Temperature dependent study of volume and thermal expansivity of solids based on equation of state

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1 Indian Journal of Pure & Applied Physics ol. 47, August 009, pp emperature dependent study of volume and ermal expansivity of solids based on equation of state Kamal Kapoor & Narsingh Dass Physics Department, College of Engineering oorkee, ardhamanp Puram, oorkee eceived, 7 October 008; revised 6 March 009; accepted June 009 Five different approaches have been used to obtain simple relations to study e temperature dependent volume and ermal expansivity of e solids. All e relations are found to be identical. he computation has been done on six geological solids and ree metals wi e help of ese relations. All e relations are quite successful in representing e volume and e ermal expansivity as a function of temperature. However, e relation obtained wi e help of ait EOS (Equation of State is found to be better. Keyword: hermal expansivity, Equation of state, hermodynamical properties Introduction he study of ermo-physical and ermodynamical properties of e solids are of great importance as far as e validity of high-temperature and high-pressure equation of state (EOS of solid is concerned. Among e ermo-physical properties, e ermal expansion coefficient plays e central role. Many relations are available for e study of volume as a function of temperature. However, ere is a lot of confusion in e sense at every one claims his own derivation as a representative of a new relation. ecently Kumar has shown at Suzuki relation as well as ait,4 EOS can be used to obtain relation for e ermal expansion coefficient of e solids. he relations obtained appear to be somewhat cumbersome. In e present paper, simple relations for e study of e volume and e ermal expansion coefficient as a function of temperature of solids have been obtained. heory For analysis of e results, five different approaches, namely, ait,4, Murnaghan 5, Suzuki, Born 7 and Kuchhal and Dass 0 have been considered.. ait approach ait EOS,4,6 is expressed as: ( 0, P, = ln + A ( C P ( where A and C are constants, (P, and (0, represent e volume at pressure P=P and P=0, at temperature, respectively. Successive differentiation of Eq. ( wi pressure will give e relations for bulk modulus and its first pressure derivative at pressure P and temperature. Putting P = 0 in ese relations, one obtains e value of A = B (0, + and A C = B ( 0,. o convert Eq. ( into temperature-dependent EOS, e ermal effect is introduced into e pressure as: P( = P P ( where P is e ermal pressure and at ambient temperature it is expressed as: ( 0, ( 0, P = α B. Substituting Eq. ( into Eq. (, one gets ( 0, P, ln CP( = A +. ( In Eq. (, A B ( = 0, + and C A =. B 0, Putting P = 0 in Eq. (, one gets

2 KAPOO & DASS: OLUME AND HEMAL EXPANSIIY OF SOLIDS 59 ( ( 0, 0, ln A 0, = A α (4 Kumar claimed at Eq. ( is a new EOS obtained by him which is beyond e tru as presented here [also see ef. 6]. Following approximations will be used as and when needed. X ln X (5 / 4KE + K Q = ( 0, K A B 0, 0, where K = and Q = γ 0, Here γ (0, is e Gruneisen parameter. Furer relation for e ermal energy is expressed as:. and ( Y / ln( Y (6 Using Eq. (5, Eq. (4 can finally be written as: P γ ( 0, E = ( A A ( 0, ( (7 = α Differentiation of Eq. (7 wi temperature gives e expression for e ermal expansion as: α( 0, A α 0,. Murnaghan approach Murnaghan isoermal EOS 5 is written as: ( A A P, = + P 0, B 0, (8 (9 Converting Eq. (9 to temperature dependent EOS wi e help of Eq. ( and putting P = 0, one gets: ( A A ( 0, ( (0 = α Differentiation of Eq. (0 wi temperature gives: α( 0, ( A ( ( α 0, (. Suzuki approach Suzuki et al. have given a relation to study e temperature dependence of volume in solids. his relation at P = 0 is expressed as: Putting e values of gets: K, E and Q into Eq. (, one { ( A ( } / α 0, = 0, ( A Using of Eqs (5 and (6, one obtains: (4 ( A A ( 0, ( (5 = α Differentiation of Eq. (5 wi temperature gives e relation: α( 0, ( A ( ( α (0, = α 0, (6 Eq. (6 is a simple relation as compared to relation given by Kumar..4 Born approach Shanker et al. 8 have introduced higher order terms for e change in volume in e expansion of potential energy of Born model 7 to obtain e temperature dependence of e volume of solids and is expressed as: A P B (0, = (7 A ( 0, /

3 594 INDIAN J PUE & APPL PHYS, OL 47, AUGUS 009 Making use of Eqs (, (5 and (6, one gets: A A ( 0, ( (8 = α It is clear from e above discussion at Eqs (8 and (7 are e same. Differentiation of Eq. (8 wi respect to temperature gives e same result for ermal expansion coefficient as represented by Eq. (8..5 Kuchhal and Dass approach Kuchhal and Dass 0 isoermal EOS: have given e following P, ξ B 0, + ξ ξ = ln + P 0, B 0, + ξ B 0, Here ξ =, which is a constant. Above EOS can be converted into temperature dependent EOS wi e help of Eq. ( and is given as: (, ξ = ( 0, ( 0, P B + ξ B 0, + ξ P α 0, ξ ln + B B ( P, ( (9 his relation at P = 0 is expressed as: ξ = ( 0, ( 0, B + ξ B 0, + ξ ln α B ( B ( 0, { B ( } = / ξ 0, + ξ { B } / ln α ( 0, ( ( 0, + ξ( Using e following approximation ln ( X = X ( ( ξ + α( 0, B ( 0, = + ξ ( B + ξ( = + ξα 0, 0, = 0, 0, + α ( ( (0 his is e simplest relation and gives volume as a function of temperature. he importance of Eq. (0 is due to e fact at ere is only one adjustable able oot mean square deviations (MSD of volume and ermal expansion coefficient of geological solids Solid MgO Mg SiO 4 Al O CaO Grossular Garnet Pyrope Garnet emperature range (K α(0, (0 5 K ρ(0, (gm/cc B (0, Eq. (7 MSD in ρ( Eq. (8 MSD in α( Eq. (0 MSD in ρ( Eq. ( MSD in α( Eq. (5 MSD in ρ( Eq. (6 MSD in α( Eq. (0 MSD in ρ(

4 KAPOO & DASS: OLUME AND HEMAL EXPANSIIY OF SOLIDS 595 parameter. Differentiation of Eq. (9 wi temperature gives e relation: α( 0, ( B ( ( α 0, 0, 4 Conclusions From Figs and, it is clear at computed values of density/volume expansions have good agreement wi e experimental data. Also, calculated values of aking A = B (0, +, above relation changes to: α( 0, ( A ( ( ( α 0, which is similar to e Eq. (, obtained by Murnaghan approach. esults and Discussion he present relations are applied in six geological solids and ree metals because e relevant data required for computation are readily available. For e geological solids data are taken from ef. (9 and for metals from ef. (. he density of six geological solids and volume expansion / 0 of ree metals computed wi e help of Eqs (7, (0, (5 and (0. Similarly e ermal expansion coefficient of geological solids is also computed wi e help of Eqs (8, ( and (6. he required parameter for density and ermal expansion coefficient computation along wi root mean square deviation (MSD are given in able for geological solids. he required parameters for volume expansion along wi root mean square deviation (MSD for ree metals are given in able. Fig. ariation of density wi temperature using different approaches in six geological solids able oot mean square deviations (MSD in volume expansion of metals Metals Copper Gold anadium emperature range (K α(0, (0 5 K B (0, Eq. (7 MSD in / 0 ( 0 Eq. (0 MSD in / 0 ( 0 Eq. (5 MSD in / 0 ( 0 Eq. (0 MSD in / 0 ( Fig. ariation of volume expansion / 0 wi temperature using different approaches in ree metals

5 596 INDIAN J PUE & APPL PHYS, OL 47, AUGUS 009 ermal expansion coefficients have reasonable agreement wi experimental data. oot mean square deviation (MSD, as given in able for geological solids and in able for metals, is minimum in case of ait approach. It has been found at all e relations are quite successful in representing e volume and e ermal expansion coefficient. However, e relation given by ait approach appears to be e best among all e relations. Acknowledgement One of e auors (KK expresses his sincere anks to e Director General, College of Engineering oorkee, oorkee for providing him financial support to carry out is research work. eferences Kumar M, Indian J Pure & Appl Phys, 4 ( Suzuki I, Okajima S & Seya K, J Phys Ear, 7 ( Gibson E & Loffer O H, J Am Chem Soc, 6 ( ait P J, Phys Chem, ( Murnaghan F D, Finite deformation of an elastic solid (John Wiley and Sons, Inc., New York, 95, Chap 4. 6 Schlosser H & Ferrante J, J Phys: Condens Matter, ( Born M & Huang K, Dynamical heory of Crystal Lattice (Clarendon,Oxford, England, 954, Chap. 8 Shanker J, Kushwaha S S & Kumar P, Physica B, ( Anderson O L, Issaak D & Oda H, ev Geophy, 0 ( Kuchhal P & Dass N, Pramana J Phys, 6 ( McQueen G & Marsh S P, J Appl Phys, ( Kumar M, Physica B, (9959.

NEW RELATION BETWEEN VOLUME THERMAL EXPANSION AND THERMAL PRESSURE FOR SOLIDS

NEW RELATION BETWEEN VOLUME THERMAL EXANSION AND THERMAL RESSURE FOR SOLIDS Sumiti Narayan Tewari 1, Anita Shukla 1, Department of hysics, ranveer Singh Institute of Technology, Bhauti, Kanpur 8 U.. (India)