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The solubilities of carbon dioxide, hydrogen sulphide and propane in some normal alkane solvents—II
THE SOLUBILITIES OF CARBON DIOXIDE, HYDROGEN SULPHIDE AND PROPANE IN SOME NORMAL ALKANE SOLVENTS-II CORRELATION OF DATA AT 25°C IN TERMS OF SOLUBILITY PARAMETERS AND REGULAR SOLUTION THEORY M B KING and K KASSIM Department of Chemical Engmeermg, Umverslty of Blrmmgham, England and H
AL-NAJJAR
Iraq Petroleum Institute, Baghdad, Iraq (Recerved 24 December 1975, accepted 25 January 1977) Abstract-Solublhty data at 25°C for CO+, H,S and C,Ha and also for C,H, m the n-alkanes C,H,, to C&I, were consldered These were correlated usmg two actlvrty coefficient equations of the “regular solution” type, the first being the one including a “Flory-Huggins” entropy term (evaluated from molar volumes) which 1s normally used The second equation was slmllar except that the “Flory-Huggms” term was evaluated from molecular geometry instead of from molar volumes and, also a higher order approximation was used for this term This equation, m its composltlon-dependent form IS a variant on existing ones It was used m conlunctlon with an analytic expression for the number of segments r in a n-alkane chain as a function of the number of carbon atoms The correlation procedures were based on attrlbutmg a constant value to the mteractlon parameter, I,,, for a given solute dissolved m the entue range of alkane solvents In this way It was found possible to collate the data for CO, virtually wthm the experlmental error usmg either of the above equations together with experimentally determmed solute and solvent solublllty parameters The fit obtamed for the remauung solutes was not so close though the modltied regular solution equation then gave noticeably better results Finally, by adjusting the solute solubdlty parameters, it was found possible to collate the data for all the solutes within the experimental error m this way
lTWRODUCTlON
According to regular solution theory m its latest form [l], the activity coefficient for component 1 m a binary solution with component 2 1s given, after rearrangement, by
RT In y, = T[R(1 + In (Y/(x2 + Yxl) - Y/(x2 + Yx,))] + (&V
v, I((& - S*F + 211*&&)
where Y 1s the ratio ponents 1 and 2
(1)
of the molar volumes
of com-
Y = (V,,/V,*) 92’
42”
= X2Vln2/(Xl
vm,
RT In yz= Q” = R,E - TL?f
(2)
1s the volume fraction of component
2,
+ xzVm2)
S1 and Sz are the solublhty parameters and 2, defined by s = ((U,” - U,)/ V,)“’
chain structure and the relevant forces are presumably between segments on these chams and the solute molecules Qualitatively at least these segment/solute mteractlons would be expected to be independent of the chain-length of the solvent and the correlations developed below are m terms of I,, values which are constant for a gven solute dissolved m the alkane series Equation (1) was uutlally deduced[l] by considering the thermodynamic ldentlty
of components
(3) 1
(4)
(U IS the molar internal energy of the liquid and U,,,” that of the vapour at the same temperature but a hmltmg low pressure) 1,* 1s a parameter which allows for deviations from the geometric mean mixing rule for the attractive forces between the unlike molecules 1 and 2 The solvents considered m the present paper have a long-
(5)
where @, H,E and s” are partial molar values of the excess Gibbs function, excess enthalpy and excess entropy for component 1 m solution y, IS the actlvlty coeficlent of this component The final term m eqn (1) 1s equal to the partml molar excess enthalpy of component 1 as given by various models (Ref [l], p 82) The terms wlthm the first set of square brackets are approximately equal to the negative of the partial molar excess entropy of component 1 as given by the quasi-lattice model of solutions (Ref [l], p 75) Although this approxlmatlon 1s convenient, its use is not entirely necessary since exact expressions for the entropy of mixing have been deduced m terms of this model [2,3, lo] Of these, the earlier expressions of Huggms[lO], Guggenhelm[2] and other authors were suitable only for systems composed of linear molecules and/or monomers More recently Staverman[S], has shown how these expressions may be
1247
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KING et a/
modtied If rmg compounds are present and Chappelow and Prausmtz[5] have given an equation (eqn 10 m their paper) for the partial molar entropy of mlxmg based on Staverman’s work The models used m estabhshmg the “partial molar
entropy” and “partial molar enthalpy” terms m the regular solution treatments are admittedly overslmphfied ones However, the argument has been advanced that errors m fitE and Ts,E ~111tend to cancel (Ref [l], p 92)
average number of nearest nelghbour molecules surrounding a monomer The value Z = IO was used m the present work This follows general practlce[5] and appears to be reasonably m accord with mformatlon deduced from X-ray measurements on simple fluids [4-8] 4, and & m eqns (7) and (8) are the segment fractions of components 1 and 2 being given by
so that a fair approxlmatlon for ‘yt may be obtamed by substltutmg the values of H,E and TS,E given by simple models mto eqn (5) EQUATIONS USED FOR CORRELATING soLr_mLITIEs IN TEE ALKANRS
GAS
The mole fraction solublltles and ideal solubllmes listed m Table 1 of Part I are related to the solute activity coefficients by the equation
The activity coefficients for the solutes gwen m this table were correlated both using the usual regular solution
equation
(eqn 1) and usmg the equation
r,=O90+0283(n-1)
RT In yl = T[R(ln (d,/xJ -(Z&2)
I,,, 6, and 6, m eqn (7) are defined as m the usual “regular solution” equation (eqn 1) The present work was concerned solely with alkane solvents and the number of segments (rz) constituting an alkane cham contammg n carbon atoms was inferred from molecular geometry by dlvldmg the length of the alkane chain by the average cross-section diameter The lowest energy configuratlon for an alkane chain corresponds to a planar zig-zag[13] Takmg the van der Waals radms of the methylene group to be 2A0 and the C-C covalent radius to be 0 771 A” [ 121, the average cross section diameter of this chain 1s found to be 4 4SA0 and the length (4 + [(n - 1)/2] x 2 52)A0, whence
In (1 + (2&/Zq,)((r,/r,)
- 1)))l
+ (&)* vmdts, - &Y + 2112&W
All three solutes were taken to be monomers,
($EIR)
=-ln
from eqn (1) mainly
1e
(7) r, = 1
This differs
(11)
(12)
m that the expresston
(&lx,) + (Z/2)q, In (It- (2&/Zq,)((r1/rz)
- 1)) (8)
has been inserted for the partial molar entropy m place of the more f amlhar result (~L?,~,EIR) = -In (&/xl) + (+,/xl) - 1 gven by the Flory-Huggms approxlmatlon Equation (8) follows from an exact development of the quasi-lattice theory of solutions as apphed to mixtures composed of linear and/or spherical molecules It would not be applicable if cychc molecules were present [3,5] It may be derived by partially dlfferentlatmg Guggenheim’s expression (eqn (10 10 8) m Ref 2) for the excess entropy of mlxmg m a system composed of lmear molecules with respect to the number of moles of component 1 Alternatively it may be obtamed, after considerable rearrangement, by combmmg eqn (9) (below) wth the more general equation given by Chappelow and Prausmtz [5] In eqns (7) and (8) q, 1s the number of surface sites
available on a molecule of type z For linear molecules this 1s related to the number of segments r, of which the molecule 1s constituted by the equation (Ref [2], p 415) q, = r, - (Z/2)(r, - 1)
(9)
Zq, IS the number of monomer molecules (1 e molecules for which r = 1) which may be placed around a molecule of type z Z IS the “co-ordmatlon number”, Le the
Conslderatlon of the van der Waals and covalent radu[l2] shows that this 1s not unreasonable for CO2 and H,S, but that It 1s more debatable for C,H,, (according to eqn 11 r, = 1 47 for this substance) The necessary parameters for Insertion m eqn (7) for correlating the actlvlty coefficients were taken from eqns (9) to (12) A closely inter-related treatment for handling gas solubihties has been given by Chappelow and Prausmtz[S] These authors give an expression for the partial molar entropy of mixing which reduces to (8) when mixtures composed of linear molecules are considered When turning to actlvlty behavlour, however they consider only the case of Infinite dllutlon No equation equivalent to (7), glvmg the composltlon dependence of the activity coefficients arising from an exact treatment of the combmatory entropy term appears to have been tested previously In this context it IS noteworthy that, even m gas solublhty calculations, the calculated activity coefficient at infinite dilution can differ significantly from that m the actual solution In the present work it was found that neglect of this dtierence led to dlscrepancles of up to 5% when the solublhty of H,S was consldered m the entire range of solvents In their treatment Chappelow and Prausmtz deduce their values of r and q from tables of Increments which are taken from earher work by Bondl[I l] The values thus obtained are rather larger than those deduced from the molecular mode1 used m the present work and which are given analytically by eqns (11) and (9)
The solubllltles of CO,, H,S and &H. m some normal alkane solvents-II THE soLuBILrrY
PARAMETERS
The solubdlty parameters and hqmd molar volumes used are listed m Tables 1 and 3 The solublldy parameters for carbon dloxlde and propane were calculated from the defining eqn (4) usmg values of U and U” taken from Dm’s Tables[15] Those for the alkanes were calculated from the expresslon 8, = (((L --RT)I V,,) + pz”( T(dB,/d T) - & + VA/ VA”’ to which (4) reduces when the vapour pressure IS sufficlently low The molar latent heats and the vapour pressures at 25°C were taken from Ref [14] and the second vinal coefficients B. for hexane, heptane, octane and nonane were calculated from McGlashan and Potter’s eqn (18) The contrlbutlon from the terms m the second vlrlal coefficient became progressively smaller as the alkane series ascended and was neglected for alkanes higher than C, Less complete data were avadable for H2S and m this case an estimate for 8 obtamed from the law of correspondmg states was used (Ref [l], p 210) Table 1 Solublllty parameters and molar volumes 25°C for the solutes
((cal/cm3)“‘)
Hexane Octane Nonane Decane
Dodecane Tetradecane Hexadecane Carbon dloxlde Hydrogen sulphlde Propane Ethane
V#”
From eqn (7)
Hexane Heptane Octane Decane Dodecane Tetradecane Texadecane Mean absolute error m x0&= Average II2
X
x
00119 00119 0 0121 0 0125 0 0129 0 0136 0 0142
0 0120 00119 00119 0 0123 0 0129 0 0137 0 0144
(2 = IO) se1.z
X
I 12 0 0915 00905 0 0892 0 0892 0 0910 00911 0 0925
0 0 0 0 0 0 0 09x
1 1x1o-4 0 0907
Solute (1) CO* HS C,H, C,H, CO* H,S C,H, C,H, t“Optrmrsed”
S, ((cal/cm’)“‘) eqn (1) 5 71 5 60t 6 50t 6 70t eqn (7) 5 71 6 50t 600t { 6 13 6 20t
I,,
00907 0 1048 OOCSS
0 0145 0 0910 00866
0 0055 0 0097 0 0189
Mean absolute error m predicted mole fraction solublhty
1 1x 2 3x 0 7x 20x
1o-4 1o-4 lo-” lo-.
0 9x 2 1x 07x 09x 20x
lo-4 1o-A 1o-3 1o-3 3 lo-”
value
In the case of propane, the use of the optlmlsed
solubdlty parameter (600) m conlunctlon with eqn (7) does not offer a clear-cut advantage over the use of the normally determmed one (6 13) Indeed the latter may be preferable (Fig 3)
alkanes
increase
CORRELATION OF GAS SQLUBILlTIES IN n-ALKANE SOLVENTS AT WC USING EQN (1) (THE USUAL REGULAR SOLUTION EQUATION) AND EQN Q
131 6 147 5 163 5 179 6 195 9 228 6 261 3 294 1 6174 43 70 89 25 93 6
From eqn (1) Solvent
parameter I,, and of the solute parameter (8,) for msertlon m eqns (1) and (7) for gas solubditles m the alkanes (values of S, are aven In Table 1)
The solublhty parameters of the slowly with cham length (Table 1)
Table 2 Correlation of mole fraction solubdlttes for carbon dioxide at 25°C usmg (a) the standard reguiar solution eqn (l), (b) eqn (7) (solubdlty parameters and molar volumes as listed m Table 1)
SdC
solubdlty predlctmg
(cm’lmole)
7 28 744 7 55 764 7 72 7 83 7 91 7 97 5 71 8 36 6 13 443
Heptane
Table 3 Values of the mteractlon
at
and solvents
s Substance
1249
0121 0120 0120 0123 0129 0135 0143
0 0926 0 0915 0 0899 0 0894 0 0907 00903 0 0913
1o-4 0 0910
x IS the experlmentally determrned mole fraction solublllty of CO, The values of I,, have been back-calculated usmg eqns (1) and (7) respectively to give an exact fit to the solubdlty m each alkane xCsa IS the mole fraction solublhty of CO, m each alkane calculated from eqns (1) and (7) using the average value for I,,
The correlations were based on the solublhty data for CO, and H2S obtamed m Part I and on data for C3H8 and C2H6 reported by Hayduk et al [19,23] The determmatlons on C,Hs reported m Part I did not extend to temperatures as low as 25°C and could not, therefore, be used The value used for the ideal mole fraction solublhty of ethane at 25°C was 0 0399 The correlation procedure was based on attnbutmg a constant value to the mteractlon parameter, l,*, for a given solute dissolved m the entire range of alkane solvents In this way it was found possible to collate the data for CO* vutually wlthm the experlmental error either usmg eqn (1) or using eqn (7) together with solute solublhty parameters determmed from pure component data The closeness of the fit obtained for CO, 1s mdlcated m Table 2 The fit obtamed for the other solutes was not qmte so close when the above parameters were used though the modified regular solution equation (eqn 7) then gave noticeably better results than (1) (Figs 2-4) However it was found that the data for these solutes could also be correlated wlthm the experlmental error with a constant value for 112provided an adjusted solub&y parameter was used for the solute (Curve 1 on Ftgs 2-4) (Although not Illustrated here, an equally close fit may also be obtained by multlplymg the solvent solublhty parameters by a factor which 1s constant for a given solute) The need for some flexlblllty m the value to be attributed to the solute solublhty parameter 1s not perhaps surprlsmg m view of the high vapour pressures of some of the solutes at 25°C and the consequent ambiguity m the “hqmd” propertles attributed to them at ambient pressure
1250
M
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KING et al
Alkane
solvent
-
I+ 1 Mole fraction solulxhtles of carbon dmxlde m n-alkane solvents at 25°C (0) compared wrth predlcted values (curves 1 and 3) Curve 1 Calculated from eqn (7) with solute solubdlty parameter and I,, gtven m Table 3 Curve 3 Calculated from eqn (1) with solute solublhty parameter and 1,, given m Table 3
I 0 060
I
I
-
4
ll35r ‘SH14
, C6H16
,
,
clO% Alkane
‘1ZH26 solvent
, c14H30 -
, C16H34
Fig 2 Mole fraction solubdrtles of hydrogen sulphlde m nalkane solvents at 25°C (0) compared with predtcted values (curves 1, 2 and 3) Curve f Caiculated from eqn (7) with “optlmlsed” solute solubdlty parameter and l,, given m Table 3 Curve 2 Calculated from eqn (7) with unadjusted solute solublhty parameter (Table 1) and I,, = 0 0746 Curve 3 Calculated from eqn (1) with same solute solublhty parameter and I,, = 0 1005
Values of I,, and of the solute solublhty parameter required to fit the solublhty data for each of the solutes m the n-alkanes Cs to CX6 are given m Table 3 together with the mean absolute differences between the experrmental and calculated solublhtles These differences amount to less than 1% DISCUSSION AND CONCLUSIONS
The regular solution equations enable the solublhtles of CO*, HzS, C,Hs and C3Hs m the higher n1
ALkane
solvent
-
I
Rg 3 Mole fractton solubtitttes propane tn n-alkane solvents at 25°C (0) compared wtth predicted values (curves 1, 2 and 3) Curve 1 Calculated from eqn (7) with ‘optlmlsed” solute solubrhty parameter and I,, given m Table 3 Curve 2 Calculated from eqn (7) with unadjusted solute solubdrty parameter and 1,2 given m Table 3 Curve 3 Calculated from eqn (1) with same solute solublhty parameter and l,, = 0 0044 alkanes at 25°C to be correlated vutually within the experimental error using, at most, two adJustable constants for each solute These are 1,? and, m some instances, the solute solublhty parameter 6, 2 The value of 112 required for CO2 IS m good agreement with values of a slmllar parameter derived from an analysis of second vlrlal coefficient data (Ref [4], p 416 (c = 1- 1,*) and Ref [25], p 159) Rather szmdar, though shghtly higher values are also obtamed when phase data for COJalkane systems are correlated using equations of state at normal temperaturesrl6, 171 3 Several previous workers have correlated the phase behavlour of COJalkane and H,S/alkane systems using
The solublhties
of CO,, H,S
and C&I, m some normal
alkane
solvent+11
1251
\ St
C6HlL
Wl6
ClOH22 Atkane
‘12”26 solvent
ClLH30
C16H3L
-
Fig 4 Mole fractton solub~lltles of ethane m n-alkane solvents at 25°C (a) compared with predicted values (curves 1, 2 and 3) Curve 1 Calculated from eqn (7) with “optlmlsed” solute solublllty parameter and I,, gven m Table 3 Curve 2 Calculated from eqn (7) with unadjusted solute solublllty parameter (Table 1) and I,, = -0 0944 Curve 3 Calculated from eqn (1) with same solute solubdrty parameter and I,, = -0 1161
the usual regular solution equation (eqn 1) though their work has 1argeIy been confined to sohdlhqmd eqmhbna[20,21] and to vapour/hqmd eqmhbna for a few systems at elevated pressures[22] The alkane cham lengths considered by these workers were mostly lower than those m the present work and less tendency was observed for iI2 to be constant for a given solute For example Preston and Prausmtz[21] obtam values of -0 02, +0 08, +0 08 and +0 09 respectively from solld/hqmd eqmhbrmm data for CO, m CH,, CzHs. C,Hs and n-C,H,, respectively This form of varlatlon appears to be consistent urlth the average value of 0 09 obtained m the present work for CO2 with the n-alkanes C6 to Cl6 It does suggest, however, that the constancy m iB2may not extend to alkanes lower than Cq A study of the 112values derived by Robinson and Chao[22] from vapour/hqmd eqmhbrmm data for CO, and H,S m some of the lower members of the alkane series at elevated pressures leads to a similar conclusion 4 It appears from the present work that varlattons of 1,* with chain length for a given solute are comparatively small m n-alkane solvents higher than C6 and that, If smtably adlusted solute solublhty parameters are used, these varlatlons can be brought vn-tually to zero In view of the use m the petroleum Industry of correlations of the type proposed by Robinson and Chao[22], which involve 1,* m alkane systems, this findmg IS of some significance
solution over that predicted by the ideal solution law l?,” partml molar excess enthalpy of solute (%= = (aHE/aN,)r~i& L molar latent heat numbers of moles of components 1 and 2 N2 (N,+N,= N) P total pressure R gas constant excess of actual entropy of N moles of soluSE tlon over that predicted by the ideal solution law partial molar excess entropy of solute (3,” = 3,” (aS”/~N,h~ivJ T absolute temperature urn molar internal energy of pure component m the hquld state urn0 molar mtemal energy of same pure component as a vapour taken m the hmlt as
N,,
P+O
NOTATION
B2 GE
G,” HE
second vulal coefficient of solvent vapour excess of actual Gibbs function of N moles of solution over that predicted by the Ideal solution law partial molar excess Gibbs function of solute (@ = (aGE/aN,h.,J excess of actual enthalpy of N moles of
rl.
r2
molar volumes of solute and solvent as hqmds at given temperature average number of nearest nelghbours surrounding a monomer parameter allowmg for deviations from the geometric mean mlxmg rule for the attractive potential between molecules of types 1 and 2 number of carbon atoms m alkane cham vapour pressure of the solvent Zql and Zq2 are the numbers of mteractlons with nearest nelghbour molecules made by smgle molecules of types 1 and 2 respectlvely the numbers of segments constitutmg molecules of types 1 and 2
M
1252
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KING et al
mole fractions of solute and solvent m hqmd solution (x, = x1’@ in the present instance) 760 mole fraction of solute in solution at a solute Xl partial pressure of one atmosphere x:6pDAL Ideal mold fraction of solute at a solute parteal pressure of one atmosphere XI, x2
Greek
s ytnbols
YI 8 d,, &
activity coefficient of solute ( y, - (*)) solublltty parameter defined by eqn (4) segment fractions of components 1 and 2
( ( +I
41 y, 42”
xlrl
=
x,rl + xzr2) fractions of components
volume &’
=
1 and 2
XlVnlL x1vm,+x2vm* >
REFERENCES HI Hddebrand J H , Prausmtz J M and Scott R L , Regufar and Related Solut#ons Van Nostrand Remhold, New York 1970
PI Cuggenhelm
E
Press, Oxford [31 Staverman A [41 Kmg M B , Press, Oxford PI Chappelow C 1091 161 Clayton G T 171 Stupe D and
and Heaton L , Phys Rev 1%1 121 649 Tompson C W , J Chem Phys 1%2 36 392
A, Mrxtures, p 197 Oxford Umverslty 1951 3 , Ret Trav Chrm Pays-Bas 1950 69 163 Phase Equdlbnum m Mrwtures Pergamon 1%9 C and Prausrutz J M , A I Ch E J 1974 20
[S] Petz J I, J Chem
Phys I%5 43 2238 [9] Blanks R F and Prausmtz J M , Id Engng Chem FundI 19643 1 [lo] Hueems M L and Ann N Y . Acad Set 1942 43 1 riii Bond, A, Molecular Crystals,‘Lrqutds and Glasses Whey, L’ -’ New York 1968 WI Paulmg L , The Nature of the Chemrcal Bond Oxford Umverslty Press, Oxford 1948 of Polymer Chemrstry, p 418 Cornell [I31 Flory P J , Pnnc#es Umversity Press 1953 [141 Kmg M B and al-l\laJJaT H , Chem Engng Sn 1974 29 1003 [IS] Dm F, Thermodynamrc Fun&tons of Gases, Vols l-3 Butterworth, London 1956 [161 Hirate M , Ohe S and Nagahama K , Computer AIded Data Book of Vapour-Lquld Equrlrbnum, pp 20-22 Elsevrer, Amsterdam 1975 iI71 Gugnom R J , Eldrldge J W , Okay V C and Lee T J , r,D, A I Ch E J 1974 20 357 M L and Potter J B , Proc Roy Sot I%2 478 1101 McGlashan A267 H91 Hayduk W , Walter E B and Simpson P J , Chem Engng Data 1972 17 59 P-I Cheung H and Zander E H , Chem Engng Progr Symp Senes 1968 64(88) 34 Pll Preston G T and Prausmtz J M , Ind Engng Chem Proc Des Develop 1970 9 264 WI Robinson R L and Chao K C , Ind Engng Chem Proc Des Develop 1971 10 221 P31 Hayduk W and Cheng S C , Canad J Chem Engng 1970 48 93 ~241 Prausnttz J M , Eckert C A, Orye R V and O’ConneIl J P , Computer Cafculatrons for Multrcomponent VapourLtqurd Equdrbna Prentice-Hall, Englewood Chffs, New Jersey 1%7 of Fi’wd Phase ~251 Prausmtz J M , Molecular Thermodynamrcs Equdrbna Prentice-Hall, Englewood Chffs, New Jersey 1969
AL-NAJJAR
Iraq Petroleum Institute, Baghdad, Iraq (Recerved 24 December 1975, accepted 25 January 1977) Abstract-Solublhty data at 25°C for CO+, H,S and C,Ha and also for C,H, m the n-alkanes C,H,, to C&I, were consldered These were correlated usmg two actlvrty coefficient equations of the “regular solution” type, the first being the one including a “Flory-Huggins” entropy term (evaluated from molar volumes) which 1s normally used The second equation was slmllar except that the “Flory-Huggms” term was evaluated from molecular geometry instead of from molar volumes and, also a higher order approximation was used for this term This equation, m its composltlon-dependent form IS a variant on existing ones It was used m conlunctlon with an analytic expression for the number of segments r in a n-alkane chain as a function of the number of carbon atoms The correlation procedures were based on attrlbutmg a constant value to the mteractlon parameter, I,,, for a given solute dissolved m the entue range of alkane solvents In this way It was found possible to collate the data for CO, virtually wthm the experlmental error usmg either of the above equations together with experimentally determmed solute and solvent solublllty parameters The fit obtamed for the remauung solutes was not so close though the modltied regular solution equation then gave noticeably better results Finally, by adjusting the solute solubdlty parameters, it was found possible to collate the data for all the solutes within the experimental error m this way
lTWRODUCTlON
According to regular solution theory m its latest form [l], the activity coefficient for component 1 m a binary solution with component 2 1s given, after rearrangement, by
RT In y, = T[R(1 + In (Y/(x2 + Yxl) - Y/(x2 + Yx,))] + (&V
v, I((& - S*F + 211*&&)
where Y 1s the ratio ponents 1 and 2
(1)
of the molar volumes
of com-
Y = (V,,/V,*) 92’
42”
= X2Vln2/(Xl
vm,
RT In yz= Q” = R,E - TL?f
(2)
1s the volume fraction of component
2,
+ xzVm2)
S1 and Sz are the solublhty parameters and 2, defined by s = ((U,” - U,)/ V,)“’
chain structure and the relevant forces are presumably between segments on these chams and the solute molecules Qualitatively at least these segment/solute mteractlons would be expected to be independent of the chain-length of the solvent and the correlations developed below are m terms of I,, values which are constant for a gven solute dissolved m the alkane series Equation (1) was uutlally deduced[l] by considering the thermodynamic ldentlty
of components
(3) 1
(4)
(U IS the molar internal energy of the liquid and U,,,” that of the vapour at the same temperature but a hmltmg low pressure) 1,* 1s a parameter which allows for deviations from the geometric mean mixing rule for the attractive forces between the unlike molecules 1 and 2 The solvents considered m the present paper have a long-
(5)
where @, H,E and s” are partial molar values of the excess Gibbs function, excess enthalpy and excess entropy for component 1 m solution y, IS the actlvlty coeficlent of this component The final term m eqn (1) 1s equal to the partml molar excess enthalpy of component 1 as given by various models (Ref [l], p 82) The terms wlthm the first set of square brackets are approximately equal to the negative of the partial molar excess entropy of component 1 as given by the quasi-lattice model of solutions (Ref [l], p 75) Although this approxlmatlon 1s convenient, its use is not entirely necessary since exact expressions for the entropy of mixing have been deduced m terms of this model [2,3, lo] Of these, the earlier expressions of Huggms[lO], Guggenhelm[2] and other authors were suitable only for systems composed of linear molecules and/or monomers More recently Staverman[S], has shown how these expressions may be
1247
1248
M B
KING et a/
modtied If rmg compounds are present and Chappelow and Prausmtz[5] have given an equation (eqn 10 m their paper) for the partial molar entropy of mlxmg based on Staverman’s work The models used m estabhshmg the “partial molar
entropy” and “partial molar enthalpy” terms m the regular solution treatments are admittedly overslmphfied ones However, the argument has been advanced that errors m fitE and Ts,E ~111tend to cancel (Ref [l], p 92)
average number of nearest nelghbour molecules surrounding a monomer The value Z = IO was used m the present work This follows general practlce[5] and appears to be reasonably m accord with mformatlon deduced from X-ray measurements on simple fluids [4-8] 4, and & m eqns (7) and (8) are the segment fractions of components 1 and 2 being given by
so that a fair approxlmatlon for ‘yt may be obtamed by substltutmg the values of H,E and TS,E given by simple models mto eqn (5) EQUATIONS USED FOR CORRELATING soLr_mLITIEs IN TEE ALKANRS
GAS
The mole fraction solublltles and ideal solubllmes listed m Table 1 of Part I are related to the solute activity coefficients by the equation
The activity coefficients for the solutes gwen m this table were correlated both using the usual regular solution
equation
(eqn 1) and usmg the equation
r,=O90+0283(n-1)
RT In yl = T[R(ln (d,/xJ -(Z&2)
I,,, 6, and 6, m eqn (7) are defined as m the usual “regular solution” equation (eqn 1) The present work was concerned solely with alkane solvents and the number of segments (rz) constituting an alkane cham contammg n carbon atoms was inferred from molecular geometry by dlvldmg the length of the alkane chain by the average cross-section diameter The lowest energy configuratlon for an alkane chain corresponds to a planar zig-zag[13] Takmg the van der Waals radms of the methylene group to be 2A0 and the C-C covalent radius to be 0 771 A” [ 121, the average cross section diameter of this chain 1s found to be 4 4SA0 and the length (4 + [(n - 1)/2] x 2 52)A0, whence
In (1 + (2&/Zq,)((r,/r,)
- 1)))l
+ (&)* vmdts, - &Y + 2112&W
All three solutes were taken to be monomers,
($EIR)
=-ln
from eqn (1) mainly
1e
(7) r, = 1
This differs
(11)
(12)
m that the expresston
(&lx,) + (Z/2)q, In (It- (2&/Zq,)((r1/rz)
- 1)) (8)
has been inserted for the partial molar entropy m place of the more f amlhar result (~L?,~,EIR) = -In (&/xl) + (+,/xl) - 1 gven by the Flory-Huggms approxlmatlon Equation (8) follows from an exact development of the quasi-lattice theory of solutions as apphed to mixtures composed of linear and/or spherical molecules It would not be applicable if cychc molecules were present [3,5] It may be derived by partially dlfferentlatmg Guggenheim’s expression (eqn (10 10 8) m Ref 2) for the excess entropy of mlxmg m a system composed of lmear molecules with respect to the number of moles of component 1 Alternatively it may be obtamed, after considerable rearrangement, by combmmg eqn (9) (below) wth the more general equation given by Chappelow and Prausmtz [5] In eqns (7) and (8) q, 1s the number of surface sites
available on a molecule of type z For linear molecules this 1s related to the number of segments r, of which the molecule 1s constituted by the equation (Ref [2], p 415) q, = r, - (Z/2)(r, - 1)
(9)
Zq, IS the number of monomer molecules (1 e molecules for which r = 1) which may be placed around a molecule of type z Z IS the “co-ordmatlon number”, Le the
Conslderatlon of the van der Waals and covalent radu[l2] shows that this 1s not unreasonable for CO2 and H,S, but that It 1s more debatable for C,H,, (according to eqn 11 r, = 1 47 for this substance) The necessary parameters for Insertion m eqn (7) for correlating the actlvlty coefficients were taken from eqns (9) to (12) A closely inter-related treatment for handling gas solubihties has been given by Chappelow and Prausmtz[S] These authors give an expression for the partial molar entropy of mixing which reduces to (8) when mixtures composed of linear molecules are considered When turning to actlvlty behavlour, however they consider only the case of Infinite dllutlon No equation equivalent to (7), glvmg the composltlon dependence of the activity coefficients arising from an exact treatment of the combmatory entropy term appears to have been tested previously In this context it IS noteworthy that, even m gas solublhty calculations, the calculated activity coefficient at infinite dilution can differ significantly from that m the actual solution In the present work it was found that neglect of this dtierence led to dlscrepancles of up to 5% when the solublhty of H,S was consldered m the entire range of solvents In their treatment Chappelow and Prausmtz deduce their values of r and q from tables of Increments which are taken from earher work by Bondl[I l] The values thus obtained are rather larger than those deduced from the molecular mode1 used m the present work and which are given analytically by eqns (11) and (9)
The solubllltles of CO,, H,S and &H. m some normal alkane solvents-II THE soLuBILrrY
PARAMETERS
The solubdlty parameters and hqmd molar volumes used are listed m Tables 1 and 3 The solublldy parameters for carbon dloxlde and propane were calculated from the defining eqn (4) usmg values of U and U” taken from Dm’s Tables[15] Those for the alkanes were calculated from the expresslon 8, = (((L --RT)I V,,) + pz”( T(dB,/d T) - & + VA/ VA”’ to which (4) reduces when the vapour pressure IS sufficlently low The molar latent heats and the vapour pressures at 25°C were taken from Ref [14] and the second vinal coefficients B. for hexane, heptane, octane and nonane were calculated from McGlashan and Potter’s eqn (18) The contrlbutlon from the terms m the second vlrlal coefficient became progressively smaller as the alkane series ascended and was neglected for alkanes higher than C, Less complete data were avadable for H2S and m this case an estimate for 8 obtamed from the law of correspondmg states was used (Ref [l], p 210) Table 1 Solublllty parameters and molar volumes 25°C for the solutes
((cal/cm3)“‘)
Hexane Octane Nonane Decane
Dodecane Tetradecane Hexadecane Carbon dloxlde Hydrogen sulphlde Propane Ethane
V#”
From eqn (7)
Hexane Heptane Octane Decane Dodecane Tetradecane Texadecane Mean absolute error m x0&= Average II2
X
x
00119 00119 0 0121 0 0125 0 0129 0 0136 0 0142
0 0120 00119 00119 0 0123 0 0129 0 0137 0 0144
(2 = IO) se1.z
X
I 12 0 0915 00905 0 0892 0 0892 0 0910 00911 0 0925
0 0 0 0 0 0 0 09x
1 1x1o-4 0 0907
Solute (1) CO* HS C,H, C,H, CO* H,S C,H, C,H, t“Optrmrsed”
S, ((cal/cm’)“‘) eqn (1) 5 71 5 60t 6 50t 6 70t eqn (7) 5 71 6 50t 600t { 6 13 6 20t
I,,
00907 0 1048 OOCSS
0 0145 0 0910 00866
0 0055 0 0097 0 0189
Mean absolute error m predicted mole fraction solublhty
1 1x 2 3x 0 7x 20x
1o-4 1o-4 lo-” lo-.
0 9x 2 1x 07x 09x 20x
lo-4 1o-A 1o-3 1o-3 3 lo-”
value
In the case of propane, the use of the optlmlsed
solubdlty parameter (600) m conlunctlon with eqn (7) does not offer a clear-cut advantage over the use of the normally determmed one (6 13) Indeed the latter may be preferable (Fig 3)
alkanes
increase
CORRELATION OF GAS SQLUBILlTIES IN n-ALKANE SOLVENTS AT WC USING EQN (1) (THE USUAL REGULAR SOLUTION EQUATION) AND EQN Q
131 6 147 5 163 5 179 6 195 9 228 6 261 3 294 1 6174 43 70 89 25 93 6
From eqn (1) Solvent
parameter I,, and of the solute parameter (8,) for msertlon m eqns (1) and (7) for gas solubditles m the alkanes (values of S, are aven In Table 1)
The solublhty parameters of the slowly with cham length (Table 1)
Table 2 Correlation of mole fraction solubdlttes for carbon dioxide at 25°C usmg (a) the standard reguiar solution eqn (l), (b) eqn (7) (solubdlty parameters and molar volumes as listed m Table 1)
SdC
solubdlty predlctmg
(cm’lmole)
7 28 744 7 55 764 7 72 7 83 7 91 7 97 5 71 8 36 6 13 443
Heptane
Table 3 Values of the mteractlon
at
and solvents
s Substance
1249
0121 0120 0120 0123 0129 0135 0143
0 0926 0 0915 0 0899 0 0894 0 0907 00903 0 0913
1o-4 0 0910
x IS the experlmentally determrned mole fraction solublllty of CO, The values of I,, have been back-calculated usmg eqns (1) and (7) respectively to give an exact fit to the solubdlty m each alkane xCsa IS the mole fraction solublhty of CO, m each alkane calculated from eqns (1) and (7) using the average value for I,,
The correlations were based on the solublhty data for CO, and H2S obtamed m Part I and on data for C3H8 and C2H6 reported by Hayduk et al [19,23] The determmatlons on C,Hs reported m Part I did not extend to temperatures as low as 25°C and could not, therefore, be used The value used for the ideal mole fraction solublhty of ethane at 25°C was 0 0399 The correlation procedure was based on attnbutmg a constant value to the mteractlon parameter, l,*, for a given solute dissolved m the entire range of alkane solvents In this way it was found possible to collate the data for CO* vutually wlthm the experlmental error either usmg eqn (1) or using eqn (7) together with solute solublhty parameters determmed from pure component data The closeness of the fit obtained for CO, 1s mdlcated m Table 2 The fit obtamed for the other solutes was not qmte so close when the above parameters were used though the modified regular solution equation (eqn 7) then gave noticeably better results than (1) (Figs 2-4) However it was found that the data for these solutes could also be correlated wlthm the experlmental error with a constant value for 112provided an adjusted solub&y parameter was used for the solute (Curve 1 on Ftgs 2-4) (Although not Illustrated here, an equally close fit may also be obtained by multlplymg the solvent solublhty parameters by a factor which 1s constant for a given solute) The need for some flexlblllty m the value to be attributed to the solute solublhty parameter 1s not perhaps surprlsmg m view of the high vapour pressures of some of the solutes at 25°C and the consequent ambiguity m the “hqmd” propertles attributed to them at ambient pressure
1250
M
B
KING et al
Alkane
solvent
-
I+ 1 Mole fraction solulxhtles of carbon dmxlde m n-alkane solvents at 25°C (0) compared wrth predlcted values (curves 1 and 3) Curve 1 Calculated from eqn (7) with solute solubdlty parameter and I,, gtven m Table 3 Curve 3 Calculated from eqn (1) with solute solublhty parameter and 1,, given m Table 3
I 0 060
I
I
-
4
ll35r ‘SH14
, C6H16
,
,
clO% Alkane
‘1ZH26 solvent
, c14H30 -
, C16H34
Fig 2 Mole fraction solubdrtles of hydrogen sulphlde m nalkane solvents at 25°C (0) compared with predtcted values (curves 1, 2 and 3) Curve f Caiculated from eqn (7) with “optlmlsed” solute solubdlty parameter and l,, given m Table 3 Curve 2 Calculated from eqn (7) with unadjusted solute solublhty parameter (Table 1) and I,, = 0 0746 Curve 3 Calculated from eqn (1) with same solute solublhty parameter and I,, = 0 1005
Values of I,, and of the solute solublhty parameter required to fit the solublhty data for each of the solutes m the n-alkanes Cs to CX6 are given m Table 3 together with the mean absolute differences between the experrmental and calculated solublhtles These differences amount to less than 1% DISCUSSION AND CONCLUSIONS
The regular solution equations enable the solublhtles of CO*, HzS, C,Hs and C3Hs m the higher n1
ALkane
solvent
-
I
Rg 3 Mole fractton solubtitttes propane tn n-alkane solvents at 25°C (0) compared wtth predicted values (curves 1, 2 and 3) Curve 1 Calculated from eqn (7) with ‘optlmlsed” solute solubrhty parameter and I,, given m Table 3 Curve 2 Calculated from eqn (7) with unadjusted solute solubdrty parameter and 1,2 given m Table 3 Curve 3 Calculated from eqn (1) with same solute solublhty parameter and l,, = 0 0044 alkanes at 25°C to be correlated vutually within the experimental error using, at most, two adJustable constants for each solute These are 1,? and, m some instances, the solute solublhty parameter 6, 2 The value of 112 required for CO2 IS m good agreement with values of a slmllar parameter derived from an analysis of second vlrlal coefficient data (Ref [4], p 416 (c = 1- 1,*) and Ref [25], p 159) Rather szmdar, though shghtly higher values are also obtamed when phase data for COJalkane systems are correlated using equations of state at normal temperaturesrl6, 171 3 Several previous workers have correlated the phase behavlour of COJalkane and H,S/alkane systems using
The solublhties
of CO,, H,S
and C&I, m some normal
alkane
solvent+11
1251
\ St
C6HlL
Wl6
ClOH22 Atkane
‘12”26 solvent
ClLH30
C16H3L
-
Fig 4 Mole fractton solub~lltles of ethane m n-alkane solvents at 25°C (a) compared with predicted values (curves 1, 2 and 3) Curve 1 Calculated from eqn (7) with “optlmlsed” solute solublllty parameter and I,, gven m Table 3 Curve 2 Calculated from eqn (7) with unadjusted solute solublllty parameter (Table 1) and I,, = -0 0944 Curve 3 Calculated from eqn (1) with same solute solubdrty parameter and I,, = -0 1161
the usual regular solution equation (eqn 1) though their work has 1argeIy been confined to sohdlhqmd eqmhbna[20,21] and to vapour/hqmd eqmhbna for a few systems at elevated pressures[22] The alkane cham lengths considered by these workers were mostly lower than those m the present work and less tendency was observed for iI2 to be constant for a given solute For example Preston and Prausmtz[21] obtam values of -0 02, +0 08, +0 08 and +0 09 respectively from solld/hqmd eqmhbrmm data for CO, m CH,, CzHs. C,Hs and n-C,H,, respectively This form of varlatlon appears to be consistent urlth the average value of 0 09 obtained m the present work for CO2 with the n-alkanes C6 to Cl6 It does suggest, however, that the constancy m iB2may not extend to alkanes lower than Cq A study of the 112values derived by Robinson and Chao[22] from vapour/hqmd eqmhbrmm data for CO, and H,S m some of the lower members of the alkane series at elevated pressures leads to a similar conclusion 4 It appears from the present work that varlattons of 1,* with chain length for a given solute are comparatively small m n-alkane solvents higher than C6 and that, If smtably adlusted solute solublhty parameters are used, these varlatlons can be brought vn-tually to zero In view of the use m the petroleum Industry of correlations of the type proposed by Robinson and Chao[22], which involve 1,* m alkane systems, this findmg IS of some significance
solution over that predicted by the ideal solution law l?,” partml molar excess enthalpy of solute (%= = (aHE/aN,)r~i& L molar latent heat numbers of moles of components 1 and 2 N2 (N,+N,= N) P total pressure R gas constant excess of actual entropy of N moles of soluSE tlon over that predicted by the ideal solution law partial molar excess entropy of solute (3,” = 3,” (aS”/~N,h~ivJ T absolute temperature urn molar internal energy of pure component m the hquld state urn0 molar mtemal energy of same pure component as a vapour taken m the hmlt as
N,,
P+O
NOTATION
B2 GE
G,” HE
second vulal coefficient of solvent vapour excess of actual Gibbs function of N moles of solution over that predicted by the Ideal solution law partial molar excess Gibbs function of solute (@ = (aGE/aN,h.,J excess of actual enthalpy of N moles of
rl.
r2
molar volumes of solute and solvent as hqmds at given temperature average number of nearest nelghbours surrounding a monomer parameter allowmg for deviations from the geometric mean mlxmg rule for the attractive potential between molecules of types 1 and 2 number of carbon atoms m alkane cham vapour pressure of the solvent Zql and Zq2 are the numbers of mteractlons with nearest nelghbour molecules made by smgle molecules of types 1 and 2 respectlvely the numbers of segments constitutmg molecules of types 1 and 2
M
1252
B
KING et al
mole fractions of solute and solvent m hqmd solution (x, = x1’@ in the present instance) 760 mole fraction of solute in solution at a solute Xl partial pressure of one atmosphere x:6pDAL Ideal mold fraction of solute at a solute parteal pressure of one atmosphere XI, x2
Greek
s ytnbols
YI 8 d,, &
activity coefficient of solute ( y, - (*)) solublltty parameter defined by eqn (4) segment fractions of components 1 and 2
( ( +I
41 y, 42”
xlrl
=
x,rl + xzr2) fractions of components
volume &’
=
1 and 2
XlVnlL x1vm,+x2vm* >
REFERENCES HI Hddebrand J H , Prausmtz J M and Scott R L , Regufar and Related Solut#ons Van Nostrand Remhold, New York 1970
PI Cuggenhelm
E
Press, Oxford [31 Staverman A [41 Kmg M B , Press, Oxford PI Chappelow C 1091 161 Clayton G T 171 Stupe D and
and Heaton L , Phys Rev 1%1 121 649 Tompson C W , J Chem Phys 1%2 36 392
A, Mrxtures, p 197 Oxford Umverslty 1951 3 , Ret Trav Chrm Pays-Bas 1950 69 163 Phase Equdlbnum m Mrwtures Pergamon 1%9 C and Prausrutz J M , A I Ch E J 1974 20
[S] Petz J I, J Chem
Phys I%5 43 2238 [9] Blanks R F and Prausmtz J M , Id Engng Chem FundI 19643 1 [lo] Hueems M L and Ann N Y . Acad Set 1942 43 1 riii Bond, A, Molecular Crystals,‘Lrqutds and Glasses Whey, L’ -’ New York 1968 WI Paulmg L , The Nature of the Chemrcal Bond Oxford Umverslty Press, Oxford 1948 of Polymer Chemrstry, p 418 Cornell [I31 Flory P J , Pnnc#es Umversity Press 1953 [141 Kmg M B and al-l\laJJaT H , Chem Engng Sn 1974 29 1003 [IS] Dm F, Thermodynamrc Fun&tons of Gases, Vols l-3 Butterworth, London 1956 [161 Hirate M , Ohe S and Nagahama K , Computer AIded Data Book of Vapour-Lquld Equrlrbnum, pp 20-22 Elsevrer, Amsterdam 1975 iI71 Gugnom R J , Eldrldge J W , Okay V C and Lee T J , r,D, A I Ch E J 1974 20 357 M L and Potter J B , Proc Roy Sot I%2 478 1101 McGlashan A267 H91 Hayduk W , Walter E B and Simpson P J , Chem Engng Data 1972 17 59 P-I Cheung H and Zander E H , Chem Engng Progr Symp Senes 1968 64(88) 34 Pll Preston G T and Prausmtz J M , Ind Engng Chem Proc Des Develop 1970 9 264 WI Robinson R L and Chao K C , Ind Engng Chem Proc Des Develop 1971 10 221 P31 Hayduk W and Cheng S C , Canad J Chem Engng 1970 48 93 ~241 Prausnttz J M , Eckert C A, Orye R V and O’ConneIl J P , Computer Cafculatrons for Multrcomponent VapourLtqurd Equdrbna Prentice-Hall, Englewood Chffs, New Jersey 1%7 of Fi’wd Phase ~251 Prausmtz J M , Molecular Thermodynamrcs Equdrbna Prentice-Hall, Englewood Chffs, New Jersey 1969