hrmodynamical insight on th rol of additivs in shifting th quilibrium btwn whit and gry tin Nikolay Dmntv Dpartmnt of Chmistry, mpl Univrsity, Philadlphia, PA 19122 Abstract In this study mthods of statistical thrmodynamics wr applid to whit tin gry tin quilibrium to find if thr is any corrlation btwn Dby tmpratur of an additiv and its ffct on th quilibrium. Gibbs fr nrgy of th transition was calculatd for a composit systm comprisd of tin and th additiv in assumption of Dby tmpratur of th composit systm bing a linar combination of Dby tmpraturs of th componnts with thir rspctiv molar fractions as wighing cofficints. Calculations showd that additivs which Dby tmpraturs ar abov 214 o K should stabiliz gry tin, whil th additivs with Dby tmpraturs blow 214 o K should stabiliz th whit on. Dvisd rul corroborats with all prviously rportd xprimntal data on th ffct of diffrnt additivs on th transition. Prdictions of th rul about non rportd additivs provid a man for its xprimntal vrification. KEYWORDS: whit tin, gry tin, Dby tmpratur, quilibrium, transition, statistical thrmodynamics, additivs, alloys 1
1. Introduction It is wll known that at tmpraturs blow 13.2 C, -Sn (whit tin) transforms into -Sn (gry tin).[1,2] Abrupt incras of th tin s spcific volum (~26%) during th transition usually lads to disintgration of a spcimn, an ffct also known as a tin plagu or a tin pst.[1-3] Potntial dangr a tin pst might prsnt in th constructions hld by th tin basd alloys, inspird an xtnsiv sarch for additivs that might inhibit th disadvantagous transition.[4-10] It has bn found that lad (Pb) is th bst inhibitor amongst th known ons.[4] Howvr, nvironmntal and halth considrations call to halt us of Pb, thus urging on to sarch for its rplacmnt.[4] Studis in this ara, howvr, wr mostly mpirical and basd on a try-and-s principl so far, as no lading rul in sarching for th additivs was dvisd yt.[4-10] Rsults of considration of th ffct of additiv on a whit to gry tin transition with rspct to th additiv s Dby tmpratur ar rportd hr. Gibbs fr nrgy associatd with th transformation was calculatd using th mthods of statistical thrmodynamics. It was found that additivs with Dby tmpraturs abov and blow 214 o K should stabiliz gry and whit tin, rspctivly. 2. Outlin of calculations Firstly, in ordr to chck if th calculation path is corrct, mthods of th statistical thrmodynamics will b applid to th transition without any additivs: Sn Sn (1) Scondly, th tmpratur of th zro chang of Gibbs fr nrgy will b calculatd for a composit systm with an additiv: ( Sn additiv) (2) Calculations will b conductd in a similar mannr as for an additiv fr systm, assuming that th Dby tmpratur of ach of th componnts of 2
th composit systm can b prsntd as a composit proprty constructd as a linar combination of Dby tmpratur of pur componnt and th Dby tmpratur of an additiv (Fig.1): and tin, rspctivly; A A x (3) A x (4) # of additiv atoms x (5) # of additiv atoms # of tin atoms ar Dby tmpraturs of a pur whit tin and a gry is Dby tmpratur of a pur additiv;, ar ffctiv Dby tmpraturs of (gry tin + additiv) x and (whit tin + additiv) systms, rspctivly; is a molar fraction of an additiv in composit systms. Finally, som major conclusions about th corrlation btwn Dby tmpratur of th additiv and its ffct on th transition will b mad. Figur 1. Dpndnc of th ffctiv Dby tmpratur on additiv s contnt 3
3. ransition without additivs Chang of th Gibbs fr nrgy of th systm during th hating at constant prssur for ach of th modifications can b xprssd as: G G k ln Q (6) G is Gibbs fr nrgy at tmpratur ( o K); G is Gibbs fr nrgy at 0 K; k Q is Boltzmann s constant; is a canonical partition function is an absolut tmpratur hus, th chang of th Gibbs fr nrgy for gry tin will b: And for th whit tin: G G k ln Q (7) G G k ln Q (8) And, finally, for th phas transition Sn Sn : Q G G G k ln G G (9) Q Canonical partition function, in gnral, can b xprssd as a product of th molcular partition functions: Q q q q q (10) transl rot vib lc vibrational, q transl, q rot, q vib, and q lc ar th translational, rotational, Sinc th phas transition and lctronic partition functions, corrspondingly. Sn Sn occurs in solid stat at rlativly whit modrat tmpratur, th only contribution from th vibrational mod qvib can b considrd. gray hus, on can writ down for diatomic molcul: 4
Q q vib h 1 h is Planck s constant; Eq. (11) can b rwrittn as: 2k h k is th frquncy of oscillations Q q vib 1 2 h is Dby tmpratur of th substanc k hr ar thr vibrational mods for vry atom in th crystal. hat is why partition function for 1 mol of atoms in th crystal can b prsntd as: N a Q (11) (12) 3 N a (13) q vib is Avogadro s numbr hus, q. (9) can b rwrittn as: qvib( ) Gm G ) G ) 3R ln G ) G ) (14) q vib( ) G m 1 R ln 1 2 3 G ) G ) R is a gas constant (8.31 J/mol K); subscript m dnots molar functions; is a Dby tmpratur of a whit tin (200 o K);[3,11] (15) is a Dby tmpratur of a gry tin (230 o K) [3,11] In ordr to do calculations basd on q.(15), it is ncssary to find out valu of th ) G ) G trm. his trm can b drivd from q.10 and 5
th fact that quilibrium tmpratur for th whit tin-gry tin systm is 13.2 o C ( o K) [3]: 1 2 3 ln R G ( ) ( ) 0 1 m Gm (16) G ) G ) 1 3 R ln 1 2 (17) G ) G ) 1044. 08 J mol (18) Eq.(15) can b rwrittn as: G m 1 3 ln R 1 2 1044.08 (19) h function Gm ( ), calculatd from q.(19), is shown in Figur 2. As on can s (Fig.2) th dpndnc has a positiv slop, maning that gry tin is mor stabl than th whit on at tmpraturs blow o K ( and whit tin is mor stabl than th gry on at tmpraturs abov o K ( xprimntal obsrvations.[1,2]. hs rsults ar in full agrmnt with th 6
Figur 2. Dpndnc of fr Gibbs nrgy of th transition tmpratur, calculatd from q.(19) Sn Sn on 4. ransition with an additiv In ordr to dscrib transition should b transformd into th following on: ( Sn additiv), Eq.(19) 1 3 R ln 1 2 1044.08(1 x) Eq.(20) allows on to dtrmin th chang of th fr Gibbs nrgy of th transition ( Sn additiv) at K as a function of (20) molar fraction x and th Dby tmpratur outlin of calculations). Rsults of th calculations of A of th additiv (s an as a function of A for x 0.01; 0.025; 0.05; 0.1and 0.2 ar prsntd in th Fig.3. 7
Figur 3. Dpndnc of th chang of fr Gibbs nrgy of th transition ( Sn additiv) ( Sn additiv) on Dby tmpratur of an additiv for its diffrnt molar fractions (0.2, 0.1, 0.05, 0.025, 0.1) at K On can notic that all curvs plottd in Fig.3 hav ngativ slops and intrsct A axis at ~214 o 214 K should stabiliz a whit tin ( A with 214 K should stabiliz a gry tin ( A 5. Vrifications and prdictions K, maning that all additivs with, and all additivs < Dby tmpraturs of diffrnt additivs with rspct to 214K ar prsntd in Fig.4.[11] As on can s, th prdictions of th rul, drivd abov, ar in full agrmnt with all of th xprimntal rsults found in litratur (Figur 4, solid lins). It was found that Pb, Bi, Sb, and Cd stabiliz whit tin [1,4,8], whil Al, G, Cu, Zn, and Ag stabiliz gry tin. [4,8] h rul also allows on to suggst that additivs: l, In, Yb, La, h, Au, Gd, U and Lu (Figur 4, dashd lins), which ffct on Sn Sn transition was not invstigatd yt (to th bst of th author s knowldg), will inhibit Sn Sn transition. 8
Figur 4. Dby tmpraturs of additivs, which ffct was (solid lins) and was not (dashd lins) xprimntally vrifid with rspct to 214K (dottd lin) Rfrncs 1. Emsly, J., Natur's building blocks. 2001. 2. Grnwood, N.N., Earnshaw, A., Chmistry of th lmnts. 1984. 3. Lid, D.R., CRC Handbook of Chmistry and Physics. 1997-1998(78). 4. Joo, Y.J. and. akmoto, ransformation of Sn-Cu alloy from whit tin to gry tin. Matrials Lttrs, 2002. 56(5): p. 793-796. 5. Vnuk, F., Prparation of compact alpha-in Spcimns. Journal of Crystal Growth, 1980. 48: p. 486-488. 6. Zimmrmann, H., t al., Growth of Sn thin films on Cd(111). Surfac Scinc, 1997. 377(1-3): p. 904-908. 7. Yun, W.., t al., h Growth And Elctronic-Proprtis Of Alpha-Sn hin- Films Grown On Insb(100) And (111bar) Substrats By Molcular-Bam Epitaxy (Mb). Journal Of Crystal Growth, 1991. 111(1-4): p. 943-947. 8. Kariya, Y., t al., in pst in Sn-0.5 wt.% Cu lad-fr soldr. Jom-Journal Of h Minrals Mtals & Matrials Socity, 2001. 53(6): p. 39-41. 9. Gomz, J.A., t al., hortical and xprimntal study of alpha-sn dpositd on Cd(001). Physical Rviw B, 2003. 67(11). 10. Styrkas, A.D., Growth of gry tin crystals. Inorganic Matrials, 2003. 39(7): p. 683-686. 11. Kittl, C., Introduction to solid stats physics, 2005. 9