904
T. BOUBLIK AND G. C. BENSON
Estimation of the Excess Thermodynamic Functions of Nonelectrolyte Solutions from the First-Order Perturbation of a Hard-Sphere System’ by T. Boublik2and G. C. Benson Division of Pure Chemistry, National Research Council of Canada, Ottawa, Canada
(Received March 17, 1969)
Expressions for the excess functions of a binary mixture of nonpolar molecules were derived from a first-order perturbation treatment using the equation of state and distribution function for a system of hard spheres and assuming a square-wellinteraction potential. Application of the equations to data for the system cyclopentanecarbon tetrachloride at 25” provided a satisfactory correlation between the excesg thermodynamic functions and properties of the pure component liquids.
Introduction relations were formulated for calculation of the thermodynamic functions of a pure liquid, assuming the effect of the attractive part of the intermolecular forces (together with the contribution due to softness of the repulsive part) to be a perturbation of a system of hard spheres. The equation of state and radial distribution of the latter is relatively well known either from the solution of the Percus-Yevick equation6r7or from the scaled-particle theorya8 Y o ~ i m ~used , ’ ~ the resuIts of the scaled particle theory to calculate the heat of vaporization of the pure liquid and the excess entropy of a binary solution. These calculationsexploited the ideaof a thermodynamic cycle in which “charging” arid “discharging” of the hard spheres by intermolecular forces occurred. This is also the starting point of the following treatment.
Theoretical Considerations The process of mixing the pure components (numbers of molecules N i ) to form a mole of solution at constant temperature T and pressure p can be imagined to take place in five isothermal steps. (i) The pure components (molar volumes V,) are “discharged” to hardsphere systems at constant volume. (ii) Each hardsphere system is expanded from volume Vi to some large volume V* for which the behavior is ideal (p*V* = E T ) . (iii) The components are mixed at constant pressure p*. (iv) The solution of hard spheres is compressed to volume V, where V,is the molar volume of the real solution with composiiion X, = N i / N a ( N A is Avogadro’s number) at temperature T and pressure p . (v) The hard spheres are “charged” with attractive forces at constant volume 8,. The entropy of mixing per mole, AS”, calculated from these steps is
The Journal of Physical Chemistry
where Si and S , are the entropies of component i and of the solution, respectively. The superscript zero denotes the hard sphere system and AXo is t8hemolar entropy of mixing of the hard sphere components at high dilution. Similarly, the energy change associated with the mixing (ie., the excess energy, essentially at constant pressure) is given by
Abro
+ Jv:
(%)”If T
+ U , - Urno ( 2 )
where Ui and U sare the energies per mole of component i and of the solution, respectively. AUo is the energy of mixing of the hard sphere systems (at V* -+ 0 0 ) and is obviously equal to zero. Also since for hard spheres the ratio p / T is a function only of the volume it follows that (dV/dV),O vanishes and that T(dS/dV)Tois equal to the pressure. Thus eq 1 and 2 can be simplified to the forms
TSE = T [ S , - S,O
-
Si(Si
i
-
+
Sio)]
and
(1) Issued as NRCC No. 11194. (2) Kational Research Council of Canada Postdoctorate Fellow, 1967-1968. Institute of Chemical Process Fundamentals, Czechoslovak Academy of Science, Prague, Czechoslovakia. (3) R. W. Zwansig, J.Chem. &%US., 22, 1420 (1954). (4) J. A. Barker and D. Henderson, ibid., 47, 2856 (1967). (5) J. A. Barker and D. Henderson, ibid., 47,4714 (1967). (6) J. L. Lebowitz, Phys. Rev., 133, A895 (1964). (7) J. S. Rowlinson, Mol. PhUs., 7,349 (1964). (8) H. Reiss, Advan. Chem. Phys., 9, 1 (1965). (9) S. J. kosim and B. B. Owens, J. Chem. Phys., 39, 2222 (1963). (10) 5. J. Yosirn, ibid., 43, 286 (1965).
ESTIMATION OF THE EXCESS THERMODYNAMIC FUNCTIONS In preparation for evaluating the molecular parameters of the pure components, we consider two different isothermal processes for evaporating a pure liquid initially a t pressure pisat (saturation vapor pressure) to produce a gas a t very high dilution (pressure p* -t 0). I n the first process the liquid is “discharged” a t constant volume to produce a hard-sphere system and then expanded to volume V* (p*V* = RT). In the second process, the liquid is evaporated at constant pressure pisat and then expanded to volume V*. From a comparison of the changes in energy, and of the changes in entropy, associated with these processes, it follows that
905
( N A / ~ V , ~ T ) zixj J m uij(r)gij0(r)4ar2dr (10) i,i
Uii
and
+
U , / N A ~ T= U s o / N ~ k T
( N A / ~ V ~ ~ TX i)Z j J m u i j ( ~ ) g i j ~ ( ~ ) 4(12) ~r~d~ 19
1
Uii
It is clear from eq 3, 8, and 11 that the excess entropy of mixing is given by the hard sphere contribution. /.V*
and
Here, AUicohis the cohesion energy, AHiVaP the heat of vaporization, and Vi’ the vapor phase volume, of component i. Assuming that the energy due to attractive forces is a sum of pair interactions which can be treated as a perturbation of the potential of hard spheres, it has been shown4 that the Helmholtz free energy of a pure component is given by
+
This expression is identical with that derived by YosirnlO but has been obtained as a consequence of the limitation to first-order perturbation terms, without recourse to the original assumption that the “charging” and “discharging” terms cancel. The pressure of a system of hard spheres at given temperature, volume, and numbers of molecules of different kinds is a function of the radii R1,Rz. . of the hard spheres. For a binary mixture, Lebowitz, Helfand, and PraestgaardI2 derived the following equation of state from the scaled particle theory 6 kT - a
351b
50
[E) (1 -
where the variables
+
+
&)2
352a (1 -
]
(14)
En are defined by
Fi/NAkT = F i o / N ~ k T
Jm
(N~/2vikT)
Uii
~ i i ( ~ ) g i i ~ ( ~ ) 4(7) ar~d~ In the case of a one-component system, eq 14 becomes
where only the first-order term in the perturbation treatment is retained and the proper choice of hardsphere diameter dii is used.ll I n this equation, k is the Boltzmann constant, uii(r) is the potential between a pair of molecules of species i with centers separated by a distance Y, giio(r) is the radial distribution function of the unperturbed system, i.e., the hard-sphere system, and cii is the separation for which uii(r) vanishes. It can be seen from differentiation of eq 7 with respect to temperature that si = s i 0
where
The integrals in eq 13 can be evaluated after substituting the hard-sphere pressures from eq 14 and 16. The formula for the excess entropy can then be written
(8)
and hence that Ui/”AkT = 7Jio/N.&T
+
J
(N~/2vikT)
u i i (Y)gii0(~)47+d~ (9)
uii
Following arguments similar to those given for pure liquids it is possible to obtain first-order perturbation relations for the solution. Thus
(18) The hard-sphere integral in eq 6 can be handled in a, (11) Cf.ref 5 , eq 12. (12) J. L. Lebowitz, E. Helfand, and E. Praestgaard, J. Chem. Phys., 43, 774 (1965).
Volume 74, Number 4 February 19, 1970
906
T. BOUBLIK AND G. C. BENSON
similar manner. Evaluation of the second integral in eq 6 is based on the simple virial expression vig
RT
= -
P
+
Bii
(19)
where Bii is the second virial coefficient. Using Haggenmacher's formulala to estimate the temperatme variation of Bii, the formula for tJheheat of vaporization becomes AHivw = 2Biipisat -
and eq 12 becomes
us0 - usRT
(NA/V,)
xixjoij2a(Ri i,i
+ Rj)'gijO(Ri + Rj)
(28)
The expression for the excess energy, obtained by combining eq 4,21,27, and 28 is
The radial distribution functions are needed for determining the excess energy (or free energy). The form of these functions is known from the scaled particle theory only for particles at closest approach; thus in a binary solution
(--RiRj )'
12522
(1 -
Ri
+ Rj
(21)
which becomes
(22) for a one-component system. Equations 21 and 22 are sufficient for considering those cases in which interactions which are significantly different from those of hard spheres occur only at intermolecular distances differing very slightly from closest approach; in other cases, the solution of the Percus-Yevick equation is required. In the present treatment, the interactions are represented by the simple square-well potential
r
? A1,). ( )
< aij a i j < r < aij + a r > uij + u r
= +a
= -eij
=o
It appears unlikely that the contributions of hard spheres and of the attractive forces to the volumetric behavior of the system can be separated and derived from the thermodynamic cycle described above. Thus, in subsequent numerical calculations, eq 18 and 29 are used t o express TSE and UE as functions of the temperature, volume, and certain molecular parameters R and o. Numerical Calculations The utility of the equations derived in the preceding section was investigated by applying them t o the system cyclopentane-carbon tetrachloride. This system was chosen because the component molecules are approximately spherical, and because data for the thermodynamic functions HE, VE, and GE (ie., the excess enthalpy, volume, and Gibbs free energy) at 25" are a~ai1able.l~"Experimental" values of TSE and CJE can be obtained from these results using the thermodynamic relations
TEE
=
HE -
b'E = HE = d;j = Ri
+ Rj
(24)
and it is assumed that the distance a is very small. The integrals in eq 10 and 12 can then be approximated
J: uij(r)gij0(r)4nr2dr= + Rj)4n(Ri + Rj)%
-cijBij'(Ri
Defining the molecular parameter wij
=
wij
eq 9 can be rearranged to The Journal of Physical Chemistry
(25)
- pVE
(31)
The term pVE is relatively small in the present case, and for practical purposes it is unnecessary to distinguish between UE and HE. Several different ways of establishing the values of the molecular parameters (Ri and wii) of the pure component liquids were investigated. I n all cases, the semiempirical mixing rule w122
by
~ij~/kT
(30)
and
where uij
GE
(26)
=
WlW22
(32)
(13) J. E.Haggenmaoher, J . Amar. Chem. SOC.,68,1123 (1946). (14) T.Boublik, 17. T. Lam, S. Murakami, and G. C. Benson, J . Phy3. Chem., 73,2356 (1969).
ESTIMATION OF THE EXCESS THERMODYNAMIC FUNCTIONS
\
loo
90
ao
t
L
* * * e
907
0.05 were calculated from the equations representing the smoothed experimental data.14 It appears from comparison of these values that the theoretical estimates of both the excess energy and entropy are too small. ~~
e e
I
~
~~~~
~~
~~
Table I: Properties of the Component Liquids at 25"
-1
0
E
W
60
Torr Vi, cm3 mol-' Bii, cm3 mol-' AHiVnp,J mol-' AUiaoh, J mol-'
pinat,
50
3 LT
0
40
I-
Carbon tetrachloride
317.5" 94.71 - 1054" 28540' 26200'
113.8b 97.10 - 167jd 32450' 30050"
" American Petroleum Institute Research Project 44. "Selected Values of Properties of Hydrocarbons and Related Compounds,'' Carnegie Press, Carnegie Institute of Technology, Pittsburgh, Pa., 1953, and later revisions. * J. A. Barker, I. Brown, and F. Smith, Discussions Faraday Soc., 15, 142 (1953). R. W. Hermsen and J. M. Prausnitz, Chem. Eng. Sci., 18,485 (1963). P. G. Francis and M. L. McGlashan, Trans. Faraday Soc., 51, 593 (1958). Calculated by method described in the text.
W
cn
Cyclopentane
30 20
IO n "
0.0
0.2
0.4
0.6
0.8
I .o
X , MOLE FRACTION OF C,H,
Caloulationb
Figure 1. Comparison of theoretical and experimental values of excess functions of system cyclopentane (1 )-carbon tetrachloride (2) at 25". Dotted and solid curves correspond, respectively, to calculations (a) and (b) described in the text. Points represent experimental data: 0,TSE; 0, UE.
was used t o evaluate the cross coefficient w12, and theoretical values of the excess functions TSE and U Ewere computed from eq 18 and 29 for volumes of the mixture given by
v,
=
i
zivi
+
VE
Table 11: Values of the Molecular Parameters"
(33)
The values adopted for the properties of the pure component liquids are summarized in Table I. The heats of vaporization and cohesion energies were estimated from the temperature derivative of the vapor pressure (represented by an Antoine form) with correction for nonideality of the vapor phase approximated by the second virial contribution (again, the temperature variation of Bii was obtained from Haggenmacher's f ~ r m u l a ' ~ ) . I n our initial calculation (a), Ri for each component was computed from eq 20 using the data for AHiVaPfrom Table I. The corresponding w i i was then obtained from eq 27, using the values of Ri and AUiwh. The resulting molecular parameters are listed in Table 11. Using these values, the dotted curves for TSEand U Ein Figure 1 were computed from eq 18 and 29. The points plotted in Figure 1 at mole fraction intervals of
(a) (b) a
RI
2.6530 2.6417
Ri and w i i in .&.
WIl
Ra
w21
1.4687 1 ,5353
2.7114 2.7270
1 .4666 1.3736
' Calculations (a) and (b) described in the
text.
However, the results computed for the excess functions are fairly sensitive to the values employed for the radii. This is illustrated by the solid curves in Figure 1 which were obtained in calculation (b). The values of Ri used in the latter were altered to achieve a leastsquares fit of the experimental TSE results, instead of the wii were obtained being determined from AHiY&p; from AUiCoh as in calculation (a). The fit of TSEin this case is within the uncertainty of the experimental results, and the agreement between the theoretical and experimental excess energies is much better than that obtained in calculation (a). It can be seen from Table I1 that this improvement has been achieved by changes of less than 1% in the radii. Using eq 20, it can be shown that the changes in Ri correspond t o variations of between 1 and 2% in AHivap; these are similar in magnitude to estimates of the accuracy of AHiY&D.
Discussion The agreement between experiment and theory found for calculation (b) shows that the real behavior of a system of simple nonpolar molecules can be obtained quite well from the perturbation of a hard-sphere system. Advantages of this treatment are the relative Volume 74, Number 4
February 19, 1070
908
R. A. FRIEDEL, J. A. QUEISER,AND H. L. RETCOFSKY
simplicity of the expressions obtained and the small number of molecular parameters required. Since only first-order perturbation terms are retained and an oversimplified model of the molecular interactions is assumed, it appears that the theory should work better for energetical functions than for functions derivable more directly from the equation of state. The latter equation can be obtained from the Helmholtz function (eq lo), but attempts to use it to formulate a description of the volumetric behavior of the system, and to
calculate VE (at constant pressure), were unsuccessful.
It seems likely that this failure is due to the neglect of higher order perturbation terms and to the assumption that the attractive forces are independent of the separation between the molecules over some range
a. Application of the present expressions is limited to systems composed of nearly spherical nonpolar molecules. Extension of the equations to consider multicomponent systems is straightforward.
Coal-Like Substances from Low-Temperature Pyrolysis at Very Long Reaction Times by R. A. Friedel, J. A. Queiser, and H. L. Retcofsky U.8. Department of the Interior, Bureau of Mines, Pittsburgh Coal Research Center, Pittsburgh, Pennsylvania 16119 (Received Septmber 2, 1969)
The importance of long reaction times in the coalification process has been demonstrated in the laboratory. Samples of cellulose and pine sawdust in evacuated, sealed glass vials have been heated at 200" for 2 years. Black coal-like chars were produced; conventional heating at 200" for hours produces very little chemical change. The chars were characterized by infrared and electron spin resonance spectra and by ultimate analysis. The infrared spectra of the pine sawdust char and of a subbituminous coal are nearly identical, except for differing carbonyl absorption. The spectrum of the cellulose char is also similar to that of the subbituminouscoal. The electron spin resonance (em) results show that g values, line widths, and spin concentrations are very similar for the two chars, and resemble closely the corresponding values for subbituminous coal. 200" is considered a reasonable coalification temperature; a feasible depth of burial can provide such temperatures. The experimental coalification obtained in only 2 years supports the hypothesis that geologic time could produce all ranks of coaIs, from lignite to anthracite, at temperatures below 200".
Introduction I n the study of the structure and properties of coal, attempts have been made to prepare synthetic coals, principally by pyrolysis methods. Chars having properties identical with those of coals have not been produced. Initial attempts at studying the infrared spectra of coal-like chars prepared by pyrolysis of pure materials were carried out several years ago.'p2 The pyrolysis method used was that of Smith and Howard, who had prepared from cellulose, chars that had some of the properties of coals.a We studied the infrared spectra of chars prepared from cellulose over a range of temperatures; the spectrum of a 400" cotton char somewhat resembled that of high-volatile bituminous coal.lP2 The spectrum of this char showed a definite carbonyl absorption band which is not present, except as a weak shoulder, in infrared spectra of coals. Carbohydrate chars prepared a t 300, 400, and 500" produced absorption The Journal of Physical Chemistry
bands in the aromatic region that were identical with the aromatic bands produced, respectively, by lignite, higbvolatile bituminous, and anthracite coals ; less similarity was observed in other parts of the ~ p e c t r a . ~Similarities of reaction of coal and the carbohydrate, sucrose, were illustrated by the nearly ident,ical spectra of products obtained in hydrogenation ~ t u d i e s . ~Friedman and coworkers studied many coal-like chars prepared by the pyrolysis of pure oxygenated compounds, mostly at 400" and reaction times of 1 hr; several (1) R. A. Friedel and M. G. Pelipetz, J . Opt. SOC.Amer., 43, 1061
(1953). (2) R. A. Friedel and J. A. Queiser, Anal. Chem., 28,22 (1966). (3) R. C. Smith and H. C. Howard, J . Amer. Chem. SOC.,59, 234
(1937) (4) R. A. Friedel, Proceedings of the Fourth Carbon Conference, University of Buffalo, Pergamon Press, New York, N . Y . ,1960, p 321. (6) M. G. Pelipetz and R.A. Friedel, Fuel, 38, 8 (1969). I
