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Specific heat anomaly of sulphuric acid with dilution
CHEMfCA&
Volume32.oumbcr3
PHYSICS
1 fyky 1975
LETTERS
,.
SPECIFIC HEAT kNOMALY OF SULHWRIC iCID WITH DILUTION
O.P. SINHA’and O.P. PURI C7ark Collzge. Atlanta, Georgia 30314, USA Received 23 July 1974
The specific heat and the specific volume of sulphuric acid show aslight decreve
with diMan.
at est. before their
expected monotonic increase. We theorize that these effects arise from the tendency to fonn the stoichiometic HpS04*H20 composition which puts a squeeze on the free volume. Exploiting this idea and the Debey-Htickel theory for the ionized radicals, we derive a fommula which simulates the specific heat behaviour.
1.
accompany the dilution of the acid with water: its ionization and the formation of the hydrate (or hydrates).
Introduction
There is considerable interest irr the themodynamic properties of the oxides of sulphur and their hydrates because of their relevance to the problem of atmospheric pollution [l-6] .Thermodynamic properties of stdphuric acid and its hydrates have been otherwise also very extensively investigated [7-lo]. Any anomaly found in such an extensive and reliable data is, therefore, quite challenging and has some value for applied research, One interesting behaviour found in the data on sulphuric acid is with regard to the variation of the specific volume of liquid SO3 with hydration. There are two relative minima, one on either side of the ab-
The ionizaticn is initi2lly a!? the szme magniiude as +he number of moles of HZ0 introduced as one would expect from the reaction: H,SO, + H,O --t HSO; f H30* . The ionization falls off slightly, reaching the 100% level when the molar fractions of HZ0 and H,SO4 in the mixture are approximately [13,14] as 3 : 2. The ionization contributes to the free energy and hence the specific heat in accordance with the Debye-Htickel ,, theory [ 151. This contributes an additional increase in the specific heat beyond what would be expected from the replacement of the acid by water in a homo-
genizcd Cpowder-like) mixture of the two. On the other hand, it should be expected that the addition of water to the sulphuric acid should squeeze out some of the free volume of the liquid through the saturation of some of the socalled dangling bokis of-the amor-
solute H,SOd composition [ll, 121. There is an apparent cusp-like maximum at the I-QSO, composition. A similar th@g is noticed with regard to.the’specific heat variation. It has a similar maximum [S] at the H2S04’composition. The specific heat thus, decleues
slightly
with
dilution,
reaching
phous
h minimum‘at
state with the formation
ofH2S04
- Hz0 lo-
about 9836 by weight comxntration of the acid, In this co~un.ication,we propose to quantitatively :
tally.
examine the affects of dilution on &lphuric acid and develop a formula relating the specific heat with the dilution of su1phut-k acid on a theoretical basis,‘,
trations. of water, the addition of water is one hundred per cent.effective in the formation ofH2Sbd aHz0 (at least locally so in the amorphous sample) and thereby putting 2 squeeze on the free volume*. This would
We can justifiably assume that, for very low concen-
:
mere arF.twG signifi&t
‘.
: T This k,found’to be true for the gzeoas state (see ref. [IO]) *here nearly all (U~O~~d) E2% eats a~ hydrate. We should expect this for the Liquid even more so.
,.
2. The derivation &=+-al .
ch~g,as,y~,-h
: .
..
:
, :.
;
..
.,’
.,--
.‘.
_
.’
495
’ CHEMICAL PHYSICS FETE@
., .~olume 32, numper3
: -,
,l
May
1975
.. ,’
,-
:
t&d to red.ke the specific’heat since some gas-like degrees of freedom would apparegtly be converted into s&d like, reducing th? specific heat per degree fr,om .’ ‘. Sk to,an E&feinrosci&tor like v&e [16]. With these ideas, we proceed to calculate the specific heat of the mixture.as fellows: let c and I.--c be.the molar fractiom of H20 and H,SO, respedtively in the mixture. :We can.cdnkrt c a’nd 1 -c into the weight fractions w and 1 -‘w through the comers&n formula**: c= 98w/(lR+ROw).
where Jo k the density of the s&qle tid K the dielectric constzint of the acid, the other’symbols having their. : usual 7neariing. ’ ’ :
We further assume.‘tat all t&eNw/lB molecules of water (for the s&U w under consideration) attach tbems&es to neutral H2S04 I9 foim H,SO, mHiO. This places a’restriction on the upper limit of w for this assumption to be. true, viz.; Nw/18 should not exceed
ci>
;
We assume that a fraction o! of the total nk-nber of (both-type?) of molecules, ex.ists,as HSO, and H+ ions. For this IY,guided by the exper&nental result [13] that
the number of neutral H2S0, in G-&sample. The number, N,, of unionized; &attached (to H20), H2S04 ~molecuks is given by, 1
the number of ionized H2S04 mo!eoules lags only aiittle behind.the nu’mber of the Hz0 molecules intro-.
N- Al--4 il
-.$!(;;f;;;;)GK.
(7)
duced, we try the relation (empirically): CY=.c_ac2,
hooting that the specific heat of pure acid is about 0.338 :al/g “C, the average specific heat contribution per molecule of neutral, pure’H2S,04 should be d-338 X.98/Ncal/‘C. Hence theN, neutral molecules of H2SO& contribute to the spe&fic heat:
(21
the positive constant a is determined froni the obserwtion that at c = 9, all HzSO4 exists in the ionized HSOT and Hf form. Hence LY= a foI c = $ tibich gives
a=$,or &&&2.,
C, =N,(O.338 X98/N) = 0.338 - 0338~
(3)
I It iito be nqted that our eq. (3) is valid only for c
ke’should
is less than 1 -c,
also tote
that p as defied
by (5)
We estimate the specific heat contribution ofH SO,
the frac’tion of acid in the mixture:
ions (bayond
that included in the Debye-Hiickel term), them as if it is exactly the same as the n&u tral H2S04 contribution per molecule except for the loss of 3 degrees of freedom by the.removal of H+ i&s.
Converting c into W,we obtzin, 98w 18+8ow
“=
A mixture
’ 1-5
98w 918+8ow
containing
treating
-
(4)
This is perhaps a lower estimate, since the ions would
l-w
gram of the acid and w gram of water has a total number N(l - w)/98 +.
be freertwith less long range order) than the HiSO4 molecules. Fsr the present we treat H+ ions as entirely
Nw/18 molecules. Multiplying this total number by eq.,(4), ive obtain the number of HSO, and Ht
free. Hence the contributions
‘a of
H+ respectively are:
where Nis the Avogadro number and the relations (+,. ‘and (5) are tme fork <$ or w a,$,-as not&d earlier. mese.equ’al numbers of charged HSOT and H+ ions !, will contribute to the Debye~&kel free energy, and ’ ,,., ,, .thereby’ to .&e.specific heat, zn amount [ 1.51, * wedo: ti~~&atIer ol.convenience,routid off moIecuIar w$htz, : ,ar;d other expected integral quantities to the nearest integer. .’ ‘, ‘:_ .. “. .,’ & ‘T,~ ; : ., .., ..‘,. ., .,_ .,,,_ : :. -’ :,.., : .: : .‘. ,.
Cs aqd C4 of HSOq and’
where Et is the specific heat of an Einstein ogcillatai per degree freedom at the temperature 1. For the (hydrated) H2S04, we assume that the spe- ” cific heat conffibution’per molecul! is the s,&ie as th’at for the $rk
..
:
H2S04
. . :.
.’ ;’
., ._ ‘.
except fot theriine : ‘. ; _’ -, : .. ‘.) .: .. :.
_
.,
.’
additional,
. .
‘: : .‘.
‘.
: :
VoIume 32, numbei 3
%,SO,
in the mixture;
thus,
,_
instead of contributing
.w calories to t,he specific heat, cont~bu~es,,~~y an ini )’ : : sigtimt.
9~5, per mblemle. T@ specific
mount
heat, c~nt~but~o~ of the hydrated acid (NvfI8 molecules) is, b;ya &milar way of Alculatitin,‘ .’
c, =
:
:
0.338 x&i&w
+ BE,(*/li3))N
:
may rat~er.seern’~~re~t~~ b&t they are quite co&s-
tent with pur $ctur&: aI H,O initially goes to satuiate: any‘loafe bo~ds’ay~a~~~ @or the very sri~allvalues:of. IV) foxming H~SO, - E$O. Free volume is partially squeezed’out reqlting in soLid Eke contrih$ion to the’
specific heat. If should be noted that akpiding to the,
expe~~n~~ data I’?], the specific heat of H$O4 Hz0 is about 32 calfmofe just before melting, that of, H$W4 is also abotit this value, btit the specific heat ‘. of molten H$ZO.+-I-&+ is around (33 + 18) calfmole, “C, The xpoIten hydrate should contain very little ,%$33~ *Hz0 armoring to our stj~ul~~on;sin~e more &an 213 .of the HaSt)i: will be ionized and not available l
.’ for H$@4’H20
fdrmatidn, The stipulated W-$3& *
I&O~~~t~~ut~~n C, according toourmodel, istk
..
‘soIid.likevalue just before meeting. From this point of view, Et in eq. (11) could even be negative. Ignonjlg any difference between C, and Ci for the liquid and adding the contributions, Cl thrpugh C,, the specific heat of the mixture C is given by t ,’
?% regard ji , which depends on the stip”:“lted Einstein oscillator frequency as a parameter to be adjusted to obtair~ a,minimum for the c at the observed experimental value. ftur cafculations’show it to be 0.065 approximately. However, we simpIi@ the expression by dropping ‘the middle term. The ~o~~~bution of the dropped term,is partfy absorbed in a redefined A to obtzin a z-A&&n at the same ur ag&, The &q&f&i .formuIa,, which corre~poRds to treating the Hc ion not
,a.sf&z but reattached to I&# ~d.the~~ f_fiO,@es Gas:
.&eldctriccons&t for the &i@at Tk 298K as 82. Sub-: [email protected] G&es of the cd@t&.s in eq. _ ,._’ .,-, ‘.(12) iscalotiy -. ‘1 :, _, : ,_ stituting ..
-.. ‘.. 1,‘: : ,’ ,( ,:.’ ,: ..,. ,. ,. ;. ,~ ,( _,I ‘. ,’ : ., .,.‘. .,)’ : ,: ., :. I ,,., .,,,‘. -, ;.,’ 1’ :,. ,‘( ,,,.,
1) I. .I,,. :. ” 1, ,, . . .. . -; ., _.,-.’ ,. -, ; .: ._ .~.(...’ ._” ,:,,“.,I ,,. (, ‘. -. ,, ,‘, ,._ ‘: ‘, ,, -(’ )’’ .‘,. :-‘,..‘. .,,, ‘,.,
to H,SO, *
.vc&ume,f2, number.3, __ . . ,-.
.:
-’
C~~~i~A~.PH~SI~ ,:’ ‘,_
..
,+ETTE@S’ : ‘., ..:’
I_ ,_ :’
,l Miy 1975
-: .‘.;.
‘io Mrs.;I.. b&Gill.,ofthe a&&&ii w&data, bothtl and.8 bfeq. (12) cbuld”‘~.” fo&e ‘. - helpf+$mentH; ,:. be ~~g~~~d’~ ~~j~tabl~ ~~~rnet~r~.The Vduss of ., r : Atlant;i~~v~r~~ corn~~~~~n~er for help with. .’‘&l&dB’to tit the ex~e~eRt~ &rve were found to i ‘: ~mpu~atio~.,~e thank Mrs. V~Mer~~ea~er for preparing the rnanus4~~itu;ith,suc~.care. Partial financid sug~pciktfrom N.&H:Grant w8006 is grateful&
‘.‘be’urir&.listic;about 3.times hi&&r than the values’ ,~~her&d on atheoretic~ basis, Inorder to ‘riccou&for this piiftiald&cmPancy, we note$at ln our treatmentij W$‘ha%? c~rriplet&
Irg;o,+ed the possible ‘&ssocietion ,oftheacidaccordingto: 1: ,_._: ,,.
-, the
specific beat n&r, the ~bsoIu~eH-$XQ ~orn~osi~ .’
;
.
‘_
ackn&ledged. .’
; .~.‘,..
,,
:
‘.
, ,.
: ;’
.- .:
..
‘.. ,
‘. mfc Press, New York, 1961) pp. 235-53. (. .. ... “ 13J J. 13ricard et a!.,, in: Aeroscls and atmospheric chemist, ed. G. Hidy (Academic Press, New York,.19721 p. 27. ‘, [4] C.S. K.&n~ana D: Stauffer, ~Cu~io~s~F~day SC&.7
. .
(15173)‘26.
[S] P. Mirabel and f.L. Katz, 3. &em Phys. 60‘(1974)1138. [6] W..L Shug;lrd? ?.I-I. Heist and H. Reiss, J.
‘oti either side df Vie absolute H$Q co~~~ositi~~, ’ This indi,cates that the moleclular arrangements pf
Chem. P’hys.,
‘..ta5e pubI~hed. “’
‘. [7] TX Rybin and WI?.. Giauque, .I. Am; Cbem. Sot. 74 H3SO4 *.H$J and H-&07 are both better Packed ‘. (f952) 8Ot.k. . : ’ . [Bi J.E;KKunaIer and W-F. Ciauque~ 3. Am. Chem. Sec. 74 @ compared. to HgSO4. Approaching either mini-. :, (19.52)‘1472. : mum from’-the maxirriu~ Glue, &seems natural [3i W.F.Giauque’et al., J. AmI Cbem. Sot. 82 (1960) 62. for the‘ SpeciEc’k$i+ v&us composition curye 1[lOj RX-l. Heist and H. R&F, J. &hem. Phys., to be punished. .-‘fe be’concave upward. Injecting SO, (or HiO) ,@: [l If J;V, ?my, ed., chtiir& e’ngineers handbook CMcGrw- : tft& &Z+hous. (~~~id~ sample shoutd prqgrsvive2y : : m, Ne~‘~ork,-lg63) @.j-79. 112) Ha&book of Physics &I Chemistry, 49th Ed. overnice become.less a$ 1zSseffective in plug&g in the eicess’ yolunie &we a~proac~~e optimum composition. This would explain the cusp-like appe&ran& of f.he
:
specific volume curve and-hence, perhi@, for the .specifioheat curve; at. the &S04. composition. ” ; :
,‘,
Rubber C~.,,~eve~nd,
1969) table F-8.
[13] JcE.Prue, in: Ionicequilibria. International Encyclopedia
~ ..
df ChemieaI I’byaics, \iol. 16, ed.s. EA. Gugggir&&n et al., p. 21; .,“. (’ 1141 T.F. Young, LF, M~an~e’and H.M. Smith in: The s&cturf; of elec&oly’tic solutions, cd. W.JLHamer fWiley, .. New York, 1959) p. 3.6; -.
1171 In~emati~n~ C$Itiul Tables, %I.‘6 ~~cGm~-~;Ne~, York;1929)p. 74. : : ., are due to Drs: G.H. WaBrerand G.R. E&own .’ . . ~‘. . _. ‘-, . ‘. ‘,, ,’ .. ._,_ .: .._.’ ,’ .I ., : ‘. .,. .., ‘(’ .’ : “, ;. ; ,“’ ,L ,: _‘. (., : .’ ., : ,.’ . ,, ,‘, .” ., ._. .‘. _’ 2’ ,.’ ; ‘.’ _,. .:‘: 1 ‘_ I. .,: “.;,_,. .‘._. : .. ,’ .,; .-: ‘, ._ “.‘. ” .’ ; . .. .’ .: ‘. : . . ‘. ,, : : ‘__ ;, ‘. .._‘. ‘I. I.,.. . .. .‘, . ._. ‘.... . ‘. ,”,,:. ., ,‘,:’: .. ..I, ,.., ._:: .: .‘, ._’ . . ‘, ” .. . -. ‘. .: _, .I .“‘. ;j .,‘.I. :, ; ___.; ‘,, -. (, ,. : ,:~ ‘: .. :. . . .,. ,.;‘.)’ _‘: ..:: ‘, : .:, __ .‘,,‘,. ,-’ ,: 1:“.._ ..’ ‘.‘- ‘_;. ‘. : i ,, : ( ‘~ ‘. .. .( ;. .. .. ‘: ., 1,. :. ~.,,. -. ,;. : ,. ‘: .,,, I_ ‘,::.. .. . ” . ,_ :,. _. .’ ‘_ I... ;, . . . :..,“, _.._ ‘._ .‘. ‘. : .:. ..’ _. : ,’ .:. (( ‘,‘1.. .., ., ,_ ,:, .,r. ;“:. : ‘. ..’ ...’ ., .a&&
Volume32.oumbcr3
PHYSICS
1 fyky 1975
LETTERS
,.
SPECIFIC HEAT kNOMALY OF SULHWRIC iCID WITH DILUTION
O.P. SINHA’and O.P. PURI C7ark Collzge. Atlanta, Georgia 30314, USA Received 23 July 1974
The specific heat and the specific volume of sulphuric acid show aslight decreve
with diMan.
at est. before their
expected monotonic increase. We theorize that these effects arise from the tendency to fonn the stoichiometic HpS04*H20 composition which puts a squeeze on the free volume. Exploiting this idea and the Debey-Htickel theory for the ionized radicals, we derive a fommula which simulates the specific heat behaviour.
1.
accompany the dilution of the acid with water: its ionization and the formation of the hydrate (or hydrates).
Introduction
There is considerable interest irr the themodynamic properties of the oxides of sulphur and their hydrates because of their relevance to the problem of atmospheric pollution [l-6] .Thermodynamic properties of stdphuric acid and its hydrates have been otherwise also very extensively investigated [7-lo]. Any anomaly found in such an extensive and reliable data is, therefore, quite challenging and has some value for applied research, One interesting behaviour found in the data on sulphuric acid is with regard to the variation of the specific volume of liquid SO3 with hydration. There are two relative minima, one on either side of the ab-
The ionizaticn is initi2lly a!? the szme magniiude as +he number of moles of HZ0 introduced as one would expect from the reaction: H,SO, + H,O --t HSO; f H30* . The ionization falls off slightly, reaching the 100% level when the molar fractions of HZ0 and H,SO4 in the mixture are approximately [13,14] as 3 : 2. The ionization contributes to the free energy and hence the specific heat in accordance with the Debye-Htickel ,, theory [ 151. This contributes an additional increase in the specific heat beyond what would be expected from the replacement of the acid by water in a homo-
genizcd Cpowder-like) mixture of the two. On the other hand, it should be expected that the addition of water to the sulphuric acid should squeeze out some of the free volume of the liquid through the saturation of some of the socalled dangling bokis of-the amor-
solute H,SOd composition [ll, 121. There is an apparent cusp-like maximum at the I-QSO, composition. A similar th@g is noticed with regard to.the’specific heat variation. It has a similar maximum [S] at the H2S04’composition. The specific heat thus, decleues
slightly
with
dilution,
reaching
phous
h minimum‘at
state with the formation
ofH2S04
- Hz0 lo-
about 9836 by weight comxntration of the acid, In this co~un.ication,we propose to quantitatively :
tally.
examine the affects of dilution on &lphuric acid and develop a formula relating the specific heat with the dilution of su1phut-k acid on a theoretical basis,‘,
trations. of water, the addition of water is one hundred per cent.effective in the formation ofH2Sbd aHz0 (at least locally so in the amorphous sample) and thereby putting 2 squeeze on the free volume*. This would
We can justifiably assume that, for very low concen-
:
mere arF.twG signifi&t
‘.
: T This k,found’to be true for the gzeoas state (see ref. [IO]) *here nearly all (U~O~~d) E2% eats a~ hydrate. We should expect this for the Liquid even more so.
,.
2. The derivation &=+-al .
ch~g,as,y~,-h
: .
..
:
, :.
;
..
.,’
.,--
.‘.
_
.’
495
’ CHEMICAL PHYSICS FETE@
., .~olume 32, numper3
: -,
,l
May
1975
.. ,’
,-
:
t&d to red.ke the specific’heat since some gas-like degrees of freedom would apparegtly be converted into s&d like, reducing th? specific heat per degree fr,om .’ ‘. Sk to,an E&feinrosci&tor like v&e [16]. With these ideas, we proceed to calculate the specific heat of the mixture.as fellows: let c and I.--c be.the molar fractiom of H20 and H,SO, respedtively in the mixture. :We can.cdnkrt c a’nd 1 -c into the weight fractions w and 1 -‘w through the comers&n formula**: c= 98w/(lR+ROw).
where Jo k the density of the s&qle tid K the dielectric constzint of the acid, the other’symbols having their. : usual 7neariing. ’ ’ :
We further assume.‘tat all t&eNw/lB molecules of water (for the s&U w under consideration) attach tbems&es to neutral H2S04 I9 foim H,SO, mHiO. This places a’restriction on the upper limit of w for this assumption to be. true, viz.; Nw/18 should not exceed
ci>
;
We assume that a fraction o! of the total nk-nber of (both-type?) of molecules, ex.ists,as HSO, and H+ ions. For this IY,guided by the exper&nental result [13] that
the number of neutral H2S0, in G-&sample. The number, N,, of unionized; &attached (to H20), H2S04 ~molecuks is given by, 1
the number of ionized H2S04 mo!eoules lags only aiittle behind.the nu’mber of the Hz0 molecules intro-.
N- Al--4 il
-.$!(;;f;;;;)GK.
(7)
duced, we try the relation (empirically): CY=.c_ac2,
hooting that the specific heat of pure acid is about 0.338 :al/g “C, the average specific heat contribution per molecule of neutral, pure’H2S,04 should be d-338 X.98/Ncal/‘C. Hence theN, neutral molecules of H2SO& contribute to the spe&fic heat:
(21
the positive constant a is determined froni the obserwtion that at c = 9, all HzSO4 exists in the ionized HSOT and Hf form. Hence LY= a foI c = $ tibich gives
a=$,or &&&2.,
C, =N,(O.338 X98/N) = 0.338 - 0338~
(3)
I It iito be nqted that our eq. (3) is valid only for c
ke’should
is less than 1 -c,
also tote
that p as defied
by (5)
We estimate the specific heat contribution ofH SO,
the frac’tion of acid in the mixture:
ions (bayond
that included in the Debye-Hiickel term), them as if it is exactly the same as the n&u tral H2S04 contribution per molecule except for the loss of 3 degrees of freedom by the.removal of H+ i&s.
Converting c into W,we obtzin, 98w 18+8ow
“=
A mixture
’ 1-5
98w 918+8ow
containing
treating
-
(4)
This is perhaps a lower estimate, since the ions would
l-w
gram of the acid and w gram of water has a total number N(l - w)/98 +.
be freertwith less long range order) than the HiSO4 molecules. Fsr the present we treat H+ ions as entirely
Nw/18 molecules. Multiplying this total number by eq.,(4), ive obtain the number of HSO, and Ht
free. Hence the contributions
‘a of
H+ respectively are:
where Nis the Avogadro number and the relations (+,. ‘and (5) are tme fork <$ or w a,$,-as not&d earlier. mese.equ’al numbers of charged HSOT and H+ ions !, will contribute to the Debye~&kel free energy, and ’ ,,., ,, .thereby’ to .&e.specific heat, zn amount [ 1.51, * wedo: ti~~&atIer ol.convenience,routid off moIecuIar w$htz, : ,ar;d other expected integral quantities to the nearest integer. .’ ‘, ‘:_ .. “. .,’ & ‘T,~ ; : ., .., ..‘,. ., .,_ .,,,_ : :. -’ :,.., : .: : .‘. ,.
Cs aqd C4 of HSOq and’
where Et is the specific heat of an Einstein ogcillatai per degree freedom at the temperature 1. For the (hydrated) H2S04, we assume that the spe- ” cific heat conffibution’per molecul! is the s,&ie as th’at for the $rk
..
:
H2S04
. . :.
.’ ;’
., ._ ‘.
except fot theriine : ‘. ; _’ -, : .. ‘.) .: .. :.
_
.,
.’
additional,
. .
‘: : .‘.
‘.
: :
VoIume 32, numbei 3
%,SO,
in the mixture;
thus,
,_
instead of contributing
.w calories to t,he specific heat, cont~bu~es,,~~y an ini )’ : : sigtimt.
9~5, per mblemle. T@ specific
mount
heat, c~nt~but~o~ of the hydrated acid (NvfI8 molecules) is, b;ya &milar way of Alculatitin,‘ .’
c, =
:
:
0.338 x&i&w
+ BE,(*/li3))N
:
may rat~er.seern’~~re~t~~ b&t they are quite co&s-
tent with pur $ctur&: aI H,O initially goes to satuiate: any‘loafe bo~ds’ay~a~~~ @or the very sri~allvalues:of. IV) foxming H~SO, - E$O. Free volume is partially squeezed’out reqlting in soLid Eke contrih$ion to the’
specific heat. If should be noted that akpiding to the,
expe~~n~~ data I’?], the specific heat of H$O4 Hz0 is about 32 calfmofe just before melting, that of, H$W4 is also abotit this value, btit the specific heat ‘. of molten H$ZO.+-I-&+ is around (33 + 18) calfmole, “C, The xpoIten hydrate should contain very little ,%$33~ *Hz0 armoring to our stj~ul~~on;sin~e more &an 213 .of the HaSt)i: will be ionized and not available l
.’ for H$@4’H20
fdrmatidn, The stipulated W-$3& *
I&O~~~t~~ut~~n C, according toourmodel, istk
..
‘soIid.likevalue just before meeting. From this point of view, Et in eq. (11) could even be negative. Ignonjlg any difference between C, and Ci for the liquid and adding the contributions, Cl thrpugh C,, the specific heat of the mixture C is given by t ,’
?% regard ji , which depends on the stip”:“lted Einstein oscillator frequency as a parameter to be adjusted to obtair~ a,minimum for the c at the observed experimental value. ftur cafculations’show it to be 0.065 approximately. However, we simpIi@ the expression by dropping ‘the middle term. The ~o~~~bution of the dropped term,is partfy absorbed in a redefined A to obtzin a z-A&&n at the same ur ag&, The &q&f&i .formuIa,, which corre~poRds to treating the Hc ion not
,a.sf&z but reattached to I&# ~d.the~~ f_fiO,@es Gas:
.&eldctriccons&t for the &i@at Tk 298K as 82. Sub-: [email protected] G&es of the cd@t&.s in eq. _ ,._’ .,-, ‘.(12) iscalotiy -. ‘1 :, _, : ,_ stituting ..
-.. ‘.. 1,‘: : ,’ ,( ,:.’ ,: ..,. ,. ,. ;. ,~ ,( _,I ‘. ,’ : ., .,.‘. .,)’ : ,: ., :. I ,,., .,,,‘. -, ;.,’ 1’ :,. ,‘( ,,,.,
1) I. .I,,. :. ” 1, ,, . . .. . -; ., _.,-.’ ,. -, ; .: ._ .~.(...’ ._” ,:,,“.,I ,,. (, ‘. -. ,, ,‘, ,._ ‘: ‘, ,, -(’ )’’ .‘,. :-‘,..‘. .,,, ‘,.,
to H,SO, *
.vc&ume,f2, number.3, __ . . ,-.
.:
-’
C~~~i~A~.PH~SI~ ,:’ ‘,_
..
,+ETTE@S’ : ‘., ..:’
I_ ,_ :’
,l Miy 1975
-: .‘.;.
‘io Mrs.;I.. b&Gill.,ofthe a&&&ii w&data, bothtl and.8 bfeq. (12) cbuld”‘~.” fo&e ‘. - helpf+$mentH; ,:. be ~~g~~~d’~ ~~j~tabl~ ~~~rnet~r~.The Vduss of ., r : Atlant;i~~v~r~~ corn~~~~~n~er for help with. .’‘&l&dB’to tit the ex~e~eRt~ &rve were found to i ‘: ~mpu~atio~.,~e thank Mrs. V~Mer~~ea~er for preparing the rnanus4~~itu;ith,suc~.care. Partial financid sug~pciktfrom N.&H:Grant w8006 is grateful&
‘.‘be’urir&.listic;about 3.times hi&&r than the values’ ,~~her&d on atheoretic~ basis, Inorder to ‘riccou&for this piiftiald&cmPancy, we note$at ln our treatmentij W$‘ha%? c~rriplet&
Irg;o,+ed the possible ‘&ssocietion ,oftheacidaccordingto: 1: ,_._: ,,.
-, the
specific beat n&r, the ~bsoIu~eH-$XQ ~orn~osi~ .’
;
.
‘_
ackn&ledged. .’
; .~.‘,..
,,
:
‘.
, ,.
: ;’
.- .:
..
‘.. ,
‘. mfc Press, New York, 1961) pp. 235-53. (. .. ... “ 13J J. 13ricard et a!.,, in: Aeroscls and atmospheric chemist, ed. G. Hidy (Academic Press, New York,.19721 p. 27. ‘, [4] C.S. K.&n~ana D: Stauffer, ~Cu~io~s~F~day SC&.7
. .
(15173)‘26.
[S] P. Mirabel and f.L. Katz, 3. &em Phys. 60‘(1974)1138. [6] W..L Shug;lrd? ?.I-I. Heist and H. Reiss, J.
‘oti either side df Vie absolute H$Q co~~~ositi~~, ’ This indi,cates that the moleclular arrangements pf
Chem. P’hys.,
‘..ta5e pubI~hed. “’
‘. [7] TX Rybin and WI?.. Giauque, .I. Am; Cbem. Sot. 74 H3SO4 *.H$J and H-&07 are both better Packed ‘. (f952) 8Ot.k. . : ’ . [Bi J.E;KKunaIer and W-F. Ciauque~ 3. Am. Chem. Sec. 74 @ compared. to HgSO4. Approaching either mini-. :, (19.52)‘1472. : mum from’-the maxirriu~ Glue, &seems natural [3i W.F.Giauque’et al., J. AmI Cbem. Sot. 82 (1960) 62. for the‘ SpeciEc’k$i+ v&us composition curye 1[lOj RX-l. Heist and H. R&F, J. &hem. Phys., to be punished. .-‘fe be’concave upward. Injecting SO, (or HiO) ,@: [l If J;V, ?my, ed., chtiir& e’ngineers handbook CMcGrw- : tft& &Z+hous. (~~~id~ sample shoutd prqgrsvive2y : : m, Ne~‘~ork,-lg63) @.j-79. 112) Ha&book of Physics &I Chemistry, 49th Ed. overnice become.less a$ 1zSseffective in plug&g in the eicess’ yolunie &we a~proac~~e optimum composition. This would explain the cusp-like appe&ran& of f.he
:
specific volume curve and-hence, perhi@, for the .specifioheat curve; at. the &S04. composition. ” ; :
,‘,
Rubber C~.,,~eve~nd,
1969) table F-8.
[13] JcE.Prue, in: Ionicequilibria. International Encyclopedia
~ ..
df ChemieaI I’byaics, \iol. 16, ed.s. EA. Gugggir&&n et al., p. 21; .,“. (’ 1141 T.F. Young, LF, M~an~e’and H.M. Smith in: The s&cturf; of elec&oly’tic solutions, cd. W.JLHamer fWiley, .. New York, 1959) p. 3.6; -.
1171 In~emati~n~ C$Itiul Tables, %I.‘6 ~~cGm~-~;Ne~, York;1929)p. 74. : : ., are due to Drs: G.H. WaBrerand G.R. E&own .’ . . ~‘. . _. ‘-, . ‘. ‘,, ,’ .. ._,_ .: .._.’ ,’ .I ., : ‘. .,. .., ‘(’ .’ : “, ;. ; ,“’ ,L ,: _‘. (., : .’ ., : ,.’ . ,, ,‘, .” ., ._. .‘. _’ 2’ ,.’ ; ‘.’ _,. .:‘: 1 ‘_ I. .,: “.;,_,. .‘._. : .. ,’ .,; .-: ‘, ._ “.‘. ” .’ ; . .. .’ .: ‘. : . . ‘. ,, : : ‘__ ;, ‘. .._‘. ‘I. I.,.. . .. .‘, . ._. ‘.... . ‘. ,”,,:. ., ,‘,:’: .. ..I, ,.., ._:: .: .‘, ._’ . . ‘, ” .. . -. ‘. .: _, .I .“‘. ;j .,‘.I. :, ; ___.; ‘,, -. (, ,. : ,:~ ‘: .. :. . . .,. ,.;‘.)’ _‘: ..:: ‘, : .:, __ .‘,,‘,. ,-’ ,: 1:“.._ ..’ ‘.‘- ‘_;. ‘. : i ,, : ( ‘~ ‘. .. .( ;. .. .. ‘: ., 1,. :. ~.,,. -. ,;. : ,. ‘: .,,, I_ ‘,::.. .. . ” . ,_ :,. _. .’ ‘_ I... ;, . . . :..,“, _.._ ‘._ .‘. ‘. : .:. ..’ _. : ,’ .:. (( ‘,‘1.. .., ., ,_ ,:, .,r. ;“:. : ‘. ..’ ...’ ., .a&&