Internal Pressure Measurements and Evaluation of ExternaI Molecular Vibrational Modes in the Liquid State Edward B. Bagley,l Theodore P. Nelson,2 Show-An Chen,3 and Joel W. Barlow4 Department of Chemical Engineering, 1T-ashington I-nicersity, St. Louis, M o . 63130
Internal pressure data (P,)for n-heptane over a 70°C range in the liquid state, along with literature values of P I for a number of other liquids, are compared with energies of vaporization. It i s possible to evaluate the change in the effective number of classical external degrees of freedom in going from the gas to the liquid state. The effect of molecular structure is shown b y comparison of results for linear alkanes, branched alkanes, linear alkenes, and other common solvents. The significance of these observations in computing a correct solubility parameter from energy of vaporization data i s emphasized.
Receiit, iiieasiireiiieiits of the iiiteriial pressure, 11, = [ i a ~ / a 70°C raiigc iiitlicatetl that iii the liquid state the cyclohesniie molecules caii lie coiisidcred to have a~iprosiiiintelysix esteriial degrees of vibratioiid freedoni (Ihigley et al., 19iO). X molecule i? tliiii v i h i t i i i g as a whole iii a poteiiti:d w l l in the saiiie as 1i niuleciile iii the solid state, rvhich obeys the law of Duloiig :idPetit’. T h e importaiice of theqe esterii:il degrees of freedoni iii the liquid state hn.: lxeii emli1l:isized by Boiidi (1962) i i i coilsidering thermal coiiductivity :nid by I3ondi aiid Sinikiii (19GO) in work oil :i coriwpoiidiiig states correlation for liquids of higher nio1ecul:ir w i g h t . Prigogiiie :ilso dihcussed the influelice of the rstcriial dcgrces of freedom oii the evaluatioii of the 1)artition fiiiictioii i i i liquid theory (l’rigogiiie, 1 9 5 i ) . Thih :isliect of liquid statr licli:ivioi, has ofteii lieeii iicglected, li:.:trtirul:irly iii coiisideiiiig iiit,ei,ii:il Im’wire nie:isureiiieiits aiid solubility par:mieters. Giiggciiheini, however, iioteti (Hiltielirniid aiicl Scott, 19621)) “ t h a t the cohcsivc energy, -E, of :i liquid consist,s of two ii:ii,ts: the coiifiguratioiinl 1ioteiiti:il energy, E’ o h bility parnineter from eiie of valiorizntioii is nl)lirosiiii:itely correct for wine orgaiiic molecules-e.g., cyclohesaue-it is iii poor npreeiiieiit with maiiy niore--e.g., tetradec:iiic. Be-
+
v)1’2.
cause of the \vide iiidustrid use of the soliihility 1):irnmeter concept, ( H x i i w i ~1967) it 1,eeoiiies iiiiliortmt to iiivcstignte the iiiagnitiidc of tlic 1)oteiitial tiicrgy of vihitioii tcrm and its effect 0 1 1 the coni1iut:itioii of solubility pai’aiiirters from eiiergy of vnl)oriz:itioii d:itn. ~
Theory
Ileceiit siiccw>e:, i i i trcntiiig wlutioii :tiid lili:i niay I)c w i t t r i i :i% ~
E T S =
E,,
+
+ E,,?
(1 1
Here En,represeiits the equililiri~iiii~ or zcro-lioiiit e i i t ~ g y of , the atonis in the iolitl 1:i ttiw. Ei,It :iiitl E,, i ~ c l i i , r m i ttliv 110teiitial aiid kiiietic eiicrgics of the :itoiiis i,ilirtitiiig :il)out their equilibrium lat,tice 1)o.itioii.. Iii tlic cl:i>hicnl liniit, wlicrc equipartit’ioii of eiiergy holdthe tliffereiice Iict\vetii C,. for the liquid : ~ i i dthe solid i* siii:i11 (Roberts; 1947). l’o tlic cstciit t’liatthib differeiicc i. iinall, the lattice niodrl gives :i good xi>prosiiii:ttioii to the> ciicrgctic- of the liquitl st:itc. For orgaiiir iiioleciilcs i i i the liqnitl .-t:ite the bittintioil is inorc c ~ o i i i l i l r s T. h c w \vi11 1)r :I coiiti,il)iition>I?,,p. to the totnl liquid qt:itr ciici’gy. twictly :IS i i i thc mlitl KISC. ‘I’hcrc will 1)e otlicr coiiti,il~utioii.t o thc tot:il eiiei’gy, :iw)tkitcd ivitli w1i:it may be tei,iiietl iiitvriial a i d cstri,iial inode-. EL,Lt nil1 Le the Ind. Eng. Chem. Fundom., Vol. 10, No. 1 , 1971
27
cont,ribution to the total energy due to interatomic vibrations, bond bending and rotation, etc., within the molecule. The contributions to the energy due to external modes, E,,,, will be due to motions of the molecules as a whole. This would include such contributions as those due to rotation of the molecule (which for complex molecules may well be grossly hindered in the liquid state) and vibrat’ioiis of the molecules around quasi-latt’ice site positions [analogous to the ( 6 R T / 2 ) contributions as in the case of t8he monatomic solids cited above]. Of coursc, not all molecules will be trapped on quasilattice sites. Some may be translat.ing only, and a detailed consideration of this could be pursued using the concepts of Eyriiig’s “significant liquid structure” (Eyring and Jhon, 1969), but this aspect of the problem is iiot pursued here. The total liquid state energy will be given as
sure. This is suggested by the cyclohesane internal pressure data of Bagley et al. (1970). X possible method of determining the values of these last two terms might well be by using Eyring’s theorj of “significant liquid structure,” as noted earlier. For the purposes of this paper, though, these last two terms are taken as zero, so
Pi
=
Err:
= Eint’
+
+
Erot’
(4)
Etrans’
Here the van der Waals forces are neglected ( E n p= 0) aiid superscript g is used to emphasize that t’he terms refer to the gas state. The enrrgy of vaporieatioii, AEV,is t,heii given by the difference between the gas stat,e and liquid state eiiergies. In evaluating t,his difference, it, is noted that at iiormnl temperature and prewure E,,,‘ = Eintz, so the expression for LET’ simplifies t o LEV = --EnD
+ (Erot’- E d 3 +
(Etrnrlg
-
(5)
E,ib2)
If the rotatioiial energiei in the liquid and gas statcs are the same, the term = (ErOto - E,,,’) \vi11 be zero. If t,lie rotational behavior is different iii the two +at’es-for example, free rotation iu the gns with hindered rotation in the liquidAEFUt will iiot he zero. 111any event, if t’he solubility paranieter, 6, refer.: to the magnitude of the nonpolar Lontlon dispersion force., it follows from Equation 5 that’
-AEmt -
Etrsn’
P
+
Ev,bl
]
(6)
Only for materifils for which AErOt= 0 and for which E,,,,,,’ = 3RT/’2 and E , > b ’= 6RT;2 will the correctioii term to 6* of Equ:ltioli 6 be ( 3 R T ,2). X q noted by Bagley ef al. (19iO), the corrrctioii term for cyclohexane is al)proximately ( 3 R T l 2), but for i n m y substances thc correction term can be much larger. The internal pressure, now, is found by differpntiating Equation 3! SO that
Pi
=
(\p)T
=
bE,,
(c-)T
+
Enp
ETG =
+ Eint + ( R T / 2 ) f , + rRT/2)fo
Eint
(9) (10)
and
AE”
= -Enp
+ ( R T / 2 ) ( f o- f ~ )
(11)
Hence, Equation 6 can be written as
where A j = ( f ~ j G )is the general correction term needed to calculate :I true solubility parameter from energy of vaporizatioii and represents the change in effective number of classical degrees of external freedom in going from the gas to liquid st,ates. Seglectiiig the small volume dependence of f L and using E n p = ( - a ‘T’), we get Pi1’ = --Enp. Then from Equat,iori 12 it is seen t’hat Af can be evaluated rsperimeiitally from AS = 2(P1T”’- AEV)/RT
(13)
111interpreting Aj: tIvo limiting cases c a n be considered: Rot:itioii:il eiiergy cont~ributioiisare the same in the liquid and the gas phase.; there are free rotation in t’he gas and hindered rotatioii ill the liquid. The number of degrees of freedom in the gas, fo, will be
and .ftransg is equal to 3. For linear molecules froto will be equal to 2, ivhile for branched molecules it will be equal to 3 (nIoel~~-yii-Hu~lies, 1961). Planar molecules will have a froto value of 2 . 111 some caw.:, there n-ill be uncert8:iiiity in the value to choose for jrotB and in these c a m the moments of inertia in all three dimensions will be required. This ~ o r is k confined to classical cases oiily with the equipnrtitioii of energy limit. The nuriilier of esteriial degrees of freedom in the liquid will also be taken as the sum of the various cont’ributions,so that
bEd
(+)1+
&)*
+( 3 s ’ )
(7) T
At the temperatures mid relatively l o x pressures (less than 500 psi) used in the work described here, Eint’will lie a function of temperature only niid the sccoiid term of Equation 7 \vi11 be zero. Tlie lart tn o terms of Equation 7 probably make oiily a small coiitrihution (ibL =
~j
+3+
(jrot'
- jrot')
(16)
For free i,ot:itioii (or rotntioti eqwilly Iiiiidcrcd iii both lih:ise>i, .f, > I , ' = AS 3. For liiiitieiwl rot:itioii iii the liquid state ivitli fi,ec rot:itioii iii the piis j", \vi11 lie gre:itc,r th:iii (Af 3).
+
+
Experimental Procedure
Table 1. Internal Pressure, Energy of Vaporization, and A f = fL - fG for Cyclohexane as Functions of Temperature
r,
"K
298 309 319 329 338 349 358 369
87 71 84 30 79 23 84 02
Pt,
76 73 71 69
68 65 63 60
A€'.,
Ptf,
Cal/Cc
292 974 9-10 867 365 684 079 984
Cal/Mole
Col/Mole
8299 0 8154.9 8032 8 7898. .5 7827, 2 7629.3 7426.3 7286.4
7290 7130 69SO 6850 6710 6550 6410 6270
Af
3 3 3 3 3 3 2 2
40 33 31 21 32 11
85 77
Table II. Internal Pressure, Energy of Vaporization, and I f = f L - f(, for n-Heptane as Functions of Temperature
r .
1lie iisc of :i T e s : i ~Iii-trumeiit prewire gage, i i i ivhicli prc+ sure is iiie:i-uretl t o 1 1nrt in 500,000, :mi1 :I Hewlett-I':ickard q u i r t z thcrinomete~,,\\-itli digital rendout to 0.0001'C', iniikes t1ie-e n i e ~ i ~ r e i i i e i i of t s y relatively >iiiil)leand 1)recisc. T h e recent result. of Orn.oll : ~ n dFlory (1967) 011 the iiiteriid p r c ~ i r e of n-liesaiie nnti n-oct:iiie. again o1it:iiiiecl by e.+ seiiti:illj. the iiicthotl of Weqtn-ater et 01. (1928), :iw esaiiiiired h c t ~:iloiip ivith the older d:it:i of 13eiiiiiiig:i aiicl Scott (1955) 011 c:iilion tctr:ichloritle. Eiie1,pie.s of val)oriz:ition wed in the calciilation of Af froni Ifrom :in exi)ie-ioii given I J ~ Boiidi (1968) as
+
..
n-here S , 5 is the tot'nl number of skeletal uiiit 1)cr niolcciile. The first notable feature of Figwe 2 is thxt tlic.f,itJ' ~ : ~ I i i e s , Ind. Eng. Chem. Fundam., Vol. 10, No. 1, 1971
29
.
a n - alkanes 3C,n - alkanes -1-Orwoll and Flory
)-< n -heptane
A alkenes
15
\
\
\
\\
\
\
\ \\\\
\\
\
\
\
\ \\
\
\
\
\
\ \
\\
\
\ \
\
I
Y' 3 1 0 Y
I
5'
I
'
I I
1'0 I
I
I
I
15 ' I
I
I
20 ' I
I
I
'
i
n-Figure 2. Effect of number of carbon atoms in linear chains on f i t l h and 3C n-Alkane and n-alkene (Allen et al., 1 9 6 0 ) n-Hexane and n-heptane ( O r w o l l and Flory, 1 9 6 7 ) 3C values computed from Equation 1 8 and f,,hl from Equation 1 6 as Af+3
Table 111. Internal Pressure Data and Energies of Vaporization for CCI,, from Benninga and Scott ( 1 9 5 5 ) , as Functions of Temperature along with f,>,,'Values Calculated 4ssuming the Same Rotational Energy in the liquid and Gas States
-
J,
O K
268 278 288 298 308 318 328 338
16 16 16 16 16 16 16 16
PI< AEI, 3RJ/2, P,V AET Cal/Mole C o l / M o l e Cal/Mole Cal/Mole f,,t,l
8179 SO97 7978 7867 7718 7548 7351 7130
7613 7474 7335 7200 7064 6932 680 1 667 1
800 829 859 889 919 948 978 1008
30 Ind. Eng. Chem. Fundam., Vol. 10, No. 1, 1 9 7 1
566 G'23 643 667 654 616 550 459
5 5 5 5 5 4 4 4
13 25 25 25 14 95 69 63
tioii wliirh i.. kiion.1~to occu fi,oin lioth vihcoqity a i d cliffusioii stiidirs. Fiirtliri, btiitlics oil dkaiies :nid alkeiies u1)to n = 25 could lie very iriforiii:lt8ive.Such work, however, requires vel'!- 1xrvi.c d:it:i. 'l% i. I)erli:i1),eful iii showing the wide variation of Af from the single value of 3 originally suggested 1)s Guggeiiheim. From these Af vnlues, f v l I , ' could be estimated, a.ssuming either free or hindered rotation. The halogeiiated niet,hnnes and ethanes have very low Af values, a i d this i. probably in-
Aj
= = = = = =
K,? n
=
P
=
Pi
R
= =
T V,Ti 3C
= =
6
=
=
=
fL
- fc
C
"
number of external degree\ of freedom in gas number of estcriial degrees of freedom in liquid number of rotatioiial degree.: of freedom number of tranrlational ckgree.. of freedom number of vibrational degrees of freedom (external modes) number of skeletal units per niolecuje constant in potential law E = - a / V T L pressure internal pres,.ure universal g:is constant temperature volume, nioIar volume number of external degrees of freedom solubility parameter Ind. Eng. Chem. Fundam., Vol. 10, No. 1 , 1971
31
SUPERSCRIPTS 9 1
= =
gas state only liquid state only
literature Cited
Allen, G., Gee, G., Wilson, G. J., Polymer 1 (4),456 (1960). IND.ENG. Bagley, E. B., Kelson, T. P., Chen, S-A, Barlow, J. W.) C H l 3 I . F v s D a h I . 9, 93-7 (1970). Beiininga, €I., Scott, R . L., J . C h e ~ nPhys. . 23, 1911 (1955). Blanks, 11. F., Prausnitz, J. &I.,ISD. ENG.CH16RI. FL-ND.W. 3, 1 (1965). Bondi, .4.A , , A.I.Ch.E. J . 8, 610 (1962). Boiidi, A,, “Physical Propertie- of Rlolecular Crystals, Liquids and Glasses,” Wiley, New York, 1968. Roiidi. A. A , . Simkin. I). .J.. A.I.Ch.E. J . 6. 191 11960). Chen,’r\: < I . , ’ j . Chcn;. Eng.‘D& 10, 207 (1666). Eyi,iiig, H., Jhon, 11.S.,“Significant Liquid Structures,’’ Wiley, Xew l o r k , 1969. ~I
Hansen, C. hl., “Three-Dimensional Solubility Parameter and Solvent Diffusion Coefficient,” Danish Technical Press, Copenhagen, 1967. Hildebrand, J. H., Scott, R. L., “Regular Solutions,” PrenticeHall, Englewood Cliffs, N. ,J., 1962a. Hildebrand, J. H., Scott, It. L. , “Regular Solittions,” p. 167, Prentice-Hall, Englewood Cliffs, 3;.J., 1962b. Huisman, JohnYSage, B. H., J . Chem. Eng. Data 9, 233 (1964). Makay, R. A . , Sage, B. H .*, J . Chon. Eno. Data 5 . 21 ilS6O). Moelwyn-Hughes, E. A., “Physical “Chemistry,” Perg:amon Press, London, 1961. Orwoll, R.il., Flory, P. J., J . A m r . Chem. SOP. 89, 6814 (1967). Prigogiiie, I., “JIolecular Theory of Solutions,” with A . Rellemans and T’. RIathot, Chap. XVI, Korth-Holland Publishing Co., Amsterdam, and Interscience, Kew York, 1957. Roberts. J. K.. “Heat and Thermodviiamics.” Rlackie and Son. London aiid‘Glasgow, 1947. TVestwatei, IT.,Fraiitz, I T . IT.,Hildebrand, J. II., Phys. Rev. 31, 13*5(1923). RF,CILIVI:D for review August 1, 1969 ACCEPTEDAugust 28, 1970 I
~~
\ - -
N e w Three-Parameter Equation of State Byung-lk l e e and Wayne C. Edmisterl Oklahoma State U~aiversity,Stillwater, Okla. 74074 A new three-parameter empirical equation of state has been developed for hydrocarbons (pure and mixed) in the vapor phase. The parameters are functions of temperature, critical conditions, and acentric factor. An analytical solution for density at a given pressure and temperature i s possible. This new equation i s more accurate than the original and modified forms of the Redlich-Kwong equation and the generalized BenedictWebb-Rubin equation and more easily applied than the latter. It i s recommended for the calculations of fugacities and isothermal changes in the enthalpy of vapor phase mixtures.
E i i i p i i i c a l equations of state are wecl in calculatiiig fugacities as well as deiisitieq, for pure coni1)ounds aiid aiid eiitlid1)ie,~~ niisti~req.The Kedlicli-K~voiig(1949) niid the Ueiiediet-WebbRuhiii (Ikiietlict et al., 1940, 1951) equatioiis are widely used iii swIi therriiodyiianiic calculatioiis! as are niodificatioiis of them (13ai.iier cf ol.> 1966; Ednii,sibilityfactor or (Ieiisity. A dixadvniitage is it.
