A Derivation of the Gibbs Equation and the Determination of Change in Gibbs Entropy from Calorimetry*
Denis J. Evans A , Debra J. Searles B C and Stephen R. Williams D EA Department of Applied Mathematics, Research School of Physics, Australian National University, Canberra, ACT 0200, Australia.
B Australian Institute for Bioengineering and Nanotechnology, University of Queensland, Brisbane, Qld 4072, Australia.
C School of Chemistry and Molecular Biosciences, University of Queensland, Brisbane, Qld 4072, Australia.
D Research School of Chemistry, Australian National University, Canberra, ACT 0200, Australia.
E Corresponding author. Email: swilliams@rsc.anu.edu.au
Australian Journal of Chemistry 69(12) 1413-1419 https://doi.org/10.1071/CH16447
Submitted: 1 August 2016 Accepted: 14 November 2016 Published: 1 December 2016
Abstract
In this paper, we give a succinct derivation of the fundamental equation of classical equilibrium thermodynamics, namely the Gibbs equation. This derivation builds on our equilibrium relaxation theorem for systems in contact with a heat reservoir. We reinforce the comments made over a century ago, pointing out that Clausius’ strict inequality for a system of interest is within Clausius’ set of definitions, logically undefined. Using a specific definition of temperature that we have recently introduced and which is valid for both reversible and irreversible processes, we can define a property that we call the change in calorimetric entropy for these processes. We then demonstrate the instantaneous equivalence of the change in calorimetric entropy, which is defined using heat transfer and our definition of temperature, and the change in Gibbs entropy, which is defined in terms of the full N-particle phase space distribution function. The result shows that the change in Gibbs entropy can be expressed in terms of physical quantities.
References
[1] D. J. Evans, D. J. Searles, S. R. Williams, Fundamentals of Classical Statistical Thermodynamics, Dissipation, Relaxation and Fluctuation Theorems 2016 (Wiley-VCH Verlag: Weinheim).[2] D. J. Evans, D. J. Searles, S. R. Williams, J. Stat. Mech.: Theory Exp. 2009, 2009, P07029.
| Crossref | GoogleScholarGoogle Scholar |
[3] D. J. Evans, S. R. Williams, D. J. Searles, L. Rondoni, J. Stat. Phys. 2016, 164, 842.
| Crossref | GoogleScholarGoogle Scholar |
[4] D. J. Evans, G. P. Morriss, Statistical Mechanics of Non-Equilibrium Liquids 1990 (Academic: London).
[5] S. R. Williams, D. J. Searles, D. J. Evans, Mol. Simul. 2014, 40, 208.
| Crossref | GoogleScholarGoogle Scholar | 1:CAS:528:DC%2BC3sXitVWisrfO&md5=837641100daf044359eb2207bac6e98cCAS |
[6] (a) D. J. Evans, D. J. Searles, S. R. Williams, J. Chem. Phys. 2008, 128, 014504.
| Crossref | GoogleScholarGoogle Scholar |
(b) D. J. Evans, D. J. Searles, S. R. Williams, J. Chem. Phys. 2008, 128, 249901.
| Crossref | GoogleScholarGoogle Scholar |
[7] D. J. Evans, D. J. Searles, Adv. Phys. 2002, 51, 1529.
| Crossref | GoogleScholarGoogle Scholar |
[8] P. M. Morse, H. Feshbach, Methods of Theoretical Physics Vol. II 1953 (McGraw-Hill: New York, NY).
[9] D. J. Evans, S. R. Williams, D. J. Searles, J. Chem. Phys. 2011, 134, 204113.
| Crossref | GoogleScholarGoogle Scholar |
[10] J. C. Reid, E. M. Sevick, D. J. Evans, EPL 2005, 72, 726.
| Crossref | GoogleScholarGoogle Scholar | 1:CAS:528:DC%2BD2MXhtlehurrJ&md5=2d002070341ce72ee2219a953511ef5eCAS |
[11] G. E. Crooks, Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top. 1999, 60, 2721.
| Crossref | GoogleScholarGoogle Scholar | 1:CAS:528:DyaK1MXls1Cltrc%3D&md5=7157580c180e569ca7eb9a2db91ca24aCAS |
[12] C. Jarzynski, Phys. Rev. E: Stat. Phys., Plasmas, Fluids, Relat. Interdiscip. Top. 1997, 56, 5018.
| Crossref | GoogleScholarGoogle Scholar | 1:CAS:528:DyaK2sXntFGht7c%3D&md5=83c878f3b11cc34a8b97f06243d5540fCAS |
[13] R. Clausius, The Mechanical Theory of Heat with Its Applications to the Steam-Engine and to the Physical Properties of Bodies 1867 (Taylor and Francis: London).
[14] J. H. Irving, J. G. Kirkwood, J. Chem. Phys. 1950, 18, 817.
| Crossref | GoogleScholarGoogle Scholar | 1:CAS:528:DyaG3cXlsFSmtg%3D%3D&md5=1a71ae8cfa3431d0987708812142b776CAS |
[15] I. Muller, A History of Thermodynamics 2007 (Springer: Heidelberg).
[16] P. Ehrenfest, T. Ehrenfest, The Conceptual Foundations of the Statistical Approach to Statistical Mechanics 1990 (Dover: Mineola, NY).
[17] B. D. Butler, O. Ayton, O. J. Jepps, D. J. Evans, J. Chem. Phys. 1998, 109, 6519.
| Crossref | GoogleScholarGoogle Scholar | 1:CAS:528:DyaK1cXmsFCrurg%3D&md5=cf7f0ee7a7301245c82289e7d3de4d55CAS |
[18] (a) J. L. F. Bertrand, Thermodynamique 1887 (Gauthier-Villars: Paris).
(b) W. M. Orr, Phil. Mag. Ser. 6 1904, 8, 509.
| Crossref | GoogleScholarGoogle Scholar |
(c) E. Buckingham, Philos. Mag. Ser. 6 1905, 9, 208.
| Crossref | GoogleScholarGoogle Scholar |
(d) M. Planck, Philos. Mag. Ser. 6 1905, 9, 167.
| Crossref | GoogleScholarGoogle Scholar |
[19] D. J. Evans, S. R. Williams, L. Rondoni, J. Chem. Phys. 2012, 137, 194109.
| Crossref | GoogleScholarGoogle Scholar |
[20] J. W. Gibbs, Elementary Principles in Statistical Mechanics 1981 (Ox Bow Press: Woodbridge, CT).
[21] J. L. Lebowitz, Phys. Today 1993, 47, 115.
[22] F. Bonetto, J. L. Lebowitz, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys. 2001, 64, 056129.
| Crossref | GoogleScholarGoogle Scholar | 1:STN:280:DC%2BD38%2FivFSgsw%3D%3D&md5=d50267ca9212b49f8e9333fe85806feeCAS |
[23] R. Luzzi, A. R. Vasconcellos, J. R. Ramos, Braz. J. Phys. 2006, 36, 97.
| Crossref | GoogleScholarGoogle Scholar |
[24] D. Sands, Eur. J. Phys. 2008, 29, 129.
| Crossref | GoogleScholarGoogle Scholar |
[25] S. Hilbert, P. Hänggi, J. Dunkel, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys. 2014, 90, 062116.
| Crossref | GoogleScholarGoogle Scholar |
[26] D. Frenkel, P. B. Warren, Am. J. Phys. 2015, 83, 163.
| Crossref | GoogleScholarGoogle Scholar | 1:CAS:528:DC%2BC2MXhtFKntbk%3D&md5=827d9f7bb8b539e02746d0dec4f004a1CAS |
[27] R. de Miguel, J. M. Rubi, J. Phys. Chem. B 2016, 120, 9180.
| Crossref | GoogleScholarGoogle Scholar | 1:CAS:528:DC%2BC28Xht12jsbzM&md5=8a29dfd5da04e68739e61d221dd7f430CAS |
[28] D. J. Evans, D. J. Searles, S. R. Williams, in Diffusion Fundamentals III (Eds C. Chmelik, N. Kanellopoulos, J. Karger, T. Doros) 2009, Vol. 11, pp. 367–374 (Leipziger Universitatsverlag: Leipzig).
[29] H. R. Brown, W. Myrvold, arXiv:0809.1304 [physics.hist-ph] 2008. Available at: https://arxiv.org/abs/0809.1304 (accessed 15 November 2016).
