether+isopropyl alcohol+water system. Fluid Phase Equilibria,170, 113- 126.
  • Baluja, S. (2002). Ultrasonic studies of resorcinol in protic & aprotic solvents at 40o. J. Indian Chem. Soc., 79, 142- 144.
  • Bloomfeld, V. A. & Dewan, R. K. (1971). Viscosity of liquid mixtures. J Phys. Chem., 75, 3113-3119.
  • Canosa, J., odr ´gue , ., Tojo, . (2001). Liquid-Liquid Equilibrium and Physical Properties of the Ternary Mixture (Dimethyl Cyclohexane) at 298.15 K. J. Chem. Eng. Data, 46, 846-850. + Methanol +
  • Carnahan, N. F., Starling, K. E. (1969). Equation of State for Nonattracting Rigid Spheres. J. Chem. Phys., 51, 635-636.
  • Costas, M., Tra, H. V., Patterson, D., Alonso, M. C., Tardajos, G. & Aicart, E. (1988). Liquid structure and second-order mixing functions for 1-chloronaphthalene with linear and branched alkanes. J. Chem. Soc. Faraday Trans 1, 84, 1603-1616.
  • CRC Handbook of Chem. & Phys., (1997-98). (78th Edition) New York: CRC Press, Boca Raton.
  • Dewan, R. K., Bloomfeld, V. A. & Berget, P. B. (1971). Intrinsic viscosity of short-chain normal alkanes. J. Phys. Chem., 75, 3120-3124.
  • Dey, R., Soni, N. K., Mishra, R. K., Sanguri, V. & Pandey, J. D. (2006). Thermal conductivity of ternary liquid mixtures: Application of modified Flory statistical theory and uniform gaseous model. J. Mol. Liq., 124, 102-105.
  • Dhasi, V., Wadhawani, R., Akhtar, Y., Pandey, J.D. & Vyas,V. (1998). Ultrasonic and Thermodynamic Studies of Ternary Solutions. Acustica-Acta Acustica, 84, 976-979.
  • Dominguez, A., Tardajos, G., Aicart, E., Casas, S. P., Trejo, L. M., Costas, M., Patterson, D. & Tra, H. V. (1993). Van der Waals liquids, Flory theory and mixing functions for chlorobenzene with linear and branched alkanes. J. Chem. Soc. Faraday Trans 1, 89, 89-94.
  • Eyring, H. & John, M. S. (1969). Significant Liquid Structures, New York: John Wiley & Sons Inc.
  • Flory, P. J. (1965). Statistical Thermodynamics of Liquid Mixtures. J. Am. Chem. Soc., 87, 1833-1838.
  • Flory, P. J., Orwoll, R. A. & Vrij, A. (1964). Statistical Thermodynamics of Chain Molecule Liquids. I. An Equation of State for Normal Paraffin Hydrocarbons. J. Am. Chem. Soc., 86, 3507–3514.
  • Frisch HL, (1964). The equation of state of the classical hard sphere fluids. Adv Chem. Phys, 6, 229-289.
  • Hoover, W.G. & Ree, F.H, (1964). Fifth and Sixth Virial Coefficients for Hard Spheres and Hard Disks. J. Chem. Phys., 40, 939-950.
  • Jain, D.V.S., Alastair, M. N., & Pethrick, R.A. (1974). Adiabatic compressibility of binary liquid mixtures. J. Chem. Soc., Faraday Trans 1,70, 1292- 1298.
  • Jakobson, B. (1951). Intermolecular Free Lengths in Liquids in Relation to Compressibility, Surface Tension and Viscosity. Acta Chem. Scad.,5, 1214-1216.
  • Kaulgud, M.V. & Patil, K. J. (1972). Sound propagation in some mono and disubstituted derivatives of benzene. Acustica, 26, 292-296.
  • Lebowitz, J.L., Frisch, H. L. & Helfand, E. (1961). Theory of the Two‐ and One‐Dimensional Rigid Sphere Fluids. J. Chem. Phys., 34, 1037-1042.
  • Lebowitz, J. L., Helfand, E. & Praestgaard, E. (1965). Scaled Particle Theory of Fluid Mixtures. J. Chem. Phys., 43, 774-778.
  • Mosteriro, L., Mascato, E., deConinges, B.E., Iglesias, T.P., & Legido, J.L. (2001). Density, speed of sound, refractive index and dielectric permittivity of (diethyl carbonate + n-decane) at several temperatures. J. Chem. Thermodyn., 33, 787-801.
  • Murrell, J. N. & Jenkins, A. D. (1982). Properties of Liquids & Solutions, New York: John Willy & Sons.
  • Nomoto, O. (1958). Empirical Formula for Sound Velocity in Liquid Mixtures. J. Phys. Soc. Jpn. 13, 1528-1532.
  • Pandey, J. D. & Alec David, D. M. (1981). Viscosities of binary molten nitrate mixtures. J. Phys. Chem., 85, 3151-3152.
  • Pandey, J. D., Dey, R & Dwivedi, D. K. (1999). Ultrasonic velocity of binary systems at elevated pressures. Pramana-j phys, 52, 187-193.
  • Pandey, J. D., Dubey, G. P., Dey, R. & Dubey, S. N. (1997). Temperature and Pressure Dependence of Thermoacoustical Parameters of Liquid Argon and Xenon. Acustica, 83, 90-92.
  • Pandey, J. D., Dubey, G. P., Shukla, B. P. & Dubey, S.N. (1994). A Comparative Study of Isothermal Compressibility from Ultrasonic Velocity and Flory's Statistical Mixtures. Acustica, 80, 92-96. for Various Binary
  • Pandey, J. D. & Sanguri, V. (2001). Prediction of density of liquid mixture using Flory Statistical Theory. J. Chem. Research (S), 344-345.
  • Pandey J. D. & Sanguri, V. (2008). Theoretical estimations of thermodynamic properties of liquid mixtures by Flory’s statistical theory. Phys. Chem. Liq., 46, 417- 432.
  • Pandey, J. D., Sanguri, V. & Bhatt, B. D. (2003). pplication of Flory’s statistical theory for the estimation of internal pressure of the liquid mixtures. J. Chem. Research(S), 430-432.
  • Pandey, J. D., Sanguri, V. & Dwivedi, D. K. (2008). Thermodynamic properties of pure liquids within a generalized version of the hole theory. Phys. Chem. Liq., 46, 162-171.
  • Pandey, J. D., Sanguri, V., Dwivedi, D. K. & Tiwari K. K. (2007). Computation of isothermal compressibility, thermal expansivity and ultrasonic velocity of binary liquid mixtures using hole theory. J. Mol. Liq., 135, 65- 71.
  • Pandey, J. D., Srivastava, T., Chandra P., Dwivedi, D. K. & Sanguri, V. (2010). Estimation of thermodynamic properties of multicomponent systems on the basis of generalized version of hole theory. J. Mol. Liq., 157, 158-161.
  • Pandey, J. D., Tripathi, S. B. & Sanguri, V. (2002). Thermal Expansivity of multi-component liquid systems. J. Mol. Liq., 100/2, 153-161.
  • Pandey, J. D., Vyas, V., Jain, P., Dubey, G. P., Tripathi, N. & Dey. R. (1999). Speed of sound, viscosity and refractive index of multicomponent systems: theoretical predictions from the properties of pure components. J. Mol. Liq. 81, 123-133.
  • Percus, J. K. & Yevick, G.J. (1958). Analysis of Classical by Statistical Coordinates. Phy. Rev, 110, 1-13. Means of Collective
  • Prausnitz, J. M., Lichtenthaler, R. N. & de Azevedo, (1986). Molecular Thermodynamic of Fluid Phase Equilibria, Englewood Cliffs, N J: Printice-Hall Ind.
  • Reid, R. C., Prausnitz, J. M. & Poling, B. E. (1987). The Properties of Gases and Liquids, New York: Mc-Graw Hill.
  • Rowlinson, J. S. & Swinton F. L. (1982). Liquids & Liquid Mixture, London: Butter worth scientific.
  • Sanguri, V., Dwivedi, D. K., Singh, N., Pandey, N. & Pandey, J. D. (2008). Thermodynamic properties of multicomponent systems and hole theory. J. Mol. Liq., 141, 1-7.
  • Schaffs, W. (1939). Zur Bestimmung von Molekülradien organischer Flüssigkeiten aus Schallgeschwindigkeit und Dichte. Z. Phys., 114, 110-115.
  • Tardajos, G., Aicart, E., Costas, M. & Patterson, D. (1986). Liquid structure and second-order mixing functions for benzene, toluene and p-xylene with n-alkanes J. Chem. Soc. Faraday Trans. 1, 82, 2977- 2988.
  • Thiele, E. J. (1963). Equation of State for Hard Spheres. J. Chem. Phys., 39, 474-478.
  • Van Dael, W. (1975). Thermodynamics properties and velocity of sound, London: Butterworth.

Year 2013, Volume: 16 Issue: 1, 10 - 19, 14.12.2012

J. D. Pandey Prakash Chandra Rupali Sethi Vinay Sanguri

Abstract

References

  • Abe, A. & Flory, P. J. (1965). The Thermodynamic Properties of Mixtures of Small, Nonpolar Molecules. J. Am. Chem. Soc., 87, 1838-1846.
  • Acosta, J., Acre, A., Rodil, E., & Soto, A.(2001). Densities, Speeds of Sound, Refractive indices and the corresponding changes of Mixing at 25 atmospheric pressure for systems composed by Ethyl acetate, Hexane and Acetone. J. Chem. Eng. Data, 46(5), 1176-1180. C and
  • Acree (Jr), W. E. (1984). Thermodynamic Properties of Nonelectrolyte Solutions, I N C London Ltd: Academic Press.
  • Ali, A. & Nain, A. K. (2000). Study of molecular interactions in non-aqueous binary liquid mixtures through ultrasonic measurements. J. Pure Appl. Ultrason., 22, 10-15.
  • Ali, A., & Nain, A.K. (2002).Ultrsonic & Volumetric Study of Binary Mixtures of Benzyl Alcohol with Amides. Bull.Chem. Soc.Japan, 75(4), 681-687.
  • Ali, A. & Nain, A. K. (2002). Ultrasonic study of molecular interaction in binary liquid mixtures at 30oC. Pramana 58(4), 695-701.
  • Ali, A., Nain, A. K., Sharma, V. K. & Ahmad, S. (2000). Acoustical studies of molecular interaction in binary liquid mixtures at 35o C. J. Acoust. Soc. India, 28, 283- 288.
  • Ali, A., Tiwari, K., Nain, A.K., & Chakravortty, V. (2000). Ultrasonic study of molecular interaction in ternary mixtures of dimethyl sulphoxides (1) + carbon tetrachloride (2) + aromatic hydrocarbons (3) at 308.15K. Indian J. Phys., 74B, 351-355.
  • rce. ., rce, . r., geitos, . ., odil, E., odr gue , ., Soto, A. (2000). . Physical and equilibrium properties of diisopropyl ether+isopropyl alcohol+water system. Fluid Phase Equilibria,170, 113- 126.
  • Baluja, S. (2002). Ultrasonic studies of resorcinol in protic & aprotic solvents at 40o. J. Indian Chem. Soc., 79, 142- 144.
  • Bloomfeld, V. A. & Dewan, R. K. (1971). Viscosity of liquid mixtures. J Phys. Chem., 75, 3113-3119.
  • Canosa, J., odr ´gue , ., Tojo, . (2001). Liquid-Liquid Equilibrium and Physical Properties of the Ternary Mixture (Dimethyl Cyclohexane) at 298.15 K. J. Chem. Eng. Data, 46, 846-850. + Methanol +
  • Carnahan, N. F., Starling, K. E. (1969). Equation of State for Nonattracting Rigid Spheres. J. Chem. Phys., 51, 635-636.
  • Costas, M., Tra, H. V., Patterson, D., Alonso, M. C., Tardajos, G. & Aicart, E. (1988). Liquid structure and second-order mixing functions for 1-chloronaphthalene with linear and branched alkanes. J. Chem. Soc. Faraday Trans 1, 84, 1603-1616.
  • CRC Handbook of Chem. & Phys., (1997-98). (78th Edition) New York: CRC Press, Boca Raton.
  • Dewan, R. K., Bloomfeld, V. A. & Berget, P. B. (1971). Intrinsic viscosity of short-chain normal alkanes. J. Phys. Chem., 75, 3120-3124.
  • Dey, R., Soni, N. K., Mishra, R. K., Sanguri, V. & Pandey, J. D. (2006). Thermal conductivity of ternary liquid mixtures: Application of modified Flory statistical theory and uniform gaseous model. J. Mol. Liq., 124, 102-105.
  • Dhasi, V., Wadhawani, R., Akhtar, Y., Pandey, J.D. & Vyas,V. (1998). Ultrasonic and Thermodynamic Studies of Ternary Solutions. Acustica-Acta Acustica, 84, 976-979.
  • Dominguez, A., Tardajos, G., Aicart, E., Casas, S. P., Trejo, L. M., Costas, M., Patterson, D. & Tra, H. V. (1993). Van der Waals liquids, Flory theory and mixing functions for chlorobenzene with linear and branched alkanes. J. Chem. Soc. Faraday Trans 1, 89, 89-94.
  • Eyring, H. & John, M. S. (1969). Significant Liquid Structures, New York: John Wiley & Sons Inc.
  • Flory, P. J. (1965). Statistical Thermodynamics of Liquid Mixtures. J. Am. Chem. Soc., 87, 1833-1838.
  • Flory, P. J., Orwoll, R. A. & Vrij, A. (1964). Statistical Thermodynamics of Chain Molecule Liquids. I. An Equation of State for Normal Paraffin Hydrocarbons. J. Am. Chem. Soc., 86, 3507–3514.
  • Frisch HL, (1964). The equation of state of the classical hard sphere fluids. Adv Chem. Phys, 6, 229-289.
  • Hoover, W.G. & Ree, F.H, (1964). Fifth and Sixth Virial Coefficients for Hard Spheres and Hard Disks. J. Chem. Phys., 40, 939-950.
  • Jain, D.V.S., Alastair, M. N., & Pethrick, R.A. (1974). Adiabatic compressibility of binary liquid mixtures. J. Chem. Soc., Faraday Trans 1,70, 1292- 1298.
  • Jakobson, B. (1951). Intermolecular Free Lengths in Liquids in Relation to Compressibility, Surface Tension and Viscosity. Acta Chem. Scad.,5, 1214-1216.
  • Kaulgud, M.V. & Patil, K. J. (1972). Sound propagation in some mono and disubstituted derivatives of benzene. Acustica, 26, 292-296.
  • Lebowitz, J.L., Frisch, H. L. & Helfand, E. (1961). Theory of the Two‐ and One‐Dimensional Rigid Sphere Fluids. J. Chem. Phys., 34, 1037-1042.
  • Lebowitz, J. L., Helfand, E. & Praestgaard, E. (1965). Scaled Particle Theory of Fluid Mixtures. J. Chem. Phys., 43, 774-778.
  • Mosteriro, L., Mascato, E., deConinges, B.E., Iglesias, T.P., & Legido, J.L. (2001). Density, speed of sound, refractive index and dielectric permittivity of (diethyl carbonate + n-decane) at several temperatures. J. Chem. Thermodyn., 33, 787-801.
  • Murrell, J. N. & Jenkins, A. D. (1982). Properties of Liquids & Solutions, New York: John Willy & Sons.
  • Nomoto, O. (1958). Empirical Formula for Sound Velocity in Liquid Mixtures. J. Phys. Soc. Jpn. 13, 1528-1532.
  • Pandey, J. D. & Alec David, D. M. (1981). Viscosities of binary molten nitrate mixtures. J. Phys. Chem., 85, 3151-3152.
  • Pandey, J. D., Dey, R & Dwivedi, D. K. (1999). Ultrasonic velocity of binary systems at elevated pressures. Pramana-j phys, 52, 187-193.
  • Pandey, J. D., Dubey, G. P., Dey, R. & Dubey, S. N. (1997). Temperature and Pressure Dependence of Thermoacoustical Parameters of Liquid Argon and Xenon. Acustica, 83, 90-92.
  • Pandey, J. D., Dubey, G. P., Shukla, B. P. & Dubey, S.N. (1994). A Comparative Study of Isothermal Compressibility from Ultrasonic Velocity and Flory's Statistical Mixtures. Acustica, 80, 92-96. for Various Binary
  • Pandey, J. D. & Sanguri, V. (2001). Prediction of density of liquid mixture using Flory Statistical Theory. J. Chem. Research (S), 344-345.
  • Pandey J. D. & Sanguri, V. (2008). Theoretical estimations of thermodynamic properties of liquid mixtures by Flory’s statistical theory. Phys. Chem. Liq., 46, 417- 432.
  • Pandey, J. D., Sanguri, V. & Bhatt, B. D. (2003). pplication of Flory’s statistical theory for the estimation of internal pressure of the liquid mixtures. J. Chem. Research(S), 430-432.
  • Pandey, J. D., Sanguri, V. & Dwivedi, D. K. (2008). Thermodynamic properties of pure liquids within a generalized version of the hole theory. Phys. Chem. Liq., 46, 162-171.
  • Pandey, J. D., Sanguri, V., Dwivedi, D. K. & Tiwari K. K. (2007). Computation of isothermal compressibility, thermal expansivity and ultrasonic velocity of binary liquid mixtures using hole theory. J. Mol. Liq., 135, 65- 71.
  • Pandey, J. D., Srivastava, T., Chandra P., Dwivedi, D. K. & Sanguri, V. (2010). Estimation of thermodynamic properties of multicomponent systems on the basis of generalized version of hole theory. J. Mol. Liq., 157, 158-161.
  • Pandey, J. D., Tripathi, S. B. & Sanguri, V. (2002). Thermal Expansivity of multi-component liquid systems. J. Mol. Liq., 100/2, 153-161.
  • Pandey, J. D., Vyas, V., Jain, P., Dubey, G. P., Tripathi, N. & Dey. R. (1999). Speed of sound, viscosity and refractive index of multicomponent systems: theoretical predictions from the properties of pure components. J. Mol. Liq. 81, 123-133.
  • Percus, J. K. & Yevick, G.J. (1958). Analysis of Classical by Statistical Coordinates. Phy. Rev, 110, 1-13. Means of Collective
  • Prausnitz, J. M., Lichtenthaler, R. N. & de Azevedo, (1986). Molecular Thermodynamic of Fluid Phase Equilibria, Englewood Cliffs, N J: Printice-Hall Ind.
  • Reid, R. C., Prausnitz, J. M. & Poling, B. E. (1987). The Properties of Gases and Liquids, New York: Mc-Graw Hill.
  • Rowlinson, J. S. & Swinton F. L. (1982). Liquids & Liquid Mixture, London: Butter worth scientific.
  • Sanguri, V., Dwivedi, D. K., Singh, N., Pandey, N. & Pandey, J. D. (2008). Thermodynamic properties of multicomponent systems and hole theory. J. Mol. Liq., 141, 1-7.
  • Schaffs, W. (1939). Zur Bestimmung von Molekülradien organischer Flüssigkeiten aus Schallgeschwindigkeit und Dichte. Z. Phys., 114, 110-115.
  • Tardajos, G., Aicart, E., Costas, M. & Patterson, D. (1986). Liquid structure and second-order mixing functions for benzene, toluene and p-xylene with n-alkanes J. Chem. Soc. Faraday Trans. 1, 82, 2977- 2988.
  • Thiele, E. J. (1963). Equation of State for Hard Spheres. J. Chem. Phys., 39, 474-478.
  • Van Dael, W. (1975). Thermodynamics properties and velocity of sound, London: Butterworth.
There are 53 citations in total.

Details

Primary Language English
Authors

J. D. Pandey This is me

Prakash Chandra This is me

Rupali Sethi This is me

Vinay Sanguri

Publication Date December 14, 2012
Published in Issue Year 2013 Volume: 16 Issue: 1

Cite

APA Pandey, J. D., Chandra, P., Sethi, R., Sanguri, V. (2012). Estimation of Thermodynamic Properties of Binary Liquid Mixtures on the Basis of Statistical Mechanical Theories. International Journal of Thermodynamics, 16(1), 10-19.
AMA Pandey JD, Chandra P, Sethi R, Sanguri V. Estimation of Thermodynamic Properties of Binary Liquid Mixtures on the Basis of Statistical Mechanical Theories. International Journal of Thermodynamics. December 2012;16(1):10-19.
Chicago Pandey, J. D., Prakash Chandra, Rupali Sethi, and Vinay Sanguri. “Estimation of Thermodynamic Properties of Binary Liquid Mixtures on the Basis of Statistical Mechanical Theories”. International Journal of Thermodynamics 16, no. 1 (December 2012): 10-19.
EndNote Pandey JD, Chandra P, Sethi R, Sanguri V (December 1, 2012) Estimation of Thermodynamic Properties of Binary Liquid Mixtures on the Basis of Statistical Mechanical Theories. International Journal of Thermodynamics 16 1 10–19.
IEEE J. D. Pandey, P. Chandra, R. Sethi, and V. Sanguri, “Estimation of Thermodynamic Properties of Binary Liquid Mixtures on the Basis of Statistical Mechanical Theories”, International Journal of Thermodynamics, vol. 16, no. 1, pp. 10–19, 2012.
ISNAD Pandey, J. D. et al. “Estimation of Thermodynamic Properties of Binary Liquid Mixtures on the Basis of Statistical Mechanical Theories”. International Journal of Thermodynamics 16/1 (December2012), 10-19.
JAMA Pandey JD, Chandra P, Sethi R, Sanguri V. Estimation of Thermodynamic Properties of Binary Liquid Mixtures on the Basis of Statistical Mechanical Theories. International Journal of Thermodynamics. 2012;16:10–19.
MLA Pandey, J. D. et al. “Estimation of Thermodynamic Properties of Binary Liquid Mixtures on the Basis of Statistical Mechanical Theories”. International Journal of Thermodynamics, vol. 16, no. 1, 2012, pp. 10-19.
Vancouver Pandey JD, Chandra P, Sethi R, Sanguri V. Estimation of Thermodynamic Properties of Binary Liquid Mixtures on the Basis of Statistical Mechanical Theories. International Journal of Thermodynamics. 2012;16(1):10-9.