Derivation of the Third TdS Equation in Thermodynamics

Transcription

1 Physics Journal Vol. 4, No. 2, 2018, htt:// ISSN: (Print); ISSN: (Online) Deriation of the hird ds Equation in hermodynamics Uchenna Okwudili Anekwe *, Emmanuel Hassan, Oyidi Emmanuel unde Deartment of Physics, Uniersity of Science and echnology, Aleiro, Nigeria Abstract P&V Indeendent inoles the alication of &V indeendent together with the Alication of second law of thermodynamics. he third ds equation together with the first and second ds equations has been known to many as the tedious equations due to a lot of deriations with resemblances inoled. he ds equations enables us to calculate the change of entroy during arious reersible rocesses in terms of either dv and d, or dp and d, or dv and dp, and een in terms of directly measurable quantities such as the coefficient of exansion and the bulk modulus. his Manuscrit inoles another way of deriing the hirds ds equation alying the second law of thermodynamics together with equations already deried and introduced from the deriations of &V which is also an alication of the second law of thermodynamics. Keywords Putting, Substituting, Eqn, Constant Receied: May 6, 2018 / Acceted: June 19, 2018 / Published online: August 6, 2018 he Authors. Published by American Institute of Science. his Oen Access article is under the CC BY license. htt://creatiecommons.org/licenses/by/4.0/ 1. Introduction hermodynamics is a branch of hysics concerned with heat and temerature and their relation to energy and work. It defines macroscoic ariables, such as internal energy, entroy, and ressure that artly describe a body of matter or radiation. [24] he basic idea is that objects are made u of atoms and molecules, which are in ceaseless motion. he faster the motion the hotter the object. Howeer, thermodynamics deals only with the largescale resonse of a system, i.e. resonse that can be obsered and measured, to heat flow. hermodynamics is a collection of useful mathematical relations between quantities, eery one of which is indeendently measurable. Although thermodynamics tells us nothing whatsoeer of the microscoic exlanation of macroscoic changes, it is useful because it can be used to quantify many unknowns. hermodynamics is useful recisely because some quantities are easier to measure than others. he laws of thermodynamics roide an elegant mathematical exression of some emirically-discoered facts of nature. he rincile of energy conseration allows the energy requirements for rocesses to be calculated. he rincile of increasing entroy (and the resulting free-energy minimization) allows redictions to be made of the extent to which those rocesses may roceed. hermodynamics deals with some ery abstract quantities, and makes deductions using mathematical relations. In this, it is a little like mathematics itself, which, according to Bertrand Russell, is a domain where you neer know a) what you re talking about, nor b) whether what you re saying is true. Howeer, thermodynamics is trusted as a reliable source of information about the real world, recisely because it has deliered the goods in the ast. Its ultimate justification is that it works. Confusion in thermodynamics can easily result if terms are not roerly defined. here is no room for the loose use of words in this subject. he second law of thermodynamics is arguably the most * Corresonding author address:

2 24 Uchenna Okwudili Anekwe et al.: Deriation of the hird ds Equation in hermodynamics enigmatic and roocatie fundamental statement of releance to engineering, hysics, chemistry and biology. It is instructie to recall that the discoery of this rofound rincile of nature originally emerged from the ractical concerns of nineteenth century industrialists who desired to increase the efficiency of steam engines. Although the second law of thermodynamics may be stated in a myriad of ways. he energy of the unierse is constant. he entroy of the unierse seeks a maximum. hus, the second law may be summarized by the decetiely simle inequality, 0 S Uni, ertaining to the increase in the entroy of the unierse (or of any isolated entity) roduced as a result of sontaneous (irreersible) rocesses. Although there is no question regarding the wide ranging imlications of this and other statements of the second law, it is imortant to note that none of these statements can in themseles be used to quantify the excess entroy roduced as the result of a gien irreersible rocess. ( he three ds equations hae been known to generations of students as the tedious equations though they are not at all tedious to a true loer of thermodynamics, because, among other things, they enable us to calculate the change of entroy during arious reersible rocesses in terms of either dv and d, or dp and d, or dv and dp, and een in terms of directly measurable quantities such as the coefficient of exansion and the bulk modulus. We can exress entroy in terms of any two of PV. [1] his Manuscrit inoles another method or way of deriing the hirds ds equation alying the second law of thermodynamics together with equations already deried and introduced from the deriations of &V which is also an alication of the second law of thermodynamics. roerties other than those introduced in i.e. hen since Putting eqn. (3) into eqn. (2) we e Recall that & V u u du = d + d d = d + d u u du = d d d + + indeendent u u u du = d + d + d u u u du = d d + + comaring eqn (1) with eqn (6) we'e u u u u d = d = also u u u = + (2) (3) (4) (5) (6) (7) (8) 2. Methodology his section inoles deriation of and V indeendent which is an alication of the second law of thermodynamics. he P & V Indeendent show that Solution he energy difference between two neighboring equilibrium states in which the ressure and olume differs by dand d is gien as ds = C d + C d V P u u du = d + d In equation (1) the artial deriaties do not inole any (1) C u = utting eqn (9) into eqn (7) we'e recall that u = C h = u + dh = du + d + d at constant olume and constant ressure d = 0& d = 0 eqn(12) becomes dh = du (9) (10) (11) (12) (13)

3 Physics Journal Vol. 4, No. 2, 2018, utting eqn(13) into eqn (8) we'e But we can also write h h h = + (14) s s ds = d + d (24) h at constant temerature = 0 eqn(14) becomes Comaring eqn (23) with eqn (24) we e s C = (25) h h = (15) s C = (26) recall that differentiating eqn (25) artially w.r.t we'e C h = (16) 2 2 C C s = = (27) Putting eqn ( 16) into eqn ( 15 ) we'e differentiating eqn (26) artially w.r.t we'e h = C (17) 2 2 s s C C = = = (28) From eqn ( 13 ) substituting dh for du ineqn ( 17 ) we'e equating the R.H.S of eqn (27)&(28) we'e u = C (18) 2 2 C C = (29) Hence utting eqn ( 10 ) & ( 18 ) into eqn ( 1 ) we'e From eqn ( 29 ) du = C d + C d (19) From the combined first and second law of thermodynamics we e 1 ds = + ( du d) ( ) (20) At constant olume rocess d = 0 hence eqn 20 becomes 1 ds = du Putting eqn ( 19 ) into eqn ( 21 ) we'e (21) From eqn ( 23 ) we'e C C = C = C C C ds = d + d Multilying through eqn ( 31 ) by we'e ds = C d + C d (30) (31) (32) 1 ds = C d + C d C C ds = d + d (22) (23) Laws of hermodynamics ds C d C d = + Hence eqn(33) is called the third ds equation. (33) here are four basic laws of hermodynamics, he four laws

4 26 Uchenna Okwudili Anekwe et al.: Deriation of the hird ds Equation in hermodynamics of thermodynamics define fundamental hysical quantities (temerature, energy, and entroy) that characterize thermodynamic systems at thermal equilibrium. he laws describe how these quantities behae under arious circumstances, and forbid certain henomena (such as eretual motion). he four laws of thermodynamics are: [15-19]. a. Zeroth law of thermodynamics: If two systems are in thermal equilibrium with a third system, they are in thermal equilibrium with each other. his law hels define the concet of temerature. b. First law of thermodynamics: When energy asses, as work, as heat, or with matter, into or out from a system, the system's internal energy changes in accord with the law of conseration of energy. Equialently, eretual motion machines of the first kind (machines that roduce work with no energy inut) are imossible. c. Second law of thermodynamics: In a natural thermodynamic rocess, the sum of the entroies of the interacting thermodynamic systems increases. Equialently, eretual motion machines of the second kind (machines that sontaneously conert thermal energy into mechanical work) are imossible. d. hird law of thermodynamics: he entroy of a system aroaches a constant alue as the temerature aroaches absolute zero. [16] With the excetion of non-crystalline solids (glasses) the entroy of a system at absolute zero is tyically close to zero, and is equal to the natural logarithm of the roduct of the quantum ground states. Alied Fields of hermodynamics hermodynamics has seeral Alication is different fields, amongst which we'e a. Atmosheric thermodynamics b. Biological thermodynamics c. Black hole thermodynamics d. Chemical thermodynamics e. Classical thermodynamics f. Equilibrium thermodynamics g. Industrial ecology h. Maximum entroy thermodynamics i. Non-equilibrium thermodynamics j. Philosohy of thermal and statistical hysics k. Psychrometrics l. Quantum thermodynamics m. Statistical thermodynamics n. hermoeconomics Atmosheric thermodynamics is the study of heat -to-work transformations (and their reerse) that take lace in the earth's atmoshere and manifest as weather or climate. Atmosheric thermodynamics use the laws of classical thermodynamics, to describe and exlain such henomena as the roerties of moist air, the formation of clouds, atmosheric conection, boundary layer meteorology, and ertical instabilities in the atmoshere. Atmosheric thermodynamic diagrams are used as tools in the forecasting of storm deeloment. Atmosheric thermodynamics forms a basis for cloud microhysics and conection arameterizations used in numerical weather models and is used in many climate considerations, including conectie-equilibrium climate models. Classical thermodynamics Classical thermodynamics is the descrition of the states of thermodynamic systems at near-equilibrium, that uses macroscoic, measurable roerties. It is used to model exchanges of energy, work and heat based on the laws of thermodynamics. he qualifier classical reflects the fact that it reresents the first leel of understanding of the subject as it deeloed in the 19th century and describes the changes of a system in terms of macroscoic emirical (large scale, and measurable) arameters. A microscoic interretation of these concets was later roided by the deeloment of statistical mechanics. Statistical mechanics Statistical mechanics, also called statistical thermodynamics, emerged with the deeloment of atomic and molecular theories in the late 19th century and early 20th century, and sulemented classical thermodynamics with an interretation of the microscoic interactions between indiidual articles or quantum-mechanical states. his field relates the microscoic roerties of indiidual atoms and molecules to the macroscoic, bulk roerties of materials that can be obsered on the human scale, thereby exlaining classical thermodynamics as a natural result of statistics, classical mechanics, and quantum theory at the microscoic leel. Chemical thermodynamics Chemical thermodynamics is the study of the interrelation of energy with chemical reactions or with a hysical change of state within the confines of the laws of thermodynamics.

5 Physics Journal Vol. 4, No. 2, 2018, Equilibrium thermodynamics Equilibrium thermodynamics is the systematic study of transformations of matter and energy in systems as they aroach equilibrium. he word equilibrium imlies a state of balance. In an equilibrium state there are no unbalanced otentials, or driing forces, within the system. A central aim in equilibrium thermodynamics is gien a system in a well-defined initial state, subject to accurately secified constraints, to calculate what the state of the system will be once it has reached equilibrium. Non-equilibrium thermodynamics is a branch of thermodynamics that deals with systems that are not in thermodynamic equilibrium. Most systems found in nature are not in thermodynamic equilibrium because they are not in stationary states, and are continuously and discontinuously subject to flux of matter and energy to and from other systems. he thermodynamic study of non-equilibrium systems requires more general concets than are dealt with by equilibrium thermodynamics. Many natural systems still today remain beyond the scoe of currently known macroscoic thermodynamic methods. 3. Conclusion Equation (33) show the hird ds equation deried in another way, the first, second and thirds ds equations which are alications of the second law of thermodynamics can be used to calculate using method of integration, the change of entroy between one state and another, roided that the equation of state is known in order that we can ealuate the artial deriaties. References [1] Exansion, Comression and he ds equations, Ch 13 astrowww.hys.uic.ca/~tatum/thermod/thermod13. [2] M. L. McGlashan, he use and misuse of the laws of thermodynamics, J. Chem. Educ., May 1966, Vol. 43, No. 5, [3] C. P. Snow, he two cultures and a second look, Cambridge Uniersity Press, 1964,. 15. [4] G. M. Barrow, hermodynamics should be built on energy not on heat and work, J. Chem. Educ., February 1988, Vol. 65, No. 2, [5] W. F. Harris, he lethora of equilibrium constants, Chem SA, Noember 1978, [6] G. Eriksson, hermodynamic studies of high temerature equilibria. III. SOLGAS, a comuter rogram for calculating the comosition and heat condition of an equilibrium mixture, Acta Chem. Scand., Vol. 25, 1971, [7] G. Eriksson & E. Rosen, hermodynamic studies of high temerature equilibria. VIII. General equations for the calculation of equilibria in multihase systems, Chem. Scr., Vol. 4, 1973, [8] G. Eriksson, hermodynamic studies of high temerature equilibria. XII. SOLGASMIX, a comuter rogram for calculation of equilibrium comositions in multihase systems, Chem. Scr., Vol. 8, 1975, [9] R.. Jones, Comuter simulation of rocess routes for roducing crude stainless steel, MSc (Eng) Dissertation, Uniersity of the Witwatersrand, Johannesburg, 3 October [10] R.. Jones & B. D. Botes, Descrition of non-ideal slag and metal systems by the intermediate-comound method, Proceedings of Colloquium on Ferrous Pyrometallurgy, SAIMM, Vanderbijlark, 18 Aril [11] J. W. Hastie & D. W. Bonnell, A redictie hase equilibrium model for multicomonent oxide mixtures: Part II. Oxides of Na-K-Ca-Mg-Al-Si, High em. Sci., Vol. 19, 1985, [12] J. W. Hastie, W. S. Horton, E. R. Plante, & D. W. Bonnell, hermodynamic models of alkali-metal aor transort in silicate systems, High em. High Press., Vol. 14, 1982, [13] J. W. Hastie & D. W. Bonnell, hermodynamic actiity redictions for molten slags and salts, Abstract, hird International Conference on Molten Slags and Glasses, Uniersity of Strathclyde, Glasgow, June 1988, he Institute of Metals. [14] H. Gaye, A model for the reresentation of the thermodynamic roerties of multicomonent slags, Uniersity of Strathclyde, Metallurgy Deartment, Centenary Conference, June 1984, [15] Guggenheim, E. A. (1985). hermodynamics. An Adanced reatment for Chemists and Physicists, seenth edition, North Holland, Amsterdam, ISBN [16] Kittel, C. Kroemer, H. (1980). hermal Physics, second edition, W. H. Freeman, San Francisco, ISBN [17] Adkins, C. J. (1968). Equilibrium hermodynamics, McGraw-Hill, London, ISBN [18] G., Jou, D., Casas-Vázquez, J. (2008). Understanding Non-equilibrium hermodynamics. Foundations, Alications, Frontiers, Sringer, Berlin, ISBN [19] Chris Vuille; Serway, Raymond A.; Faughn, Jerry S. (2009). College hysics. Belmont, CA: Brooks/Cole, Cengage Learning ISBN [20] H. Gaye & J. Welfringer, Proc. 2nd International Symosium on Metall. Slags and Fluxes, Ed. by H. A. Fine and D. R. Gaskell, Lake ahoe, Neada, AIME Publication, 1984, [21] Hugh D. Young and Roger A. Freedman (2012) Uniersity Physics with Modern Physics. Pg 79 13th Ed. [22] Ike E. E (2014) Essential Princiles of Physics, Jos ENIC ublishers. [23] Katz, Debora M (2016) Physics for Scientists and Engineers: Foundations and Connections, Volume 1 Published by Cengage Learning. ISBN 13:

6 28 Uchenna Okwudili Anekwe et al.: Deriation of the hird ds Equation in hermodynamics [24] O. A. Oladio (2017) PHY207 (hermodynamics) - National Oen Uniersity Aailable online at htt: // [26] Joseh M. Powers (2017) Lecture Notes on hermodynamics. [25] Lecture 3 (2017) First Law of hermodynamics Aailable online at htts://www2.h.ed.ac.uk