If n is the number of moles of the limiting reagent, the molar quantities for the reaction are

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1 VI Thermochemisry EXPERIMENTS 6. Heas of Combuson 7. Srain Energy of he Cyclopropane Ring 8. Heas of Ionic Reacon PRINCIPLES OF CALORIMETRY We are concerned here wih he problem of deermining experimenally he enhalpy change H or he energy change E accompanying a given isohermal change in sae of a sysem, normally one in which a chemical reacon occurs. We can wrie he reacon schemacally in he form A( T ) B( T ) C( T ) D( T ) inial sae final sae If n is he number of moles of he liming reagen, he molar quanes for he reacon are H H n and E E n. In pracce we do no acually carry ou he change in sae isohermally; his is no necessary because H and E are independen of he pah. In calorimery we usually find i convenien o use a pah composed of wo seps: Sep I. A change in sae is carried ou adiabacally in he calorimeer vessel o yield he desired producs bu in general a anoher emperaure: A( T ) B( T ) S( T ) C( T ) D( T ) S( T ) (2) where S represens hose pars of he sysem (e.g., inside wall of he calorimeer vessel, srrer, hermomeer, solven) ha are always a he same emperaure as he reacans or producs because of he experimenal arrangemen; hese pars, plus he reacans or producs, consue he sysem under discussion. Sep II. The producs of sep I are brough o he inial emperaure T 0 by adding hea o (or aking i from) he sysem: C( T ) D( T ) S( T ) C( T ) D( T ) S( T ) (3) (1) 145

2 146 Chaper VI Thermochemisry As we shall see, i is ofen unnecessary o carry ou his sep in acualiy, since he associaed change in energy or enhalpy can be calculaed from he known emperaure difference. By adding Eqs. (2) and (3), we obain Eq. (1) and verify ha hese wo seps describe a complee pah connecng he desired inial and final saes. Accordingly, H or E for he change in sae (1) is he sum of he values of his quany peraining o he wo seps: H HI HII (4 a ) E EI EII (4 b ) The parcular convenience of he pah described is ha he hea q for sep I is zero, while he hea q for sep II can be eiher measured or calculaed. I can be measured direcly by carrying ou sep II (or is inverse) hrough he addion o he sysem of a measurable quany of hea or elecrical energy, or i can be calculaed from he emperaure change ( T 1 T 0 ) resulng from adiabac sep I if he hea capaciy of he produc sysem is known. For sep I, H I q p 0 consan pressure (5 a ) EI qy 0 consan volume (5 b ) Thus, if boh seps are carried ou a consan pressure, and if boh are carried ou a consan volume, H H II (6 a ) E E II (6 b ) Wheher he process is carried ou a consan pressure or a consan volume is a maer of convenience. In nearly all cases i is mos convenien o carry i ou a consan pressure; he experimen on heas of ionic reacon is an example. An excepon o he general rule is he deerminaon of a hea of combuson, which is convenienly carried ou a consan volume in a bomb calorimeer. However, we can easily calculae H from E as deermined from a consan-volume process (or E from H as deermined from a consanpressure process) by use of he equaon H E ( pv) (7) When all reacans and producs are condensed phases, he ( pv ) erm is negligible in comparison wih H or E, and he disncon beween hese wo quanes is unimporan. When gases are involved, as in he case of combuson, he ( pv ) erm is likely o be significan in magniude, hough sll small in comparison wih H or E. Since i is small, we can employ he perfec-gas law and rewrie Eq. (7) in he form H E RTn (8) gas where n gas is he increase in he number of moles of gas in he sysem. We mus now concern ourselves wih procedures for deermining H or E for sep II. We migh envisage sep II being carried ou by adding hea o he sysem or aking hea away from he sysem and measuring q for his process. Usually, however, i is much easier o measure work han hea. In parcular, elecrical work done on he sysem by a heang coil (ofen referred o as Joule heang ) can be used convenienly o carry ou This could be done by placing he sysem in hermal conac wih a hea reservoir (such as a large waer bah) of known hea capaciy unl he desired change has been effeced and calculang q from he measured change in he emperaure of he reservoir.

3 Principles of Calorimery 147 eiher sep II or is inverse, whichever is endohermic. Since he work is dissipaed inside he sysem, he work is posive: el h h (9 a ) w V dq VId where V h is he volage drop across he heaer, Q is he elecric charge, and I is he elecric curren. For precise work, one should measure boh V h and I during he heang period. In many insances, however, i is possible o assume ha he resisance of he heang coil R h is consan and make only measuremens of I by deermining he poenal drop V s across a sandard resisance R s in series wih he heang coil. In such a case, one can wrie 2 R wel I Rh d R h 2 s 2 V d Noe ha he elecrical work given by Eq. (9) is in joules when resisance is in ohms, poenal (volage) is in vols, and me is in seconds. If he elecrical heang is done adiabacally, s (9 b ) HII wel consan pressure (10 a ) EII wel consan volume (10 b ) Our discussion so far has been limied o deermining H II or E II by direcly carrying ou sep II (or is inverse). However, i is ofen no necessary o carry ou his sep in acualiy. If we know or can deermine he hea capaciy of he sysem, he emperaure change ( T 1 T 0 ) resulng from sep I provides all he addional informaon we need: T0 HII Cp( C D S) (11 a ) T1 T0 EII Cy ( C D S ) (11 b ) T1 The hea capacies ordinarily vary only slighly over he small emperaure ranges involved: accordingly we can simplify Eqs. (11) and combine hem wih Eqs. (6) o obain he familiar expressions H Cp( C D S)( T1 T0 ) (12 a ) E Cy ( C D S)( T1 T0 ) (12 b ) where C p and C y are average values over he emperaure range. The hea capaciy mus be deermined if i is no known. A direc mehod, which depends on he assumpon of he consancy of hea capacies over a small range of emperaure, is o measure he adiabac emperaure rise ( T 2 T 1) produced by he dissipaon of a measured quany of elecrical energy. We hen obain Cp w (13 a ) Cy el T T 2 1 (13 b ) a consan pressure or a consan volume, respecvely. This mehod is exemplified in he experimen on heas of ionic reacon. An indirec mehod of deermining he hea capaciy is o carry ou anoher reacon alogeher, for which he hea of reacon is known, in he same calorimeer under he same condions. This mehod depends on he fac ha in mos calorimeric measuremens on

4 148 Chaper VI Thermochemisry FIGURE 1 (a) Plo of emperaure T versus me for an ideal adiabac calorimeer. (b) Plo wih inerruped emperaure axis and grealy expanded emperaure scale for a ypical calorimeer run. Such a plo is designed o show clearly he emperaure variaon before and afer he me i when he reacon is iniaed. chemical reacons he hea-capaciy conribuons of he acual produc species (C and D) are very small, and ofen negligible, in comparison wih he conribuon due o he pars of he sysem denoed by he symbol S. In a bomb-calorimeer experimen he reacans or producs amoun o a gram or wo, while he res of he sysem is he hermal equivalen of abou 2.5 kg of waer. In calorimery involving dilue aqueous soluons, he hea capacies of such soluons can in a firs approximaon be aken equal o ha of equivalen weighs or volumes of waer. Thus we may wrie, in place of Eqs. (12 a ) and (12 b ), H C( S)( T1 T0) E (14 a ) (14 b ) for consan-pressure and consan-volume processes, respecvely. In Eqs. (14 a ) and (14 b ) we have omied any subscrip from he hea capaciy as being largely meaningless, since only solids and liquids, wih volumes essenally independen of pressure, are involved. The value of C (S) can be calculaed from he hea of he known reacon and he adiabac emperaure change ( T T ) produced by i, as follows: 2 1 C ( S) Hknown T T 2 1 Eknown T 2 T 1 consan pressure consan volume (15 a ) (15 b ) This mehod is exemplified in he experimen on heas of combuson by bomb calorimery. Le us now consider sep I, he adiabac sep, and he measuremen of he emperaure difference ( T 1 T 0 ), which is he fundamenal measuremen of calorimery. If his sep could be carried ou in an ideal adiabac calorimeer, he emperaure variaon would be like ha shown in Fig. 1 a. In his case here would be no difficuly in deermining he emperaure change T T 1 T 0, since ( / d ) 0 before he me of mixing he reacans and afer he producs achieve hermal equilibrium. The only cause of emperaure change here is he chemical reacon. However, i is an unrealisc idealizaon o assume ha sep I is ruly adiabac; as no hermal insulaon is perfec, some hea will in general leak ino or ou of he sysem during he me required for he change in sae o occur and for he hermomeer o come ino equilibrium wih he produc sysem. In addion, we usually have a srrer presen in he calorimeer o aid in he mixing of reacans or o hasen hermal equilibraon. The mechanical work done on he sysem by he srrer resuls in he connuous addion of energy o he sysem a a small, approximaely consan rae. During he me required for he change in sae and hermal equilibraon o occur, he amoun of energy inroduced can be significan. A ypical emperaure me variaon is shown in Fig. 1 b, where a grealy expanded emperaure

5 Principles of Calorimery 149 scale has been used wih an inerruped emperaure axis in order o show clearly he preand posreacon emperaure changes. In his case, a more deailed analysis is needed in order o exrac he adiabac T value from he observed daa poins. Consider Fig. 2, which shows a schemac plo of T ( ) associaed wih sep I as carried ou in a ypical calorimeer. This plo is based on he assumpons ha he hea leakage hrough he calorimeer walls is small and he energy inpu due o he srrer is moderae. These assumpons are appropriae for he hermochemisry experimens described in his book. The inial emperaure T i is convenienly aken o be he value a me i when he reacon is iniaed following a period of essenally linear T, readings ha serve o esablish he inial (prereacon) drif rae ( / d ) i. The emperaure is followed afer reacon for a period long enough o achieve a roughly linear variaon and o esablish he drif rae ( / d ) f a an arbirary poin T f, f. The analysis of sep I will be given for a consan-volume calorimeer run, bu analogous resuls hold a consan pressure. The effec of hea leakage is evaluaed using Newon s law of cooling, dq kt ( T s ) (16) d where T s is he emperaure of he surroundings (air emperaure in he laboraory) and k is a hermal rae consan ha depends on he hermal conducviy of he calorimeer insulaon. The mechanical power inpu o he sysem ( dw / d ) will be denoed by P, which is assumed o be a consan independen of. Thus or C de y P k( T T d d s ) (17) 1 a b [ d C P k ( T T s )] (18) leaksr where C is C y for consan-volume reacons and C p for consan-pressure reacons. The emperaure difference T f T i shown in Fig. 2 is given by f T T T f i a d d bleak (19) sr where T T 1 T 0 is he change due o an adiabac chemical reacon and he inegral is he ne change due o hea leak and srrer power inpu. Thus i follows ha f 1 T1 T0 T ( Tf Ti) { P k[ T( ) Ts ]} d (20) C FIGURE 2 Schemac plo of emperaure T versus me observed in a ypical calorimeer experimen. The emperaure me poins (T i, i ) and (T f, f ) can be chosen somewha arbirarily as long as i is prior o or a reac and f is a sufficienly long me afer reac (see ex).

6 150 Chaper VI Thermochemisry The following are he mos imporan experimenal approaches aimed a minimizing he effec of nonadiabac condions and he effec of he srrer; hey can be used separaely or in combinaon. 1. The calorimeer may be buil in such a way as o minimize hea conducon ino or ou of he sysem. A vessel wih an evacuaed jacke (Dewar flask), ofen wih silvered surfaces o minimize he effec of hea radiaon, may be used. This corresponds o making he hermal rae consan k very small. 2. One may inerpose beween sysem and surroundings an adiabac jacke, reasonably well insulaed from boh. This jacke is so consruced ha is emperaure can be adjused a will from ouside, as, for example, by supplying elecrical energy o a heang circui. In use, he emperaure of he jacke is connuously adjused so as o be as close as possible o ha of he sysem, so ha no significan quany of hea will end o flow beween he sysem and he jacke. This corresponds o making T T s very small. 3. The srring sysem may be carefully designed o produce he leas rae of work consisen wih he requiremen of horough and reasonably rapid mixing. This corresponds o making P small. 4. We may assume ha he rae of gain or loss of energy by he sysem resulng from hea leakage and srrer work is reasonably consan wih me a any given emperaure. We may herefore assume ha he emperaure variaon as a funcon of me should be inially and finally almos linear. One simple possibiliy would be k ( T T s ) 0 and P consan, in which case ( / d ) P / C has he same value a all i and f and Eq. (19) gives T Tf Ti ( f ) if k( T Ts ) 0 (21) d As shown in Fig. 3 a, T in his case is jus he vercal disance beween wo parallel T lines. Naurally, his disance can be evaluaed a any. A more realisc possibiliy is ha k ( T T s ) is small bu no negligible and P consan. In his case he T plo looks like Fig. 3 b, where he inial leak sr drif rae ( / d ) i a i differs from he final rae ( / d ) f a f. I can be shown 1 in his case ha T is given by T ( Tf Ti) a b ( d ) a b ( f d) (22) d i d f FIGURE 3 Deerminaon of he adiabac emperaure change T from experimenal T() measuremens made in nonideal calorimeers: (a) case of no hea leak bu a consan srring power inpu; (b) case of a small hea leak as well as srrer energy inpu. The value of d is chosen so ha he wo shaded regions have equal areas; see ex for deails.

7 Principles of Calorimery 151 where d is chosen so ha he wo shaded regions in Fig. 3 b are of equal area. The derivaon of Eq. (22) from Eq. (20) requires a bi of sleigh-of-hand manipulaon. Le us define a new variable T c T s ( P / k ), in erms of which Eq. (18) gives k a b a b d C T T k ( i c) and ( d C T f T c) (23) i f and Eq. (20) becomes Noe ha f f k T ( Tf Ti) ( T Tc ) d C (24) c d ( T T ) d ( T T) d ( T T )( ) i i c d i f ( T T f ) d ( T f T c )( f d ) d By he definion given above for d, he wo inegrals in Eq. (25) cancel ou. Subsung Eq. (25) ino Eq. (24) and making use of Eq. (23) leads direcly o Eq. (22). I is clear from his derivaon ha Eq. (22) will be valid even if he T variaon prior o i and afer f is no linear. In such a case, one merely uses he angen o he T curves a T i and f. In summary, one should follow he emperaure variaon before and afer reacon for a period long enough o allow a good evaluaon of ( / d ) i and ( / d ) f. Then make a plo like Fig. 3 b, choose he poin ( T f, f ), deermine he bes value for d, and calculae T from Eq. (22). In pracce f should be chosen as close o i as is consisen wih a wellcharacerized final drif rae. Noe ha d will lie much closer o i han o f for reacons ha go o compleon quickly. Whenever he me required for a change in sae is long, as in he case of deermining he hea capaciy by elecrical heang, he posion of d will lie near he middle of he range i o f. As an approximae procedure for handling he analysis of elecrical heang plos, one can choose i as he me he heaer is urned on and f as he me i is urned off and choose d as he midpoin of his heang period. In he experimens on heas of combuson, we make use of approaches 2 (oponally), 3, and 4. In he experimen on heas of ionic reacon, we make use of 1, 3, and 4. In all cases he small emperaure changes can be measured wih adequae precision wih a relavely inexpensive mercury hermomeer. Alernavely he measuremens can be made using a sensive hermisor (see Chaper XVII), which can be moniored repevely by a compuer. Calibraon of he hermisor for improved lineariy and accuracy is needed in his case, bu his procedure iself can serve as a convenien inroducon o inerfacing a compuer o a measuremen device. (25) REFERENCE 1. H. C. Dickinson, Bull. Nal. Bur. Sand. 11, 189 (1914); S. R. Gunn, J. Chem. Thermodyn. 3, 19 (1971). GENERAL READING J. L. Oscarson and P. M. Iza, Calorimery, in B. W. Rossier and R. C. Baezold (eds.), Physical Mehods of Chemisry, 2d ed., Vol. Vl, chap. 7, Wiley-Inerscience, New York (1992).