MOLECULAR THERMODYNAMICS OF SIMPLE LIQUIDS Pure Components H E N R I R E N O N , ' C. A.
ECKERT,Z AND J.
M. PRAUSNITZ
Department of Chemical Engineering, University of California, Berkeley, Calif., and Institute f o r Materials Research, National Bureau of Standards, Boulder, Colo.
Thermodynamic properties of simple liquids are calculated from an analytical partition function based on a modification of Prigogine's cell theory and on a three-parameter theorem of corresponding states. The partition function gives an excellent representation of the configurational properties of 1 5 liquids ranging in molecular complexity from argon to neopentane. Three characteristic molecular parameters are sufficient to calculate the configurational energy and entropy, volume, coefficient of expansion, and compressibility. These parameters are a molecular size, a pair-potential energy, and a term closely related to noncentral intermolecular forces; this last parameter i s a nearly linear function of Pitzer's acentric factor. The main application of this statistical thermodynamic treatment follows from its straightforward extension to liquid mixtures.
HE thermodynamic properties of pure liquids and their Tsolutions have been a topic of interest to physical chemists and to chemical engineers for several generations (74). Although a classical thermodynamic description of fluids was established many years ago, only recently has this description been attempted with the techniques of statistical mechanics. I n this study we use such a technique to develop a workable theory for the characterization of the thermodynamic properties of simple liquids. Such a result is of practical value because processes involving liquefied gases are of rapidly increasing industrial importance. In this paper we present a statistical thermodynamic treatment of pure liquids. In the work immediately following we extend our treatment to liquid mixtures. Our attention is confined to simple liquids. These may be defined as small, uncharged, nonpolar, approximately spherical molecules such as the rare gases, nitrogen, methane, carbon monoxide, oxygen, ethane, and carbon tetrafluoride. Pure simple liquids (and their mixtures) are more amenable to theoretical treatment because there are no complicated intermolecular phenomena such as hydrogen bonds or orientation effects due to polarity, nor are there extreme differences in size and shape. All simple molecules interact primarily by London dispersion forces and in aggregates of simple molecules there are no extreme entropy effects as long as the solution is not near its critical point. Thus, by limiting attention to simple liquids, the problem of solution theory may be reduced to its most basic aspects. I n this work we use a modified cell model in conjunction with a three-parameter theory of corresponding states.
systems having specified properties. A convenient ensemble is the canonical ensemble where volume, temperature, and the number of molecules are the properties specified in each system of the ensemble. The canonical partition function, 2, is given by:
where the summation is over all systems of the ensemble, and Ek is the energy of each system. All thermodynamic properties are derived from the partition function by well known relations. Our task is to evaluate the total energy of a system of S molecules in a volume V at a temperature T. This energy may be considered as the sum of several terms. The first includes the translational kinetic energy and the energy related to the intramolecular degrees of freedom; these are the same for a liquid and its ideal gas at the same temperature and are independent of the volume of the system. The second term, the potential or configurational energy, is the one which is different for the liquid and the ideal gas. We know the properties of the substance in the ideal gas state from measurement at low pressure; it is therefore sufficient to consider instead of the canonical partition function the configurational integral, Q, which incorporates only the configurational energy. Instead of Equation 1 we therefore write
Z
=
Zkinetic
Zinternsl *
Q
(2)
TV)
If u ( c , . . . , is the potential energy of a system in which the + molecules occupy the positions T,, . . ., Y . ~ ,
Statistical Mechanics
From statistical mechanics all thermodynamic properties of a system may be derived by considering an ensemble of On leave of absence from Institut Fransais du Pttrole, RueilMalmaison, France. 2 Present address, University of Illinois, Urbana, Ill. 1
52
I&EC FUNDAMENTALS
The configurational integral, Q", of an ideal gas does not vanish but is given by
Q" = V N / N !
(4)
The difference between the properties of a system of N molecules characterized by its volume and temperature and one in the ideal gas state at the same temperature and volume is derived from Q / Q O by well known relations-for instance, the Helmholtz energy #of a liquid (relative to its ideal gas) is given by
AL
- XIo
=
-kT In ( Q / Q O )
(5)
The independent varia.bles are the volume, V , and temperature. T ; these refer to both liquid and ideal gas. The properties derived from the confipiirational partition function, Q, are called configurational properties of the liquid. They are related to the properties of the liquid (superscript L ) and the ideal gas (superscript o) by:
p = PL;
u = uL-
LlO;
H
=
HL
- HO
f
RT;
and A = A L - Ao - k T In Q0 Cell Model
The cell model of the liquid was chosen for this work because it provides a reasonable physical description of liquid properties and because it separates conveniently the configurational energy of molecules into readily calculable portions, thus simplifying the statistical mechanics. However, the main advantage of the cell model lies in its straightforward extension to mixtures ( 6 ) . -4s discussed by Prigogine (39) and others, the cell model envisions a single molecule in the liquid being surrounded by a cage or “cell” of adjacent molecules. For simple molecules the field of force of the cell may be closely approximated by an average, spherically symmetric field which is determined by the two-body intermolecu1,u potential. This permits evaluation of the configurational integral, Q,in two parts: exp (-E/kT), the energy contribution associated with putting the molecules in their cells. and $.,’ the contribution of the energy of motion within the cells. Thus
of this parameter is closely related to the noncentral forces of the molecules; for those molecules where there are no noncentral forces-e.g., the noble gases-this parameter is invariant but as the complexity of the molecule increases so does c. The introduction of a third parameter into the corresponding states treatment provides a much improved basis for the representation of thermodynamic properties. Pitzer and coworkers (38) have shown that a third parameter (the acentric factor) is required for an accurate corresponding states treatment of vapor pressures, latent heats, and other configurational properties of a large number of pure fluids. Knobler, Van Heijningen, and Beenakker (27) showed that even for the simple molecules argon, oxygen, and nitrogen no consistent set of two parameters per molecule could be found to reduce their configurational energies. Further, Sherwood (45) has shown that the second and third virial coefficients of common gases cannot be accurately represented with a two-parameter potentiale.g., that of Lennard-Jones-but require a three-parameter potential (such as that of Kihara). Thus, there seems ample evidence that a three-parameter, corresponding-states treatment is necessary even for simple molecules. I n analogy to recent applications of the cell theory to hardsphere systems, we choose the independent variables which determine the terms in Equation 6 in order to obtain an empirical configuration integral. The energy E for the placement of the molecules in their cells may be taken as a function of the volume only. Recent developments in the cell theory for dense fluids by Kac et al. (79) and by LonguetHiggins and Widom (22) indicate that the function E ( V ) is inversely proportional to V . Further, E( V ) may be reduced in terms of a characteristic energy, U*, and a characteristic volume, V*:
E = U*8(V/V*)
(7)
T h e characteristic energy, U*, represents the potential energy of N molecules with coordination number s and pair potential E:
Q
= $-” exp (-E/kT)
(6)
T h e cell concept assumes that the cell sizes are only slightly larger than the molecular diameters, so that the molecules exchange cells only very rarely. The total energy contribution due to this infrequent interchange of cells is considered negligible relative to the energy associated with the motion of the molecules within the cells; thus, the cell theory is not valid in the expanded liquid region. Corresponding States lheory
An efficient method for correlating the properties of fluids in terms of molecular parameters is provided by the theory of corresponding states. This theory is used here in conjunction with our statistical-mechanical formulation. An alternative corresponding-states treatment of liquids, independent of any statistical model, has been developed for normal paraffins by Hijmans (73). The problem of applying corresponding states theory has two facets: I t is necessary first to determine the universal functions which give the reduced configurational properties to which all the fluids conform; and second, to determine the proper reducing parameters for each fluid. LVe use a modification of the original cell model first proposed by Prigogine (40) and used by Hermsen (72), Eckert (5), and Flory ( 9 ) . This modification introduces into the cell partition function an exponent, c, which is treated as a parameter characteristic of the substance. The physical significance
u* =
1/2
N
s
(8)
E
The characteristic volume is related to the collision diameter, u> by
v* = N t u3
(9)
where 5 is a geometric factor. The cell partition function, #, depends on both temperature and volume. Recently Wertheim (47), developing a new model of fluids, showed that # can be decomposed into a function f 1 of volume only (which may be deduced from calculations for hard-sphere fluids) and a function f i of V and T. For dense fluids, because of the limited range of density, f z is given to an excellent approximation by a function of T only, A detailed discussion of f l ( V ) is given by Guggenheim (77). Using the reduced volume P = V/V* and the reduced temperature p = T/T*, we write
From Equations 5 , 7 , and 10 we obtain the configurational integral :
T h e Helmholtz energy is
A
=
-3 RcT[ln V*1’3
+ In fl(p)+ In f 2 ( p ) +] U * 8 ( p )
(12)
FEBRUARY 1967
53
VOL. 6
NO. 1
In a manner similar to that of Prigogine (39) we define the characteristic temperature by
T* = U*/lO Rc
(13)
The factor 10 is introduced only for convenience. For the configurational properties we then obtain the following reduced expressions which contain only the three parameters V*, T*, and L'*. The configurational Helmholtz energy, energy, pressure, and heat capacity at constant volume are given by :
From limited experimental data on configurational heat capacities we find that a good representation is obtained by a = 0.4. Finally then, the configurational integral is: exp
(+ q ) ]
exp
(E u* T) 1
(23)
3T The equation of state derived from this expression is the same in form as that obtained by Flory. I t is 10 FP/T =
-
P1'3/(Pli3
1)
-
lO(PT)-1
(24)
From the equation of state the coefficient of thermal expansion, a, is given by I
= aT* = ( l / p ) ( h V / h p ) p =
Also, the isothermal coefficient of compressibility, p, is The configurational enthalpy and Gibbs energy are given by
z=" U F P
(18)
Analytical Configurational Integral
Functions f l , f 2 , and 8 are chosen in such a manner as to represent adequately the experimental configurational properties of simple fluids. Almost 30 years ago Eyring and Hirschfelder (7) proposed that the configurational integral can be represented by Equation 11 withc = 1,fl = ( P I i 3- l ) , andf2 = 0. Prigogine modified the cell model to make it suitable to long-chain (polymer) molecules (40). When he assumes a square-well potential to calculate the cell partition function, I), as in the free volume theory, he obtains Equation 11, where c now represents the number of degrees of freedom per molecule. Prigogine also usesf1 = (Pli3 - 1) andf2 = 0. Flory (9) writes Equation 11 with c different from 1, but does not assign to it any specific physical significance. H e also usesf1 = ( P I i 3- l ) , f2 = 0, and sets E(P) = - constant/p. Flory applies the theory to obtain the equation of state of normal paraffins and small, nonpolar molecules (7). With essentially the same expression, Schmidt (44) obtains a fit of the volumetric properties of the polyphenyls and Simha and Havlik (46) obtain a similar equation of state for nonpolar polymers. We are interested in representing not only the equation of state but also the energy of liquefied gases. Equation 17 shows that one cannot obtain a configurational heat capacity different from zero iffz = 0, as was previously noted by Hermsen (72) ; experimentally, it has been shown that the configurational specific heat at constant volume is positive and decreases with increasing temperature. To take this into account in the present treatment we use in Equation 10:
The energetic configurational properties are obtained by substitution of Equations 20, 21, and 22 into Equations 14, 15, and 17. They are:
I
cy
0.08
= '2
T
Data Reduction
Experimental data which are readily available for liquids are the molar volume, V , along the saturation curve and the vapor pressure. The configurational energy is obtained from the variation of vapor pressure with temperature. The enthalpy of vaporization is calculated by Clapeyron's formula :
dP AHVBP= T - (V" dT
- V ) = HQ - H L
(30)
The configurational enthalpy is obtained from the enthalpy Ha, the imperof vaporization, taking into account HQ fection of the vapor. The volume of the vapor and the enthalpy of vapor imperfection are calculated from Pitzer's correlation (32) using the acentric factor.
-
H
= HL
- Ha + RT
=
-AH,,,
+P (31)
RT P
V"=-+B
54
I&EC FUNDAMENTALS
~
where B , the second virial coefficient, is obtained from Pitzer’s correlation. Finally, the configurational energy is given by
Table I.
Characteristic Properties of 15 Simple Fluids Charac-
Characteristic Temp., T*, O K . 40,05 111.5 116.9 133.8 134.3 171.9 185.6 255.3 199.7 249.2 262,2 267.9 316.3 323,7 377.2
U = H - P V = - T - ( -dP + B RT -V)+ dT P
Equations 24 and 2EI give the theoretical universal form of the equation of state and the configurational energy. Selection of the parameters can be made from a minimum of data if they are accurate enough. The characteristic parameters obtained from such data may then be used to compare other experimental results with those calculated from the partition function. From the corresponding states argument, the reduced molar volume is a function of reduced temperature and pressure
v == v* B(F,F)
Neon Nitrogen Carbon monoxide .4rgon Oxygen Methane Krypton Xenon Carbon tetrafluoride Ethylene Ethane Carbon dioxide Propylene Propane Neopentane
v*
=
P(T)
(35)
Differentiating Equaticln 35 with respect to reduced temperature p and multiplying by the ratio T / V , we obtain (36)
T)
where f’( is a dimensionless, universal function. Equation 36 may be used as a #convenient method for evaluating the characteristic parameters, as was shown previously by Prigogine (do), Hermsen (72), and Eckert (5). Data for the molar volumes of simple liquids are readily available as a function of temperature. Most often these are not reported at zero pressure, but a t saturation conditions. However, for low to moderate pressures, well below the critical, the effect of pressure on the liquid volume is negligible. All the liquids may be considered to be at identical corresponding states when they are at identical values of f ’ ( p ) . This value was arbitrarily chosen as 0.4in order to be within the range of available data for most of the liquids under consideration. Equation 24 can be written with the assumption that the effect of pressure is negligible:
?!
- =
10
p-1 - 7 - 4 1 9
= _V‘ _
‘I
‘
5
0
h
W
Solution of Equation 38 gives
T’
TI*
U‘
=
-0.88203
1
7
(38)
~
I
I
I
1
I
EXPERIMENTAL DATA A NEON b NITROGEN A CARBON MONOXIDE
(37)
_ -1 - 0.4
=
The characteristic parameters are shown in Table I for 15 simple fluids. Figures 1 and 2 show a comparison of experimental and calculated values of the reduced volume and the reduced configurational energy for these liquids at zero pressure. Table I1 gives the per cent root-mean-square deviation between calculated and experimental values for the configurational energy and the volume for each substance, together with the range of temperature in which the comparison between experimental and calculated data is made. A more severe test of the theory is provided by a comparison with experimental data of the calculated values of the coefficient of thermal expansion and the isothermal coefficient of compressibility. The theoretical values are obtained from Equations 25 and 26; the experimental results are taken from a tabulation by Rowlinson (43). The results are shown in Figures 3 and 4, which indicate excellent agreement between predicted and observed quantities. Computer programs used to calculate the molecular parameters and properties of pure liquids and more detailed results are given elsewhere (47)
Combining Equations 36 and 37 and takingf’tp) = 0.4, 7113
r*
1.3138
0 0
b ln F 4 1 _ _ _ = - _ bIn B 3 +
Characteristic Energy, U’, Cal./Mole 425.8 1393 1521 1560 1676 1977 2176 3052 3076 3387 3695 4221 4777 4867 6032
(34)
but at a constant reduced pressure, here taken as zero, it is a function of reduced temperature only:
B
terzstic Vol., V*, Cc./Mole 12.63 25.89 26.30 21.98 21.14 28.99 26.64 32.74 39.30 37.ii 41.05 25.71 50,92 55.68 85.99
1
a
METHANE KRYPTON XENON CARBON TETRAFLUORIDE ETHYLENE PROPYLENE PROPANE NEOPENTANE FUNCTION
0
” Y
0 3
=
1.3138
Then
Y 1.3t‘
? = 0.66188
3 12-
and from Equation 38,
fi
= 0.88203
Using the experimental temperature T’, volume VI, and energy C;’ when the condition b In V / b In T = 0.4 is fulfilled, we obtain the characteristic parameters of the liquid by
0.3
I
I
I
I
I
I
0.4
0.5
0.6
0.7
0.8
0.9
T/T*,
Figure 1.
REDUCED TEMPERATURE
Reduced volumes of simple liquids VOL. 6
NO. 1
FEBRUARY 1967
55
-0.6
I
I
I
I
EXPERIMENTAL DATA A NEON h NITROGEN A CARBON MONOXIDE 0 ARGON 4 OXYGEN METHANE D KRYPTON 0 XENON V CARBON TETRAFLUORIDE 7 ETHYLENE
-0.72.
0 K W
z W
I
I
I
I
I
I
I
I
li
::%~,",'lOXUJE PROPYLENE PROPANE NEOPENTANE -CALCULATED FROM R\RTITION FUNCTION h
1
w
5
1.2
-
4
:
i-0.8-
z 0
1
I
3
-
.
3 3
-1.1-
I
-1.2
0.3
T/ T',
Figure 2. liquids
I
o
t
-
-I
E
I
I
0.8
0.9
I
0.6 0.7 REDUCE0 TEMPERATURE
I
Reduced configurational energies of simple
h
2-2'o-
I
I 0.5
0.4
d
-
I
I
1
Figure 4. Reduced isothermal compressibility of simple liquids
NITROGEN ARGON OXYGEN METHANE CALCULATED FROM PARTITION FUNCTION
Lo Lo
Table II. Root-Mean-Square Deviations between Calculated and Experimental Liquid Volumes and Energies
W U
II
5 1.50
Vol. Deuia-
1
tion,
%
Temp. Range, K.
Neon Nitrogen
0.1 0.1
0.8 0.1
25-36 65-100
(70, 26) (78,29,34,
Carbon monoxide
0.07
1.2
74-101
Argon Oxygen Methane Krypton Xenon Carbon tetra-
0.03 0.1
0.3
0.1 0.1 0.1 0.1
0.3 0.5
90-122 60-120 100-155 125-170 205-235 92-175
(23, 24, 34, 35 1 ( 2 , 78) ( 7 5 , 2 8 , 36) (77, 20) (27, 33) ( 3 7 , 37)
a
7
H
U W
I F 0
'" 1.0-
/o
Energy Deciation,
0
0.8 0.5
1.2
Ref.
48 1
(4)
fluoride ... .. ..-.
/
O 0.3
0.4 0.5 0.6 0.7 0.8 T / T*, REDUCED TEMPERATURE
0.9
Ethylene Ethane Carbon dioxide Propylene Propane Neopentane
Figure 3. Reduced coefficient of thermal expansion of simple liquids
X
0.1 0.3 0.02 0.2 0.3
1.0 0.1 1.1
0.01
0.5
=
1.6
1.3
C
=
C krypton
Table I11 shows the values of
Molecular Parameters
The characteristic quantities V*, U*,and T* may be used to calculate parameters E , u, and c. They depend on a coordination number, s, and a geometric factor, t . In order to avoid specification of these two quantities we define three relative molecular parameters by comparison with krypton, a typical spherical molecule : E
=
U*krypton 56
l&EC FUNDAMENTALS
(25, 30) (3) (3) (42) ( 3 , 42) (42)
u*/T* (U*/T*) krypton cr, E ,
and X for 15 simple fluids.
The Molecular Parameter X
The molecular parameter X is closely related to Pitzer's acentric factor, w . Figure 5 shows our X values as a function of the acentric factor, w , which is defined by Pitzer: w =
U* --
128-225 100-210 223-253 163-243 93-293 253-293
-log10
(p,) PSt
-
1.000
T / T c p0.70
Both
w
and h ilicrease when the structure of the molecule
Table 111.
Molecular Plarameters Relative to Those of Krypton
Neon Nitrogen Carbon monoxide Argon Oxygen Methane Krypton Xenon Carbon tetrafluoride Ethylene Ethane Carbon dioxide Propylene Propane Neopentane
U
e
x
0.780 0.991 0.996 0.938 0.926 1.029 1 .ooo 1.071 1.138 1.117 1.155 0.988 1.241 1.278 1.478
0.196 0.640 0.699 0.717 0.770 0.909 1.000 1.403 1.414 1.557 1.698 1.940 2.195 2,237 2.772
0,907 1.066 1.109 0.994 1.064 0.981 1.000 1.019 1.313 1,159 1.202 1.344 1.288 1.282 1.363
liquid is characterized by three parameters only; each of these parameters has theoretical molecular significance. It is likely that this treatment for simple molecules may be generalized to a larger class of nonpolar substances as indicated by Prigogine (40), Hermsen (72), Holleman and Hijmans (76), and Flory (8). However, such generalization, to be successful, requires a better f i ( p ) in order to represent more adequately the configurational heat capacity for which adequate data are, unfortunately, scarce. The most important feature of this treatment is that it is sufficiently fundamental to provide a useful basis for dealing with the much more important problem of the properties of mixtures (6). Acknowledgment
The authors are grateful to the National Science Foundation and to the tYoodrow Wilson Foundation for financial support; to the Computer Center, University of California, Berkeley, for the use of its facilities; and to Z. Salsburg for helpful comments. Nomenclature
-
A
=
B
= = = = = = = = = = = = = = =
C
2
Ek
fl,f 2
H
k AT
P
Q T
S
T U X
P
Z k iue t io
= = = =
Zinternai
=
U
V Z
0
0.I ACENTRIC FACTOR w
0.2
Figure 5. Relation between molecular parameter and Pitzer’s acentric factor, w
deviates from spherical symmetry; X and w are both measures of the importance of noncentral intermolecular forces. T h e low value of h for neon is probably due to quantum effects. The correlation between X and w facilitates the prediction of the properties of a substance in the liquid state when it is sufficiently far from the critical point. The following data only are needed: the acentric factor, the liquid density, the pressure, and the heat of vaporization a t one temperature. From these data, using Equations 13, 24, 28, and 33, we may obtain T*, V*, and U*. Conclusions
We have obtained an analytical partition function for simple liquids, derived from the cell theory, which incorporates also the advantages of the theorem of corresponding states. Each
Helmholtz energy (configurational without superscript) second virial coefficient molecular shape parameter heat capacity at constant volume lattice energy total energy of system k in canonical ensemble functions, factors of cell partition function enthalpy (configurational without superscript) Boltzmann’s constant number of molecules pressure configurational integral position coordinate coordination number temperature internal energy (configurational without superscript) potential energy volume canonical partition function kinetic energy factor in Z for translational degrees OJ freedom factor in Z for internal degrees of freedom
GREEKLETTERS = coefficient of thermal expansion = isothermal coefficient compressibility = pair potential = normalized molecular shape parameter ( A = 1 for krypton) = geometric factor = collision diameter = cell partition function = acentric factor SUPERSCRIPTS 0 = ideal gas L = liquid * = characteristic property = reduced property I = configurational property d(ln V)/b(In T ) = 0.4 G = gas
-
of
the
liquid
when
literature Cited
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NO. 1
FEBRUARY 1967
57
(4) du Pont de Nemours & Go., E. I., LYilmington, Del., “Preliminary Properties on Freon 14,” 1961. (5) Eckert, C. A., Ph.D. dissertation, University of California, Berkeley, 1964. (6) Eckert, C. A., Renon, H., Prausnitz, J. M., IND.END.CHEM. FUNDAMENTALS 6, 58 (1967). (7) Eyring, H., Hirschfelder, J., J . Phys. Chem. 41, 249 (1937). (8) Flory, P. J., J . A m . Chem. SOC.87, 1833 (1965). (91 Florv. P. J., Orwoll, R. A., Vrii. A.,Ibid., 86. 3507 (1964). (10) G d y , E. R., Cryogenics 2, 22i(1962). (11) Guggenheim, E. A.,M o l . Phys. 9,43 (1965). (12) Hermsen, R. W,, Prausnitz, J. M., Chem. Eng. Sci. 21,791, 803 (1966). (13) Hijmans, J., Physica 27, 433 (1961). (14) Hildebrand, J. H., Scott, R. L., “Solubility of Nonelectrolvtes.” Reinhold. New York. 1936. (15) Hoge, H. J., J . Res. Natl. Bur. Std. 44, 321 (1950). (16) Holleman, Th., Hijmans, J., Physica 28, 604 (1962). (17) Itterbeek, van A., Staes, K., Verbeke, O., Theeuwes, F. Ibrd., 30, 1896 (1964). (18) Itterbeek, van A . , Verbeke, 0..Ibid., 26, 931 (1960). (19) Kac, M., Uhlenbeck, G. E., Hemmer, P. C., J . Math. Phys. 4, 216 (1963). (20) Keyes. F. G., Taylor, R. S., Smith, L. B., J . Math. Phys. M I T 1, 211 (1922). (21) Knobler, C. M., Van Heijningen, R. J. J., Beenakker, J. J. M., Physica 27, 296 (1961). (22) Longuet-Higgins, H. C., Widom, B., Mol. Phys. 8, 549 (1964). (23) Mathias, E., Crommelin, C. A., Ann. Phys. (Parrs) 5, 137 I
,
(19%) \ - _ - _
(24) MGhias, E., Crommelin, C. A., Bijleveld, W. J., Grigg, Ph. P., Commun. Phys. Lab. Uniu. Leiden 221b (1932). (25) Mathias, E., Crommelin, C. A., Garfit Watt, H., Ibid., 189a (lYL/).
(26) Mathias, E., Crommelin, C. A., Kamerlingh-Onnes, H., Ibid., 162b (1923). (27) Mathias, E., Crommelin, C. A , , Meihuizen, J. J., Ibid., 248b (1 937). (28) Mathias, E., Kamerlingh-Onnes, H., Ibid., 117 (1911). (29) Mathias, E., Kamerlingh-Onnes, H., Crommelin, C. A., Ibid., 145c (1915).
(30) Michels, A., Wassenaar, T., Physica 16, 221 (1950). (31) Ibid., p. 253. (32) Michels, A., Wassenaar, T., de Graaf, W., Prins, Ch. R., Ibid., 19, 26 (1953). (33) Michels, A., Wassenaar, T., Zwietering, Th. N., Ibid., 18, 63 (1952). (34) Ibid., p. 160. (35) Mullins, J. C., Kirk, B. S., Ziegler, W. T., Tech. Rept. 2, Proj. A-663, Engineering Experiment Station, Georgia Inst. Technology, Atlanta, 1963. (36) Mullins; J. C., Ziegler, W.T., Kirk, B. S., Aduan. Cryog. Eng. 8, 126 (1962). (37) Patterson, H. S., Cripps, R. C., Whytlay-Gray, R., Proc. Roy. SOC.(London) A86, 579 (1912). (38) Pitzer, K. S., Lippman, D. Z., Curl, R. F., Jr., Huggins, C. M., Petersen, D. E., J . A m . Chem. SOC. 77, 3433 (1955). (39) Prigogine, I., “The Molecular Theory of Solutions,” Amsterdam. North Holland. 1957. (40) Prigogine, I., Trappeniers, N., Mathot, V., Discussions Faraday SOC.15, 93 (1953). (41) Renon, H.; Ph.D. dissertation, University of California, Berkeley, 1966. (42) Rossini, F. D., “Selected Values of Physical and Thermodynamic Properties of Hydrocarbons and Related Compounds,” Carnegie Press, Pittsburgh, 1953. (43) Rowlinson, J. S., “Liquids and Liquid Mixtures,” Butterworths, London, 1959. (44) Schmidt, H. H., Oppdycke, J., Clark, R. K., J . Phys. Chem. 68, 2393 (1964). (45) Sherwood, A. E., Prausnitz, J. M., J. Chem. Phys. 41, 429 (1964). 86, 197 (1964). (46) Simha, R., Havlik, A. J., J . Am. Chem. SOC. (47) Wertheim, M. S., J . Chem. Phys. 43,1370 (1965). (48) Ziegler, W. T., Mullins, J. C., Tech. Rept. 1, Proj. A-663, Engineering Experiment Station, Georgia Inst. Technology, Atlanta, 1963. RECEIVED for review February 8, 1966 ACCEPTED August 8, 1966
MOLECULAR THERMODYNAMICS OF SIMPLE LIQUIDS Mixtures C. A. ECKERT’, H E N R l RENONZ, AND J . M. PRAUSNITZ De$artment of Chemical Engineering, University of California, Berkeley, Calif., and Institute for Materials Research, National Bureau of Standards, Boulder, Colo. The analytical partition function for pure, simple liquids is generalized for liquid mixtures containing any desired number of components. The properties of liquid mixtures are calculated from standard statistical mechanical relations on the basis of Scott‘s two-liquid theory coupled with a three-parameter theorem of corresponding states. The effect of three molecular parameters on solution excess function is investigated, and it is shown that these are very sensitive to the characteristic energy for two unlike molecules; in general, this energy is not sufficiently well approximated by the geometric-mean assumption but must b e determined by some mixture property such as the second virial cross coefficient, Biz. Calculated excess Gibbs energies, enthalpies, and volumes agree very well with experimental results for 17 binary systems containing simple, nonpolar molecules.
treatment of the thermodynamics of liquid mixtures contributes to our fundamental understanding of molecular processes in solutions and is of direct use in the design of typical chemical process equipment. The basis for such a treatment must stem from fundamental molecular considerations ; the methods of statistical mechanics can then be used to provide a link between microscopic molecular
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Present address, University of Illinois, Urbana, Ill. On leave of absence from Institut Fransais du Pttrole, RueilMalmaison, France. 1 2
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properties and the bulk thermodynamic properties required for practical applications. In the present work we discuss the thermodynamic properties of mixtures, and aim to discover the dependence of these properties on various molecular functions. To do this, we extend to mixtures the statistical thermodynamics of pure components, based on a three-parameter theory of corresponding states, as developed in a previous paper (26). As discussed there, we confine our attention to simple, nonpolar molecules whose intermolecular force fields are nearly spherically symmetric.
