Betfa Perlmuner-Hayman
The Hebrew University Jerusolam, lrroel
Sound Absorption and Fast Reactions in Solution
T h e mensurement of sound nbsorption hns b m m e nn important tool for the evnluntion of the mtes of very f:lst rcwtions. \Yhere.lq e dct:~ilnlnnd nuthoritntiw nnnIysisl ns well ns some rhortrr necounts? of this subject nre nvnihble, n simplr prrsent:ition of the undrrlying physic^-cheniicnl principln might bc useful. Such 11 presentntion is the object of t hc present contribution. Let us coneider :r lil:~necontinuous sound wive of frequcnry o/2r pncaing tl~mugha solution. At nny point, this consists ill 11 periodic oscill:rtion of the prrssure nmund its ambient \.;I~III-. This prrssure ehmge is fast mough to txkr pl:tcc ndiabntic:111y : ~ n dis tlirrrfore nceomp:mid by a similnr osrill:ltion of the tempernture (esrrpt in ~ v : ~ t eatr 4'C jvhrrc (31*/3T), is zrrn nncl the thrrrforr, frnm (3T,'ap), = (T/C,)(dl','bT),, rh:~nge of trmprrnture (luring RII isentmpic p r e s u r c rhnugr is rern). Sow, let our solution contnin n chemicnl system in equilibrium, nnd let us nssume for thc snke of simplicity thnt it cnn be definnl by n single rquilibrium constant K. ,teeording to rlrmmtnry Ihrrnmodynnmirs, n rhnngr in prrssurr, dp, nncl a chnagr in trmprraturr. (IT. pmdurc. :I chnngc in the rquilihrium eonstnnt which is given by -(~la/NTklp + (A?i",'IIT'hlT din K (1)
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where A1'"nncl AH"nrc the chnnges in volume and in mthnlpy ~~ccomp:mying unit rrartion uucler stnndnrd ronditions. T h r equilibrium c o ~ ~ s t n nthus t oscill:~tcs wit11;I frrqueney oPlr nrnund its v:tlur in t h r nbscnee of the sound \v:~vr. For sm:dl vnrintious, ehnnges in In K c i u ~rnsily b r srrn to rnt:~ilpmlxwtionnl rhnnges in the wluilibriun~ conccntn~tione'of the substnnros whirl1 tnkr part in t h r rmrtion. Thrsr equilibrium ronrentrntions thereforr crsrillnte in t h r s:mr \v:~yns cloes In K. 12ixing our r~ttrntionon n pnrticul:~rsuhstnner whose stoirhiomrtric rorllirirnt is nuit?., wr enn I I I ~ Vwrite
8
=
B PO* wt
12)
where 8 is the amount by whirh the instant:~nenus rquilibrium vnlur of the eonrentr:~tionof our soI~stnnce
.. \ ! ' ~ : I s s ~ ~ ~A.~),~ ~111:~meienw KII, l i h l i s h & (divi&m of h h n N'ilq' & *m*, Iw.), Sew Ymk, 1963, V d . VIII, I h t 11,
differs from thnt in t h r nbsence of the sound wnve. (Is'rom the stoichiometry of t h r reaction, we can immrcli:ltely cnlcul:~tethe 8's for nll t h r othrr substnnres.) \Ye next eonsidrr t h r beh~rvinrof the rhemic:~lsyxtem under t h e influence of the sound wnvr: the arf~ral rhemicnl couerntr:~tiona will strive to follow their changing equilibrium vnlnes but they sill not quite succenl. I t cnn hr shown' thnt 6, the nrtunl (levintion of thcconcentrntion frnm thnt in thenbsenceof t h r wave is given by
where tnn 6 = or. (Here, 7 is the time of relnxntion, i.e., the time whieh it would ttllie n drvintion 6 to fnII off to l / r of its initinl value.) \l'c* sw th:~tthe nctunl roncmtrntion osrill:~teswith the s:une frwluenry n s ~ l o wthe equilibrium conrentr;~tion,hut its nmplitwlr is rcdurwl o'r2)"'. I:urthrrmorr, we sw by n fartor of 1/(1 thnt it limps behind the oreillating quilihrium concentr:ltion by n phnsr nnplr 6 which inrrr:~srsfrnm rern to 90" ns or incrn~sesfrnm r r m to infinity.' It follo~vsfrnm the thmry of acoustics thnt this phnse shift between 6 nnd 8 should cnusc sound nbwrption.'. 2 As hns been sho\vn recently,' the formuln for the rhemiral rontribution to sound nbsorption rnn nlsn be obtninerl simply fmm a qunntit:~tiveconsi(lrn~tionof the fart that :I renrtion Inking p l : w undrr non-revemihle ronditions dissipntes energy. Thix energy most be prnvirlrtl by t h r sound \v~tvrrind thus enusrs sound nbsorption. Thc rnrrgy d i s s i p n t ~ l per unit volume. while 11rcnction p m c m l s by (16, is e q u d to Ad6, where A is the nfiinity of the re:~ction'
+
where p , is the chemienl potrntinl of the ith substance, and v , its stoirhiomrtric coefiirient. 1~'urthrrmore. nenr equilibrium t h r nflinity is known to be1A4I)mpor8. tionnl to the drvintion from equilibrium, i.c., to 6 T h e chrmicnl contribution to snrtnd nhrorption is now ohtninnl I)y integrating Ad6 ovrr one cycle i ~ n ddividing hy the energy of the sound wnve. T h e tlrtnils of this cnlcul~rtion:Ire given rlse\vherr.' T h e result shows 'eVhc sound nt>.~orptiorrto hr pmportionnl to
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.AS,>
-I/(
1
+w
W
(4)
p. X!lT, IT.
This expression is seen to pnss thmugh n mnximum' a t or = 1. \Yhrn or IX, I?. F'., "FwI Renclion$ i n Sdulio~t," Srienlifir Puhlirntionr. Oxfnnl, 1%. p. X I IT., 0,) I(., " l h ~ y r l o l m l i n Physi~s,"(Edtlor: ? ~ l W m ,S.) r : K 6 I I I 1,. 20'2 IT.; 1") l h t v , \I., e . ~ . , 18)
Hlackticll
Volume 47, Number 3, March 1970
/ 201
the nffinity is zero, nnd therefore no energy in absorbed, nlthough renetion continuously tnkcs plnce.' On thc other hnnd, when w r >> 1, exprcwion (4) ngnin b r c o m n negligibly nmnll. F4nntion (3) shows that undrr these co~~ditions, 6 0 , i.e., the prrxsure nnrl trmpernture rh:~nyrsnrc so fast thnt t h r rhenlicnl rrnction cnnnot follow thcm nt nll, and t h c n c t ~ ~roncrntr:itio~~srrmnin al nn:~firctnI. I t in the, rnngc hrtwwn thrsr two rxtrrmrs
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which in of interest t o the chemist: fmm the cleprnclrnec of muncl itbmrorption on frcull~enryit is pnssihlr to clew conrl~~sionsabut the rrlnxation timrs r~ncl mrrhnnirms of rrnetions whirh nrr too fn.st for menwrtL ment by conventionnl t r c h n i q ~ n . ~ 6 See f o o l n o l ~ lI and 2; liternturn cited in reference 20; nlso, G., Z. phyrik. Chma., S.F.,49. 163 (1M8). I k w , 31, ~sn.\l.r.r%%,
rlsh College of Sonto Fe Sonto Fe. N e w Mexico 87501
A Simple Colorimetry Apparatus
A simplr nppnrntur to impmve the techn i o i ~ rnnd m u l t s in no intmcluctorv cxperimrnt in cniorimetrs is h r r r prwrntnl. Thc cnhatr~lctiondrtnila nrr shown in t h r ncrompnnying figurr. The l)nck-honrd rvns mndr of l/,-in. prrssnl honrd; t h r cnlnmn.i n r r r of soft pine, nnd the trnnspnrrnt hnsr wns 11 stnnclnrd micmscope slide glunl to the c o l ~ ~ m nnntl s prntrrtnl frnm neridentnl h~vnkngrhy bring rovrrnl on l h r outsidr with trnnspnrrnt plnstic tnpr. T h r srnlr on the hnrk-hnnrd ~ v n smndr I y inking in npproprintr linw on grnph pnprr and mnking wrrrnl ropirs on n S r r n x mnchinr. The scnlr IWR c o v r r d xvith ~ e I I o p h n i ~t or protrrt it from ncriclrntnl spillnge during rxperimmta. Shrll vinls (84rnm sirr: 2.; n m X 9.3 mm; liiml~lc #F,MRO) wrrr 11sn1nnd t h n r d r t ~ r m i n ntlh r dimrnsions of t Iw nppnrntns. Thrflnt hnsrs of t h r vinls nlong with their rnthrr uniform thickness g i r r prncticnlity to this nrrnngemrnt. I n ncti~nlrxprrimmts the rtudentr trnnsfrrrcul atnn-
dnrd nnd unknown wli~tinnehy menns of plastic .wdn tlw strnwa nsrd ns "piprts." T o prrvrnt eonf~~sion. right hnnd vinl wns t h r stnndnrd (S) nnd the lrft hnnd v i d , t h r u n k n o w ~( S ) . T h r s y n i h l * (S) nnd ( S )wrrr printwl on t h r npproprintr Imsts. The nppnrntus with t h r vinla rontnining t h r solutions w r r plnenl ovrr "light-hoxrsU-flr~ormcr~~th u h or incnndmrrnt lamps in :I simplr h x rovrrnl with frostnl glnss. The vinls wrrr hrlcl in the nppnrntur by an rlnatir hnnd. T h r rrsults from :III rxprrimrr~twing ropprr s ~ ~ l f n t r solutions i ~ ~ d i c n t nthnt l cnnd ronaistrnt rrwlts rnn h r ohtninnl. T h r dntn in t h r tnhlr nrr from onr awtion of frrshmnn chrmistry s t r ~ l r n t s . A frw s t ~ ~ d r n trns cnr~ntrrrdsrriow clifficultim in mntrhing t h r rolors. Thrir rwults sho\vrrl ~ W - 1 0vnristion. ~ ~ Sotirr thr r m ~ ~ lnt t nt h r rncl of t h r list in t h r tnl~lr. - --
C'.II..
:\
Representative Colorimetric Measuremenk -
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few studmtr had "pmldrmr."
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.
the nffinity is zero, nnd therefore no energy in absorbed, which in of interest to the chemist: fmm the cleprnnlthough renction continuously tnkcs plnce.' On the clrnec of mund nbmrorptio~~ on frculumry it is possible to other hand, when wr >> 1, mprm-ion (4) ngnin brcomn c\r:t\v conrlueions s h u t the rrlnxntion timrs nncl negligibly nmnll. Fkpntion (3) shows that under thrsc mrrhnnisms of renetions whirh nre too fiwt for mensurtL couclitions, 6 0 , i.e., the presuure nntl tempernture ment by convention~lt ~ h n i q u e s . ~ rh:tngw nrc m fnat thnt the rhen~icnlrrnction cnnnot follow them nt d l , nnrl thcnctual roncentr:~tio~~sremnin 6 See f o o l n o l ~ lI and 2; liternturn riled in referenre 20; also, un:tflret~l. I t in thr rnngc hrtwrrn thrsr two extremes I k w , 31, ~ s n . \ l . r . r x ~ G., , Z. phyrik. Chma., S.F..49. 163 (1W~8).
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202 / l w r m l of Chemical Education
