J . Phys. Chem. 1990, 94, 8471-8482 ditional insights to the solvation or other effects.
Conclusions Explicit expressions of the chemical potential and activity coefficient of the Stell-Lebowitz perturbation theory are reported. The one-parameter equation can fit experimental data well. The improvement over the Debye-Hiickel equation comes mainly from the hard-sphere properties. For some cations, the ionic-diameter
8477
dependence of experimental data is opposite from the theoretical prediction. Correlations developed upon this simple result are possible, whereas improvement in the RPM is necessary. Work is being carried out t o apply the recent perturbation iheory of Henderson et a1.6for mixtures and compare it with experimentaldata. Acknowledgment. I thank Dr. Douglas Henderson for reading the work and pointing out some major errors.
Dependence of the Standard Thermodynamic Properties of Isomer Groups of Benzenoid Polycyclic Aromatic Hydrocarbons on Carbon Number Robert A. Alberty* and Kuo-Chih Chou Department of Chemistry, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 (Received: March 28, 1990)
The benzenoid polycyclic aromatic hydrocarbons can be organized as an infinite number of homologous series with an increment of C4H2between the successive isomer groups in each series, rather than CH, for the usual homologous series. The standard thermodynamic properties in the range 298.15-3000 K have been calculated for the isomers in 20 isomer groups in the first six series. Each of the standard thermodynamic properties has been found to be linear in carbon number n. The parameters A(A,Go) and B(A,Go) for representing the dependence of the standard Gibbs energy of formation on carbon number, A&", = A(A,Go) + B(A,G")n, are of special interest because the equilibrium distribution within a homologous series at constant T, f,PHIdepends on A(A,Go) and the equilibrium distribution at constant T, P,PCIH, PH2depends on B(A@). It is especially interesting to find that the A parameters for A C , , Ago,,, AfH",, and A&",, are the same for all six series. This has made it possible to determine parameters for representing the temperature dependences of all of the thermodynamic properties of all six series by linear regression of the data on the six series simultaneously. Thus any property for any isomer groups in the first six series at any temperature in the range 298.15-3000 K can be calculated by using the 42 parameters involved and the I O parameters for representing the temperature dependences of the properties of graphite and molecular hydrogen.
Introduction The infinite number of isomer groups of benzenoid polycyclic aromatic hydrocarbons can be organized into an infinite number of homologous series in which the increments between successive isomer groups are C4H2, rather than the C H 2 of ordinary homologous series.'v2 In making equilibrium calculations on benzenoid polycyclic aromatic hydrocarbons, it is advantageous to use isomer groups to reduce the number of species and estimate thermodynamic properties at higher carbon numbers, where data are l a ~ k i n g . ~The , ~ molecular formulas for isomer groups in the first six series are given in Table I with the ranges of carbon number n. Tables of standard thermodynamic properties of all of the isomers in the first several isomer groups in these six series in the range 298.1 5-3000 K have been calculated by using the Benson group additivity method5 with parameters of Stein and Fahr.68 The numbers of isomer groups studied are given in Table 1. The published tables of the standard properties of isomer groups show that all of the properties are linear in carbon number. This is illustrated by Figures 1-3, which show the isomer group properties A$",, AfHon,and A@", at IO00 K as a function of carbon number n. It is striking to find that the plots for these properties have a common intercept for the six series. The lines in the figures were obtained by linear regression by the plotting program. The plots for A C P nare similar but have relatively more (1) Dias, J. R. Handbook of Polycyclic Hydrocarbonr; Elsevier: New York. 1987: Part A. (2) Alberty, R. A.; Reif, A. J . Phys. Chem. Ref. Data 1988, 17, 241. (3) Alberty, R. A. J . Phys. Chem. 1983,87, 4999. (4) Alberty, R. A. J . Phys. Cliem. 1989, 93, 3299. ( 5 ) Benson, S.W. Thermochemical Kinetics; Wiley: New York, 1976. (6) Stein, S.E.; Fahr, A. J . Phys. Chem. 1985,89, 3714. (7) Alberty, R. A.; Chung, M. B.; Reif, A. J . Phys. Chem. Ref. Dora 1989, 18, 77. (8) Alberty, R. A.; Chung, M . B.:Reif, A. J . Phys. Chem. Ref. Data 1990, 19, 349.
0022-3654 f 90 f 2094-8411$02.50f 0
TABLE I: Formulas for Isomer Groups in Six Homologous Series of Benzenoid Polvcvclic Aromatic Hvdrocarbons no. of isomer series formulas values of n groups studied benzene CnHn/2t3 6, 10, ... 6 CnHn/2+2 16, 20, ... 4 pyrene CnHn/2tl 22, 26, ... 3 naphthopyrene C,H,/2 24, 28, ... 3 coronene 2 naphthccoronene CnHn12-l 30, 34, ... CnHn/2-2 32, 36, ... 2 ovalene
noise because of the small magnitude of AfC",. When there is a common intercept for certain properties for the six series, there will be different intercepts for other properties, as will be shown here. In order to discuss the linear dependences on carbon number within a given homologous series, eqs 1-9 are used. The bars AfCop, = A(AfC",) A g o , = A(ApSO)
+ nB(A&Op)
+ nB(A,,S") AfHon= A(AfHo) + nB(AfHo) A@", = A(A,Go) + nB(A@") ~ o f n= A(Cof) + n E ( C p ) So, = A(So) + nB(So) H", - Ron.298= A(Ho - R"298)+ nB(Ho - H o 2 9 8 ) - H" = A(Ro - H",,)+ nB(R" - H",,) Go, - Ron,,,= A(Go - R",,) + nB(G" - R",,) PO,
n,scr
(1)
(2) (3) (4)
(5) (6) (7) (8) (9)
indicate molar properties, but no bar is used on the formation properties because the subscript f indicates that 1 mol of the isomer group is formed. The parameters A and B are functions of temperature only for a given homologous series. The labeling of 0 1990 American Chemical Society
The Journal of Physical Chemistry, Vol. 94, No. 22, 1990
Alberty and Chou
0
500 450
-100 400
350
-200
-
300
\
250
n
E"
-300
-3
Y
W 0
-400
F
-500
200
t 50
100
-600
-700
,
- 5 0 1 " " " " ' " " " J 0
5
10
15
25
20
n
30
35
40
0
5
10
15
20
25
30
35
40
"c
C
Figure 1. Standard molar entropies of formation (in J K-' mol-') of isomer groups of polycyclic aromatic hydrocarbons at 1000 K as a function of carbon number. The lines were calculated by linear regression for the series separately. From left to right, the first members of the series are benzene, pyrene, naphthopyrene, coronene, napthocoronene, and ovalene.
Figure 2. Standard molar enthalpies of formation (in kJ mol-') of isomer groups of polycyclic aromatic hydrocarbons at 1000 K as a function of carbon number. The lines were calculated by linear regression for the series separately. From left to right, the first members of the series are benzene, pyrene, naphthopyrene,coronene, naphthocoronene,and oval-
parameters in this way is a little unusual, but the labels are too large to be subscripts. The standard thermodynamic properties have been arranged in this order because A ( A , C p ) , A(ApSO), and A(ArHo) are the same for all six homologous series at each temperature. Therefore, the following derivations are based on the prior determination of the six parameters in eqs 1-3 by linear regression of the values of Afcop~,.A$o,,and AfHonsimultaneously for a given homologous series. The 12 parameters in eq 4-9 can be expressed in terms of the first six parameters. The properties Ron- Ron,,,and Gon- H",,, are included because they have the advantage that they are calculated entirely from data commonly compiled for the substance i t ~ e l f . ~ * ' ~ So far the discussion has been focused on the dependence of the standard thermodynamic properties on carbon number, but, of course, the A and B parameters are functions of temperature and linear regressions should include both carbon number and temperature. Alberty and Chou" have shown how this can be done for the alkylbenzene homologous series. They showed how all of the standard thermodynamic properties for a homologous series can be represented by use of empirical equations with 12 parameters, which are characteristic of the isomer groups. The parameters for calculating all of the properties of all of the isomer groups in a homologous series can be determined from data on only two or three of the standard thermodynamic properties. Several choices of standard thermodynamic properties are possible. son,and H", - H",,,,, fof the Alberty and Chou used alkylbenzenes because this focuses attention on the properties of the isomer groups themselves. It is convenient to use A,Cpn,A , P , and AfHO, for the polycyclic aromatic hydrocarbons because the A parameters for these properties are the same for the first six series. However, these are not the only possible choices for the
linear regression. For example, So, and A", - H",,, or A$", and AfHo, could be used because these pairs each involve all 12 parameters (see Appendix). The parameters A ( W ) and B ( W ) are especially interesting because Alberty and OppenheimI2 showed that the equilibrium distribution within a homologous series depends only on B(prC0) when PC2", is specified, and Alberty'3,'4 has shown that the equilibrium distribution depends only on A(AI(;") when P H 2is specified.
c,,
(9) Barry, T. 1. Chemical Thermodynamics in Industry: Models and Computaiion; Blackwells: London, 1985. ( I O ) Barin, I.; Knacke, 0. Thermochemical Properties of Inorganic Subsrunces; Springer-Verlag: Berlin, I973 (1977 supplement). ( I 1 ) Alberty, R. A.; Chou, K.-C. J. Phys. Chem. 1990, 94, 1669.
ene.
Relations between the Parameters in Eqs 1-4 The first eight parameters are interrelated, and these interrelations must be respected in the process of linear regression. Since aAfHon/aT = A C p n (10) substitution of eqs 1 and 3 yields 8A(ArH")/8T + naB(AfHo)/aT = A ( A , C p ) + n B ( A C P ) (1 1)
Since this equation applies for all carbon numbers n in the homologous series, a A ( A f H " ) / a T = A(ArCOp) (12) aB(ArHo/ar) = B(AfCOP) (13) Similarly, aA$",/aT = ArCOp,/T (14) yields aA(A$")/aT A(ACp)/T (15) as(Ap)/aT = B(Afcp)/T (16) Thus the A's and B's in eqs 1-3 obey the same equations as the parent thermodynamic properties. Alberty, R. A.; Oppenheim, I. J. Chem. Phys. 1986, 84, 917. (1 3) Alberty, R. A. J. Chem. Phys. 1989,91, 7999. (14)Alberty, R. A. J. Chem. Phys. 1990, in press. (12)
The Journal of Physical Chemistry, Vol. 94, No. 22, 1990 8479
Isomer Groups of Benzenoid Hydrocarbons
A and B. The difference between the expressions for A ( e f ) and
I200
B ( C o p )for other homologous series of polycyclic aromatic hydrocarbons are discussed in the next section. The molar entropy So,,of an isomer group in the benzene series is given by So, = Apso, + (3/2)S0H2 n(Soc + S0H2/4) (24)
I100
IO00
+
900
-
n
800
E \
700
Substitution of eqs 2 and 6 yields A ( S o ) = A(A,So) + (3/2)SoHZ
B ( S o ) = B(A,S")
3
Y
W
600
0
500
(26)
The next thermodynamic property, A", - R0n,298,the enthalpy increment for the nth isomer group, can be calculated by using the thermochemical cycle in Alberty and Chou." R",,- Ron,298= AfHO,, - AfHon.298 n(R"c- A'c.298) (3/2 + n/4)(R0Hz - R0H2,298) (27)
+
.4-
a
(25)
+ S 0 c + S0H2/4
400
+
200
Substituting eqs 3 and 7 yields A(A" - H0298) = A(AfH") - A298(ArH0) + (3/2)(R0H2 - R ' H ~ , Z ~(28) ~)
100
B(Ro - R0298) = B(AfHo) - B29s(AfH0) +
300
- A°C.298) + (RoHz - RoH2,298)/41 (29) lo
l5
2o
25
30
35
40
"c
Figure 3. Standard molar Gibbs energies of formation (in kJ mol-') of isomer groups of polycyclic aromatic hydrocarbons at 1000 K as a function of carbon number. The lines were calculated by linear regression for the series separately. From left to right, the first members of the series are benzene, pyrene, naphthopyrene, coronene, naphthccoronene, and ovalene.
The standard Gibbs energy of formation was not included in the linear regression to determine the parameters for representing the temperature dependences of the isomer group properties because it is considered to be a derived property and is calculated from
A&',, = AlHO, - TAPS",,
(17)
Substituting eqs 2, 3, and 5 yields A(A&O) = A(AfHo) - TA(A,So)
(18)
B(AP0)
(19)
B(AfHo) - TB(A,So)
Thus the A and B parameters for the standard Gibbs energy of formation also obey the same equations as the Gibbs energy itself.
Expression of the Ten Parameters in Eqs 5-9 for the Benzene Series of Polycyclic Aromatic Hydrocarbons in Terms of the Parameters in Eqs 1-3 and the Properties of the Elements The formation reactions for the successive isomer groups in the benzene series are given by nC(graphite) + (n/4 + 3 / 2 ) H ~ ( d= CnHn/2+3k) n = 6, 10, 14,
... (20)
The parameters in eq 4-9 can be expressed in terms of the parameters in eqs 1-3 for a specific homologous series. Since Cop,, is given by p p n
=
A C p n
+ ( 3 / 2 ) p p H 2 + n ( C , + PPH2/4)
(21)
substitution of eqs 1 and 5 yields
A(ef) = A(AfCOf) + (3/2)pfH2
B(ef) = B ( A C f )+ p f c + efH2/4
(22) (23)
It is important to note that when the elements are introduced, the A and B parameters do not follow the statement in the preceding section that the A's and B s obey the same equations as the parent
thermodynamic properties. The elements play different roles in
The enthalpy of an isomer group relative to the standard element reference, denoted by ser, at 298.15 K can be expressed in terms of the enthalpy of formation by use of the formation reaction to obtain A", - P,,, = W o n n(R"c- R"c.298) + (n/4 3/2)(A0H2- R O H ~ , ~ )
(30) Substituting eqs 3 and 8 yields
A(Ao - Rose,)= A(AfHo) + (3/2)(R0H2 - H"~~,298) (31)
B(R0 - A",,) = B(AfHo) + [(Roc - R°C,298) + (RoHz - RoH2,298)/41 (32) The last parameters to be calculated are for the standard Gibbs energy relative to the enthalpy of the elements at 298.15 K. This property for the nth isomer group is defined by Go,- H",,,,, = Ron- Ron,,,,- TSO,, (33) Substitution of eqs 6, 8, and 9 yields A(G0 - A",,,)= A(R0 - R",,,)- TA(S0)
(34)
B(GO - A",,,) = B(Ro - R",,,)- TB(So)
(35)
Use of eqs 25, 26, 33, and 34 yields A ( c o - A",,,) = A(AfH") + (3/2)(R0H2 - R0H2,298)~ [ ~ ( ~ p+s (o3 )/ 2 ) ~ 0 ~ ,(36) 1
B(Go - R",,,) = B(AfHo) + [(Roc- Roc,298) + ( R O H ~RoH2,298)/41 - T[B(Afi0) + S ° C + S0H2/4i (37) Thus for a given homologous series, the A and B parameters in eqs 4-9 can be calculated from the first six A and B parameters and the standard thermodynamic properties of the elements in the formation reactions for the successive isomer groups.
Relations between A and B Parameters for Various Series of Polycyclic Aromatic Hydrocarbons The relations between the parameters that have been derived in the preceding section are illustrated by Figure 4, which gives the standard heat capacity for an isomer group as a function of carbon number at lo00 K, and Figure 5 , which gives the standard entropy of an isomer group as a function of carbon number at 1000 K. Equation 22 shows that A(Cop)for the successive series should each be smaller by e p H 2 / 2 , which is 15 J K-' mol-' at 1000 K. In making Figure 4 some very small adjustments of A(Cop)were made so that the intercepts are evenly spaced, but
8480 The Journal of Physical Chemistry, Vol. 94, No. 22, 1990
Alberty and Chou
TABLE 11: Residuals of Standard Gibbs Energy of Formation 298.15 C6H6 -5.53 CIoH8 2.92 C14HIO 3.14 -1.55 CisHi2 C22Hi4 0.10 C26H16 -0.58 Cl6HIo 2.25 C20HI2 -2.76 C24H14 -2.72 C28H16 2.98 CZ2Hl2 -0.46 C26H14 -1.55 1.77 CjoHl6 C24H,2 4.44 C28H14 0.57 C32H16 -3.73 C30H,4 3.04 C34H16 -2.64 C32H14 4.40 C36H16 -3.87
benzene series
pyrene series
naphthopyrene series coronene series naphthocoronene series ovalene series
300.00 -5.54 2.92 3.15 -1.51 0.09 -0.61 2.23 -2.76 -2.69 2.97 -0.49 -1.55 1.79 4.43 0.55 -3.72 3.03 -2.63 4.40 -3.87
500.00 -6.21 2.96 4.40 1.97 -0.51 -3.21 0.89 -2.41 -1.09 1.87 -2.44 -1.00 2.52 3.31 -0.53 -2.08 2.18 -2.04 4.06 -3.73
700.00 -7.24 3.00 5.83 3.19 -0.41 -5.31 0.10
-1.39 -1.51 0.95 -3.75 -0.48 2.35 2.30 -1.27 -1.04 1.16 -1.65 3.46 -3.74
T IK 1000.00 -8.88 3.29 8.35 3.83 0.52 -7.51 -0.43 0.85 -2.06 0.16 -4.98 -0.04 2.79 1.53 -1.46 -0.33 0.01 -0.63 2.90 -3.26
1500.00 -11.68 3.96 12.78 4.55 2.59 -10.14 -0.77 5.02 -2.44 -0.57 -6.26 0.42 4.55 I .42 -0.60 0.25 -1.35 1.68 2.57 -1.78
2000.00 -14.59 4.49 16.90 4.94 4.23 -12.84 -1.27 8.7 1 -3.16 -1.73 -7.65 0.37 6.1 1 1.57 0.20 0.17 -2.98 3.63 2.00 -0.74
2500.00 -16.78 5.87 21.95 6.60 7.23 -13.73 -0.68 13.48 -2.48 -1.28 -7.73 1.72 9.30 3.23 2.43 1.59 -3.18 7.15 2.93 1.77
3000.00 -20.44 5.22 24.38 5.30 6.67 -18.48 -2.53 15.16 -5.27 -4.75 -10.76 -0.40 8.48 2.10 1.12 -1.08 -6.83 6.68 0.48 0.23
1600
1400
900 800
t
1200
1000
700 n
800
600
Y
\ -3
Y
500
\
W
600
-3
Q
0
W
400
400
m 300 200
200
OVi
100
-200
0 - 1 O O L ' I ' I '
0
5
10
I
15
'
I
20
'
I
25
'
I
30
'
I
35
'
-400
j
40
5
0
10
15
20
25
30
35
40
"c
Figure 4. Standard molar heat capacity (in J K-' mol-')of isomer groups of polycyclic aromatic hydrocarbons at 1000 K as a function of carbon
Figure 5. Standard molar entropies (in J K-I mol-') of isomer groups of polycyclic aromatic hydrocarbons at 1000 K as a function of carbon
number. From left to right, the first members of the series are benzene, pyrene, naphthopyrene, coronene, naphthocoronene, and ovalene.
number. The lines were calculated by linear regression for the series separately. From left to right, the first members of the series are benzene, pyrene, naphthopyrene, coronene, naphthocoronene, and ovalene.
it is clear from the figure that these lines represent the data very well. Figure 5 shows that the shifts in the intercept for the entropy are larger. According to eq 25, the shift in intercept in going from which is 84 J K-I mol-' one series to the next higher one is SoH2/2, at 1000 K. In this figure the lines were obtained by linear regression of the data shown. Equations for Linear Regression of a Single Homologous Series Alberty and Chou" showed how all of the data on COP", so,, and H" - H",,,, for the alkylbenzene homologous series can be used to determine 12 parameters by linear regression of these properties simultaneously. These 12 parameters and the 10 parameters for graphite and molecular hydrogen can be used to calculate any standard thermodynamic property of any isomer group in the homologous series at any temperature in the range
298.15-3000 K. However, for the polycyclic aromatic hydrocarbons, it is more convenient to fit A , C p , , A P , , and AJ-I", because these properties have common intercepts for the six series at zero carbon number. Therefore, the data on a single series of polycyclic aromatic hydrocarbons are fit with &copn
= (A
k,n)
+ ( B + kbn)/T1/'+ ( c+ k , n ) / T + (D + k d n ) / p (38)
AF", = ( A + k,n) A f H o n= ( A
In T - 2 ( B + k , n ) / T 1 / *- ( C + k , n ) / T ( 1 / 2 ) ( 0 + k d n ) / p + ( E + ksn) (39)
+ k,n)T + 2(B + kbn)T'/' + (c+ k,n) In T (D+ k d n ) / T + (F + khn) (40)
The Journal of Physical Chemistry, Vol. 94, No. 22, 1990 8481
Isomer Groups of Benzenoid Hydrocarbons TABLE I11 Parameters for Graphite and Molecular Hydrogen
graphite hydrogen
A
B
C
2.68619EOI' 6.18082EOl
2.05477E02 -1.8981OE03
-1.30775E04 3.08701E04
D 1.20390E06 -2.36815E06
E
-1.60604E02 -3.51 121E02
'Read as 2.68619X IO'.
TABLE IV: Parameters of Six Series of Polycyclic Aromatic Hydrocarbons benzene pyrene naphthopyrene A
B C
D E
F
-2.72717EOl' 3.10095E03 -9.34066E04 9.47597806 2.26787E02 4.62949E05
coronene
naphthocoronene
ovalene
k,, -4.87814EOO ka2 -4.30818E00 ka3 -5.79334EOO ka4 -6.45310EOO ka5 -6.55236800 kbl 4.71385E02 kb2 kcl -1.03465E04 kc2 kdl 9.76888805 k.32 k,l 3.12659EOl ks2 khl 6.23963804 kh2
3.94291802 kb3 5.05279802 ku 5.4101OEO2 kb5 -8.20208803 kc3 -1.04046804 k,4 -1.07569804 kc5 7.27985E05 kd3 9.78383E05 kd4 8.68266E05 kd5 2.65480801 k,, 4.31728E01 k,, 5.06935801 k,s 5.10756E04 kh3 6.04794804 kh4 6.07258E04 kh5
ka6 -6.67693E00 5.48735802 kM 5.47429E02 -1 .I 1538804 kc6 -1.09595E04 9.65270805 kd6 9.32705E05 5.25508EOl ko6 5.43271801 6.28391804 kh6 6.15388804
'Read as -2.727I7 X IO'. TABLE V: Values of A (A&') (kJ/mol) and B(A6') (kJ/mol) for Six Series
4%") / T IK
B(%")l(kJlmol)
(kJ jmolj
benzene
6.21 6.27 14.65 26.25 46.14 81.76 118.38 155.31 192.37
21.53 21.57 26.02 30.53 37.27 48.38 59.36 70.24 81.06
298.15 300.00 500.00 700.00 I000.00
1500.00 2000.00 2500.00 3000.00
pyrene 20.27 20.31 24.40 28.51 34.63 44.71 54.66 64.54 74.38
naphthopyrene 19.29 19.33 23.15 26.98 32.68 42.04 51.27 60.43 69.56
coronene 18.51 18.55 22.12 25.72 31.09 39.89 48.57 57.19 65.79
naphthcoronene 17.93 17.96 21.29 24.67 29.72 38.03 46.23 54.38 62.52
ovalene 17.54 17.56 20.75 23.97 28.77 36.66 44.46 52.22 59.98
TABLE VI: Values of A (A&") (kJ/mol) and B(A,G') (kJ/mol) for Hieher Series
298.15 300.00 500.00 700.00 I000.00
1500.00 2000.00 2500.00 3000.00
6.21 6.27 14.65 26.25 46.14 81.76 118.38 155.31 192.37
17.07 17.10 20.16 23.24 27.84 35.41 42.87 50.33 57.69
16.57 16.60 19.44 22.30 26.57 33.58 40.50 47.44 54.29
16.78 16.81 19.74 22.69 27.10 34.35 41.50 48.66 55.73
where n is carbon number. It is important to distinguish the A and B used here from the A's and B's with modifiers in parentheses in eqs 1-9. I t is convenient to use similar equations for the elements cope = Ac B c / T 1 I 2 C c / T D c / P (41)
+
+
+
soc= Ac In T - 2 B C / T 1 l 2- C c / T - ( 1 / 2 ) D c / P + Ec
15.56 15.59 18.15 20.71 24.53 30.79 36.97 43.16 49.30
PpH, S O H ,
= AH2 + B H 2 / T ' I 2+ CH,/T
(44)
= AH2 In T - 2&,/T1I2 - C H , / T - ( 1 / 2 ) D ~ , / + p EH, (45)
(HoH2- H0H2,298)= AH2(T - 298) + 2BH2(T1/2 - 298Il2) + cH2~n ( ~ / 2 9 8-) D H , [ ( I / T )- (1/298)1 ( 4 6 ) where, of course, 298 represents 298.15 K. The parameters in eqs 41-46 were determined by using data from the JANAF tables. I With the 12 parameters in eqs 38-40 and the 10 parameters in eqs 41-46, it is possible to calculate any standard thermodynamic property of any isomer group in the homologous series. The number of parameters required to represent the data for the six (15) Chase,M. W.; Davies, C. A.; Downey, J. R.; Frurip, D. J.; McDonald, R. A.; Syverud, A. N. JANAF Thermochemical Tables, 3rd ed.; J . Phys. Chem. Ref.Data 1985, 14, Suppl. No. 1.
13.53 13.55 15.49 17.42 20.26 24.89 29.46 34.06 38.65
Temperature Dependences of the A and B Parameters in Eqs 1-9 The parameters A , B, C,D, E , F, k,, kb, k,, kd, k,, and kh can be used to express the temperature dependences of the parameters in eqs 1-3. However, since the A parameters in eq 1-9 are the
+ B / T 1 I 2+ C / T + D / P B(4fCop) = k, + k b / T 1 I 2+ k , / T + k d / P = A In T - 2 B / T 1 I 2- C / T - D / 2 P + E A ( A f C o p )= A
+
+DH,/P
14.15 14.17 16.29 18.40 21.52 26.62 31.65 36.71 41.74
series separately is 6 X 12 = 7 2 , not counting the elements.
(42) ( H o c - H0c,z9s) = A,-( T - 298) 2Bc( T'J2- 2 9 8 1 / 2 ) Cc In ( T / 2 9 8 ) - D c [ ( l / T ) - ( 1 / 2 9 8 ) ] ( 4 3 )
+
14.84 14.86 17.18 19.51 22.95 29.60 34.16 39.75 45.30
A(A$O)
B ( A , P ) = k, In T - 2 k b / T ' t 2 - k J T - k d / 2 P + k,
+ 2BT1I2+ C In T - D / T + F B(4,HO)= k,T + 2kbT1I2+ k, In T - k d / T + kh A ( A f H o )= A T
(47) (48) (49) (50)
(51) (52)
same for all six series, A , B, C,D, E , and F in eqs 47, 49, and 51 are the same for all of the six series. This indicates that the data on the six series can be fit simultaneously with 6 + 6 X 6 = 42 parameters, not counting the elements. Simultaneous Linear Regression of Data on Six Series of Polycyclic Aromatic Hydrocarbons Since A(A,Cp,), A(A,So,), and A(AfHo,) are the same for all six series, a linear regression has been made on AfCop,,4pS0,, and AfHo,on all six series simultaneously to determine the best values of the 42 parameters required. This is 72 - 42 = 30 fewer
8482 The Journal of Physical Chemistry, Vol. 94, No. 22, 1990
parameters than fitting the six homologous series separately. The design matrices for the linear regression of the six series separately were constructed by computer programs that were tested separately to determine that the three properties were represented well by eqs 38-40. Then the design matrices were combined to produce the design matrix (540 X 42) for the linear regression of the 540 data points for the six series simultaneously. The residuals of the standard Gibbs energy of formation are given in Table 11. These residuals have random signs and are about the same size as the uncertainties in the Benson method; Bensod has stated that the estimates from group additivity of ArHo are uncertain by about 2 kJ mol-' for small molecules and 13 kJ mol-' for larger molecules. Table 111 gives the parameters for calculating the standard thermodynamic properties of graphite and molecular hydrogen in the range 298.1 5-3000 K. Table IV gives the 42 parameters defined by eqs 38-40 for the benzenoid polycyclic aromatic hydrocarbons. The numbers 1-6 in the subscipts are used to designate the series. The I O parameters for the elements and the 42 parameters for the isomer groups can be used to calculate any property of any isomer group in any of the six series in the range 298.15-3000 K . They can also be used to calculate the A and B parameters in eqs 1-9. Since the A(A@O) and B(A,-Go) parameters determine the equilibrium distribution at specified PHz and specified PCzHzand PHI,respectively, the values of these parameters are given in Table V for a series of temperatures. These parameters make it possible to calculate A@' quickly at these temperatures. I n principle, these seven parameters for a series contain all of the thermodynamic properties of the isomer groups in the series. Since &Gois known as a function of temperature for the first members of seven higher seriess and since we expect these higher series to have the same A(A,Go) values, the B(A,G") values for these higher series are readily calculated and are given in Table VI. These higher series are identified by the formula of the first member of the series.
Discussion There are two obvious questions that cannot be readily answered. "Why do the series of polycyclic aromatic hydrocarbons have certain A parameters in common but not B parameters? Why are the A parameters independent of series for the four particular properties?" According to the Benson group additivity method, the thermodynamic properties are made up of contributions of only four types of groups and the contribution of the symmetry factor to the entropy. The contribution of the 1,5H,H interaction was made in the benzene series but not in the higher series. The numbers of the various Benson groups in a particular isomer group depend in a very complicated way on the allowed structures and the equilibrium distribution of the isomer groups, which depend on temperature. All of the thermodynamic properties of the isomer groups in a homologous series in the temperature range 298-3000 K can be expressed in terms of 12 parameters determined by linear regression. The residuals are of about the same size as the uncertainties in the Benson method. The A and B parameters defined in eqs 1-9 can be expressed in terms of these same regression parameters. After the first six A and B parameters have been calculated from the regression parameters, it is convenient to use the equations given here to calculate the remaining 12. The parameters A and B that are defined in eqs 1-9 are standard thermodynamic properties with slightly different characteristics from other standard thermodynamic properties; they obey the same thermodynamic equations as the parent properties, but, when elements are brought in through formation reactions, they play different roles in A and B. This is apparently what makes them useful in the calculation of equilibrium distributions of the isomer groups in a series at specified PHzor specified PCzHz and PHz. This analysis puts us pretty close to our goal of being able to calculate any thermodynamic property of any isomer group of benzenoid polycyclic aromatic hydrocarbons in the range 298-3000
Alberty and Chou
K. This can now be done by using the parameters in Table IV for the first six series given in Table I. For the seven series for which thermodynamic properties have been calculated only for the first members, A@' can be calculated by using parameters in Table VI. The one thing we do know about the series above CNHZ4is the formulas of the first members, since there is a regular pattern, and this of course makes it possible to write all higher formulas in each series. If the Benson method is used to estimate the thermodynamic properties of the first member, the slope parameters B can be calculated since the intercept parameters A are presumably the same as for the other series. Then any property of any isomer group in that series can be calculated without further use of the Benson method. The coronene series is like the alkene series in that the H / C ratio is independent of carbon number. Therefore, changing PHz will not affect the equilibrium distribution of the isomer groups in this homologous series. For the benzene, pyrene, and naphthopyrene series, raising PHzfavors the lower isomer groups. For the nzphthocoronene and ovalene series, raising PHzfavors the higher isomer groups. Acknowledgment. Acknowledgment is made to the donors of the Petroleum Research Fund, administered by the American Chemical Society, for support of this research.
Appendix The parameters in eqs 38-40 for A P A , ApsO,, and ArHO, were determined for the benzene series by linear regression, but the parameters for Cop",So,, and Ron- Ron,,,,could have been determined. The parameters determined for the second choice are indicated by primes in C o p , = ( A ' + k,'n) + ( B ' + k+,'n)/T'I2 + (C'+ k , ' n ) / T + (D'+ k,'n)/TZ ( A l )
so,= ( A ' +
k i n ) In 7'- 2(B'+ k ( n ) / T ' / 2 (C'+ k,'n)/T- (1/2)(D'+ kdn)/TZ + ( E ' + kdn) (A2)
+
A",,- H",,,,, = ( A ' + k,'n)T + 2(B'+ k { n ) T 1 / 2 (C'+ k,'n) In T - (D'+ k,ln)/T+ ( F ' + k',n) (A3) where n is the carbon number. If eqs 38-40 and eqs Al-A3 are substituted in eqs 21,24, and 30, we find A = A ' - (3/2)AH,
('44)
B = B'- (3/2)BHz
('45)
C
C'- (3/2)CH,
('46)
D = D'- (3/2)DH,
('47)
E = E'- (3/2)EHz
(A8)
F = F' + (3/2)H~,,298
(A91
k , = k,' - ( A ,
+ A,,/4)
(A 10)
kb = kb'- (Bc
+ BH,/~)
(All)
k, = kc'- ( c c
CH,/~)
('412)
k,j = kd' - (Dc
+ D,,/4)
k, = k,' - ( E c + E H , / ~ )
(A13) (A141
k h = khl + (HHC.298 + HH~z.298/4) (A19 where H H ~ ~ , 2 9=8 AH, X 298.15 + 2B X 298.15'/2 + C HIn ~ 298.15 - DH,/298.15 (A16)
HHc,298 = Ac
X
298.15
+ 2Bc X 298.151/2+ cc In 298.15 -
Dc-298.15 (A17) Thus if the parameters in eqs A I , A2, and A3 are determined for the benzene series, the parameters for eqs 38-40 can be readily calculated. The extension of these equations to the other homologous series is readily made with the corresponding formation reactions.
