Ind. Eng. Chem. Fundam. 1883, 22, 318-321
318
Chemical Thermodynamic Properties of Isomer Groups Robert A. Alberty Chemistry Depsrtmnt, Massachusetts Institute of Technology, CambrMge, Massachusetts 02 139
I n calculating the equilibrium composition of a system involving isomer groups it is advantageous first to calculate the mole fractions in terms of isomer groups and then to distribute the d e fraction in each isomer group among the individual members of that group. The first step in the calculation is conveniently carried out by use of the standard Gibbs energy of formation A G O 1 1 of the isomer group or equilibrium constants expressed In terms of sums of mole fractions of isomers in a group. The calculation of the corresponding standard enthalpy of formation standard entropy S01,and standard heat capacity C OPI is discussed and illustrated by calculating these quantities for four alkanes at three temperatures. I n view of the lack of data for calculating these quantities for higher isomers, the estimation of AGOtI using information on the numbers of isomers is discussed and illustrated for the alkanes.
Introduction When equilibrium calculations are made on organic systems with C4 and larger molecules, the number of isomeric species which have to be included increases very rapidly with the number of carbon atoms in reactant molecules. Some time ago, Smith (1959) pointed out that such calculations can be carried out more economically in two steps: (1)the isomers in a group are treated as a single species and the mole fractions of the isomer groups are calculated; (2) the mole fraction in an isomer group is distributed between the various individual species in the group. This is possible because in an ideal gas system the distribution of isomers within a group depends only on the temperature. Smith pointed out that calculations in the first step can be made by using an equilibrium constant KI which is the sum of the equilibrium constants Kfifor the formation of the individual isomers, of which there are NI in the group. NI
K ~=I c K f k k=1
(1)
The Kficould be for the formation from any chosen species, but here the standard thermodynamic properties are emphasized, and so Kfi is the equilibrium constant for the formation of isomer i from ita elements. After the mole fraction yI of the isomer group has been calculated, the mole fraction yi of an individual isomer may be calculated by
The reduction in the number of species that have to be considered in step 1 of an equilibrium calculation may be very large for reactions systems of practical importance. This approach may also be necessary for systems where the Gibbs energies of formation of individual isomers are not known, provided adequate estimates of AGOff can be made. For organic systems the number of isomers increases geometrically with the number of carbon atoms. There are 75 isomers of decane and over 377 for decene. Lederberg and co-workers have developed computer programs for calculating the numbers of isomers for various molecular formulas (Lederberg et al., 1969). The thermodynamic properties of all of the alkane isomers through C10H22are summarized by Stull, Westrum, and Sinke (1969), but values for the alkene isomers are apparently only known through CsH12. In view of the lack of data for calculations on equilibria in this range of carbon number and beyond, it is of interest to make estimates of standard Gibbs energies of formation AGOff for isomer groups. Derivations Equation 3 can be rearranged to a form that is more familiar. The standard Gibbs energy of formation for an isomer group is equal to the weighted average standard Gibbs energy of formation for the isomers in the group plus the Gibbs energy of mixing of the isomer group, assuming ideal gases. Nl
AGO, = Cr,AGof, k=1
where ri is the mole fraction within the isomer group. In calculating equilibrium compositions for large systems using computers, Dantzig and DeHaven (1962) and Duff and Bauer (1962) found it convenient to write the preceding two equatipns in terms of standard Gibbs energies of formation as NI
AGO, = -RT In [C exp(-AGofi/RT)] i=l Yi
- = ri = exp[(AGoff- AGofi)/RT]
YI
(3)
NI
+ R T kC= 1 r kIn r,
(5)
Since the Gibbs energy of mixing is necessarily negative, AGOfl is more negative or less positive than the weighted average standard Gibbs energy of formation given by the first term. Since Smith (1980) has given a concise derivation of eq 3 and 4, these equations may be used to derive the corresponding equations for AHofl,SoI,and CopI. The equation for the standard enthalpy of formation of the isomer group is obtained from
(4)
Here AGOfi is the standard Gibbs energy of formation of isomer i of an isomer group, and AGOff is the standard Gibbs energy of formation of the isomer group. This method of making equilibrium calculations was used by Smith and Missen (1974), and the concepts have recently been reviewed by Smith (1980) and by Smith and Missen (1982).
This differentiation yields Nl
NI
m0fi= icMofi =l exp[(AGon- AGofi)/RT] = i=l XriAHofi (7) 0 1983 American Chemlcal Society
Ind. Eng. Chem. Fundam., Vol. 22, No. 3, 1983 318
The equation for the standard entropy of formation ASo, of an isomer group is obtained from AS", = aAG,
-(a)
is related to the standard Gibbs energies of formation of isomer groups AGO, or equilibrium constants Kfjifor the formation of individual species from their elements as NR
AGO1 = -RT In K I = CujAGoffj
P
Son =
(14)
j=l
This differentiation yields NI
NI
i=l
i=l
CriSofi-RZri
Substituting eq 3 In ri
so that the standard entropy for an isomer group is the weighted average standard entropy plus the entropy of mixing the isomer group, assuming ideal gases. This equation has been written in terms of the standard entropy S O 1 of the group of isomers and the standard entropies Soi of the individual species, rather than entropies of formation, because the entropies of the unmixed elements required to form a mole appear on both sides of eq 9 and cancel. The standard heat capacity of formation ACoM of the isomer group at constant pressure is obtained from
This differentiation yields the following equation for the standard heat capacity of the isomer group. NI
C0p1 = x C o p iexp[(AG0, - AGofi)/RT] i= 1
NI,
N R
AGO1
+
= -RTzuj j=l In
In KI =
NR
NU
p 1
i-1
[cexp(-AGofji/RT)] i=l
C In [C exp(-AGofji/RT)lYi
(15) (16)
where Kui is the equilibrium constant of the formation reaction for the ith isomer of reactant j . If the Kyi's are expressed in terms of the mole fractions of the individual isomers and the elements from which it is formed, eq 3 is obtained. As an example, the conversion of methanol to butanes and C8 arenes is represented by CH4O(d = H20(d
+ 3/20C4Hio(g) + 1/20C&10(g)
(18)
where C4H1o represents the sum of n- and isobutane and C8H10represents the sum of the xylenes and ethylbenzene. The change in standard Gibbs energy for this reaction is given by
+
(11) This equation has been written in terms of heat capacities, rather than heat capacities of formation, because the heat capacities of the elements appear on both sides of eq 11 and cancel. The second term of this equation is always positive because the weighted average of the squares of the individual enthalpies of formation is always greater than the square of the weighted average enthalpy of formation. This is in accord with Le Chatelier's principle that as the temperature is raised the equilibrium shifts in the direction that causes the absorption of heat. Chemical Reactions in Terms of Isomer Groups The expressions for the thermodynamic properties of isomer groups derived above correspond with writing chemical reactions in terms of isomer groups and equilibrium constant expressions in terms of summations of mole fractions of isomers. The stoichiometric equation may be written NR
CUjIj = 0
j=l
(12)
where Ij represents isomer group j or individual reactant j (if it has no isomers), uj is the stoichiometric coefficient (positive for products and negative for reactants), and NR represents the total number of reactants. The mole fraction yj for an isomer group is equal to the summation of the mole fractions yjf of the individual isomers in the group. Thus for a reaction of ideal gases the equilibrium constant K I written in terms of isomer groups is
where P is the total pressure and PO is the standard-state pressure. This equilibrium constant expression written in terms of isomer groups rather than individual species
AGOI = -RT In KI = AGof(H20)+ 3/20AGoff(C4Hlo) f/zoAGodC8Hio)- AGof(CH40) (19) where AGon(C4Hlo)= -RT In (exp[-AGof(n-C4Hlo)/RT] exp[-AGof(i-C,Hlo)/RT]J (20) AGoff(C8Hlo)= -RT In {exp[-AGof(o-xyl)/RT] exp[-AGof(m-xyl)/RT] exp[-AGof(p-xyl)/RT] exp[-AGof(etbenz) /RT] 1 (21)
+
+
+
+
This yields the following equilibrium constant expression KI
= [YH20(Yn-C,Hlo+ Y i - C ~ H l ~ ) ~ / ~ ~ ( Y o+- x Ym-xyl yl -k Yp-xyl
+ Yetbenz)1'20(P/~)'/61/YCH,o (22)
Calculation of Thermodynamic Properties of Isomer Groups In order to illustrate the thermodynamic tables that can be produced for isomer groups, the values of AGO,, Man, SoI,and CopI for the butanes, pentanes, hexanes, and heptanes are given for three temperatures in Table I. These values have been computed from data in Stull, Westrum, and Sinke (1969) using APL programs to implement eq 3, 7, and 10. The standard entropies of the isomer groups have been calculated using AGO, = AHo, - TASoffand the standard entropies of graphite and hydrogen gas. The thermodynamic properties for a number of isomer groups may be calculated from existing data, but the usefulness of these quantities for higher hydrocarbons raises the question as to whether these quantities can be estimated beyond the range of existing data. The numbers of isomers can be calculated and these numbers may be of some help in estimating thermodynamic quantities. This question is examined here for the alkanes. Table I1 gives AGO, at 500,700, and 900 K for isomer groups of the alkanes up to Clo which have been calculated from data in Stull, Westrum, and Sinke (1969). These
320
Ind. Eng. Chem. Fundam., Vol. 22, No. 3, 1983
calculations were extended by Perry (1932) to include all carbon numbers up to Cd0and a value for Cg0. Table I1 shows the ratio of A(AGof) to -RT In NI. At each temperature this ratio approaches a nearly constant value at C9 and Clo. This suggests that AGOfl for CI1 and beyond may be estimated by using
Table I. Standard Thermodynamic Quantities for Alkane Isomer GrouDs' temp,
K
AG"f1
AH"fI
SO1
C3PI
C,H,,(g)
500 700 900
13.78 33.48 53.61
-34.57 -- 3 6.4 5 -37.54
88.50 102.44 114.71
37.43 45.58 51.98
C,H,,(g)
500 700 900
19.87 44.54 69.70
-40.84 -42.90 -44.05
101.37 118.72 133.86
46.97 56.29 64.16
C,H,,(g)
500 700 900
26.34 55.98 86.13
-46.73 -48.90 -50.15
114.25 135.10 153.08
C,H,,(g)
500 700 900
32.72 67.24 102.38
-52.26 -55.00 -56.46
128.03 151.60 172.28
57.04 67.28 75.96 63.30 76.97 87.58
AGOfl = AGOfn - R T In Nla
where a is 0.82 at 500 K, 0.69 at 700 K, and 0.63 at 900 K. The values of AGOfl for Cll through CI5 alkanes estimated in this way are shown in Table 111. These values would not make it possible to calculate the mole fraction of any particular isomer, but this becomes impractical anyway when the number of isomers becomes very large.
a A G " f 1 and A H " f 1 are in kcal mol-', and SO1 and C 0 p l are in cal K-' mol-'.
values are compared with the AGOf of the normal alkane in each case by calculating the quantity, A(AGof) that has to be added to AGOfnto obtain AGOm The normal alkane often has the least positive value of a AGOfi and is always near that extreme for the group. Alternatively, A(AGof) could be calculated by using the weighted average AGOfi. Here the normal alkane is used as a reference because AGOf values are avilable for the normal alkanes up to C20,and it is of interest to estimate AGOfl from Cll to CI5. The expected value of A(AGof) is readily calculated for the hypothetical case that all of the isomers have the same value of AGOf;. If the number of isomers is represented by NI then AGOfl = -RT In [NI exp(-AGofi/RV1 = AGOfi - RT In NI (23) This relation may also be obtained from eq 5. If all the isomers have the same value of AGOfi, they all have the same equilibrium mole fraction ri = l/NI, and eq 23 is obtained. The numbers of isomeric alkanes were first calculated accurately by Henze and Blair (1931) for carbon numbers up to C20, with also values for CZ5,C30,and C,o. These
Discussion Published tables of thermodynamic properties for isomer groups would be very useful for making various types of equilibrium calculations. In the second step of an equilibrium calculation the mole fractions of individual isomers could be calculated using the mole fractions ri of the individual isomers at that temperature. Tables of mole fractions of individual isomers can be produced in the process of calculating thermodynamic properties for isomer groups. Data is available in Stull, Westrum, and Sinke (1969) and elsewhere for calculating thermodynamic properties of a number of isomer groups. However, data is lacking for higher numbers of carbon atoms and so the question arises as to the extent to which estimates can be made for higher isomer groups, as illustrated here for Cll to CI5 alkanes. Existing calculations of the numers of isomers of the alkanes do not include chiral properties. Molecules with three different groups attached to a single carbon atom have D,L isomers, and further isomers are introduced by additional asymmetric carbon atoms. In the alkanes the existence of D,L isomers is first encountered at C, where there are D,L isomers for 3-methylhexane and 2,3-dimethylpentane. For C8 there are also two D,L pairs, and beyond that the D,L pairs increase rapidly. These chiral isomers have not been included in the current calculations,
Table 11. Standard Gibbs Energies of Formation (kcal mol-' ) of Alkane Isomer Groups A G"f1
500 K 4 5 6 7 8 9 10
2 3 5 9 18 35 75
13.78 19.87 26.34 32.72 39.77 45.83 52.24
4 5 6 7 8 9 10
2 3 9 18 35 75
33.48 44.54 55.98 67.24 79.21 90.14 101.43
4 5 6 7 8 9 10
2 3 5 9 18 35 75
53.61 69.70 86.13 102.38 119.37 135.16 151.39
14.55 21.52 28.31 35.13 42.02 48.70 55.78
-0.77 -1.65 -1.97 -2.41 -2.25 -2.87 -3.54
1.11 1.51 1.23 1.10 0.78 0.81 0.83
34.19 46.19 57.98 69.80 81.71 93.62 105.53
-0.71 -1.65 -2.00 -2.56 -2.50 -3.48 -4.10
0.74 1.07 0.89 0.84 0.62 0.70 0.68
54.33 71.44 88.30 105.19 122.20 139.21 156.20
-0.72 -1.74 -2.17 -2.81 -2.83 -4.05 -4.81
0.58 0.88 0.75 0.71 0.55 0.63 0.62
700 K 5
(24)
900 K
Ind. Eng. Chem. Fundam. 1983,22, 321-329
Table 111. Estimated Standard Gibbs Energies of Formation (kcal mol-' ) for Alkane Isomer Groups
AHofi = standard enthalpy of formation of isomer i, kcal mol-' AHofi = standard enthalpy of formation of isomer group I, kcal mol-' Ij = represents the chemical formula for the jth isomer group in a chemical reaction Kfi = equilibrium constant for the formation of isomer i from elements Kn = equilibrium constant for the formation of isomer group I from elements KI = equilibrium constant for a reaction involving isomer groups NI = number of isomers in an isomer group NR = number of reactants (expressed in isomer groups) in a chemical reaction v j = stoichiometric coefficient for reactant j in a chemical reaction P = total pressure, atm Po = standard-state pressure, atm ri = mole fraction of species i in an isomer group S0i = standard entropy of isomer i, cal K-' mol-' Sol = standard entropy of isomer group I, cal K-l mol-' yi = mole fraction of isomer i yI = mole fraction of isomer group I Literature Cited
-RT In Nc
NI
12 13 14 15
159 355 802 1858 4347
11 12 13 14 15
159 355 802 1858 4347
11 12 13 14 15
159 355 802 1858 4347
11
AGDfn
500 K 62.66 69.55 76.43 83.31 90.19 700 K 117.43 129.34 141.25 153.15 165.05 900 K 173.20 190.21 207.21 224.21 241.20
Nla
AG"f1
-4.13 -4.79 -5.45 -6.13 -6.83
58.53 64.76 70.98 77.18 83.36
-4.87 -5.64 -6.42 -7.23 -8.04
112.56 123.70 134.83 145.92 157.01
-5.71 -6.62 -7.54 -7.53 -9.44
167.49 183.59 199.67 216.68 231.76
321
but this is a subject that needs more attention.
Dantzlg, 0.B.; DeHaven, J. C. J . Chem. Phys. 1962, 36. 2620. Duff, R. E.; Bauer, S. H. J . Chem. Phys. 1982, 36, 1754. Henze, H. R.; Blair, C. M. J . Am. Chem. SOC. 1931, 53, 3077. Lederberg, J.; Sutherland. G. L.; Buchanan, B. G.; Feigenbaum, E. A.: Robertson, A. V.; Duffield, A. M.; Djerassi, C. J . Am. Chem. SOC. 1969, 91, 2973. Perry, D. J . Am. Chem. SOC. 1932, 5 4 , 2918. Smith, B. D. AIChE J . 1959, 5 , 26. Smith, W. R . Chapter in "Theoretlcal Chemistry, Advances and Perspectives"; Henderson, D. Eyring, H.,Ed.: Academic Press: New York, 1980; Vol. 5. Smith, W. R.; Missen, R. W. Can. J . Chem. Eng. 1974, 52, 280. Smith. W. R.; Missen, R. W. "Chemical Reaction Equilibrium Analysis: Theory and Algorithms"; Wiley-Intersclence: New York, 1982. Stull, D. R.; Westrum, E. F.; Sinke, G. C. "The Chemical Thermodynamics of Organic Compounds"; Wlley: New York, 1969.
Acknowledgment The financial suport of the Dreyfus Foundation is gratefully acknowledged. Nomenclature Copi = standard heat capacity of isomer i at constant pressure, cal K-' mol-l CopI = standard heat capacity of isomer group I at constant pressure, cal K-l mol-' AGOfi = standard Gibbs energy of formation of isomer i, kcal mol-' ACon = standard Gibbs energy of formation of isomer group I, kcal mol-'
Receiued for review July 1, 1982 Accepted April 27, 1983
0-Alkylation of Phenols for Upgrading of Coal-Derived Liquids. 1. Reaction of Phenols with Branched Olefins Kenneth A. Gould' and Robert B. Long Corporate Research Science Laboratories, Exxon Research and Engineering Company, Linden, New Jersey 07036
Model phenolic compounds as well as coalderived phenols were found to be readily etherified by reaction with 2-methyld-butene and an acM catalyst. The reaction is equilibrium limited with formation of ethers being favored at low temperatures and high olefin concentrations. Ring alkylation can be minimized by employing mild reaction conditions. Various aclds were employed and 40% sulfuric acid in methanol was found to be superior to 40% sulfuric acid in water, especially at high olefin concentrations. The acidlmethanol solutions were also found to be quite useful for extraction of phenols from coal naphtha. Increased steric bulk in the vicinity of the phenolic hydroxyl group was observed to retard etherification.
Introduction Naphtha boiling range liquids produced from conversion of coal by such techniques as donor solvent or catalytic liquefaction, pyrolysis, and hydropyrolysis generally contain significant (10-50% ) concentrations of phenols (Haggin, 1981). These phenols are known to be toxic and corrosive and they are suspected of contributing to coalliquid instability, high viscosity, and incompatibility with 0196-4313/83/1022-0321$01.50/0
petroleum feedstocks (Schlosberg and Scouten, 1981). One technique for eliminating phenols from coal-derived liquids is catalytic hydrodeoxygenation. This procedure converts the phenols into hydrocarbons and removes the oxygen as water. The problem, however, is that large amounts of hydrogen are consumed. An alternative approach to eliminating the hydroxyl groups is conversion to derivatives such as ethers. Inasmuch as these ethers 0
1983 American Chemical Society
