General Method for Predicting the Latent Heat of Vaporization Larry W. Fish and Janis Lielmezs* Chemical Engineering Department The University of British Coiumbia Vancouver Britmh Columbia Canada
Using the dimensionless cotrdinates S4' (S* = ( l H \ - / T ) / ( A H H v B / T B )where , AH\, = heat of vaporization at the given temperature T . K, while A H V B = heat of vaporization at the normal boiling point T B , "K and T* ( T * = ( T c / T - 1 ) / ( 7 c / T ~ - l ) ) , where Tc = the critical point temperature O K , an empirical equation, S" = (T* T*n)/(l T " m ) ,has been established to calculate the latent heat of vaporization values over the entire liquid range f r o m the normal melting point to the critical point for the separate classes of metal, quantal, inorganic, and organic liquids. The proposed method (this work) compares excellently in overall accuracy with other well known selected correlation methods over the complete range of investigation (total of 308 compounds, from the melting point to the critical point).
+
+
Introduction and the Proposed Method The differential equation describing the coexistence curve of two-phase equilibrium in one-component systems for pure substances is the Clausius-Clapeyron equation
where AHv is the molar heat of vaporization, and term (Va - V I ) is the associated molar volume change of the system between gaseous (g) and liquid (1) phases along the liquid-vapor pressure saturation curve. The subscript S refers to this liquid-vapor pressure equilibrium state. For this univariant system, subject to changes in thermodynamic states, the latent heat calculated from eq 1 is found as product between the differential pressure-temperature ratio and the temperature times the associated molar volume change. However, often the needed data, especially those of the volume change, are not readily available. Consequently, there has been proposed a flurry of theoretical and semiempirical relations (compare with Reid and Sherwood, 1966). Quite often these methods use parameters associated with the vapor pressure data, critical state data, including the acentric factor (Pitzer, 1955) and Riedel's constant (Riedel, 1954) and the molecular structural data such as molecular weight, volume and Altenburg's quadratic mean radius (Ogden and Lielmezs, 1969). The difficulties in using the vapour pressure and critical point data are well known (Reid and Sherwood, 1943). On the other hand, the use of methods based on the molecular parameters is not entirely satisfactory (Reid and Sherwood, 1966; Ogden and Lielmezs, 1969), for instance. in case of complex isomeric substances. It appears, therefore, that a direct empirical relation between the heat of vaporization and the temperature along the liquid-vapor saturated equilibrium curve of pure substances would be a useful tool obtained if we combine the following dimensionless coordinates
where AH\, = heat of vaporization for pure substances at the given temperature, cal/mol, AHVB= heat of vaporization for pure substances at the normal boiling point temperature, cal/mol and T, TR, TC are the temperatures of the given state, the normal boiling point, and the critical point, respectively, OK. Then for this coordinate system (eq 2 and 3) we can write empirical relation (Figures 1, 2, and 3; Tables I and 11) 248
Ind. Eng. Chem., Fundam., Vol. 14, No. 3, 1975
S* = (T* + T*")/(1 + T f " ) (4) with n and m representing characteristic group constants obtained from the experimental data sets (Table I) by means of nonlinear least-squares regression methods (U.B.C. Computer Centre programs (Halm, 1972)) for the three proposed general liquid groupings: the liquid metals, the quantal liquids, the inorganic and organic liquids. The behavior of the plotted experimental data indicates (Figures 1-3; Tables I-VI), that eq 4 and the three calculated group constant pair (constants n and m for metal, quantal and inorganic and organic liquids) values within the specified accuracy limits are valid over the entire liquid range from the melting point to the critical point. Indirectly, the use of T* (eq 3 and 4) is validated by Lielmezs' (1974) finding that straight line relations exist over the entire liquid range for liquids between the natural logarithm of the dimensionless group Dp$ (where D is the liquid self-diffusion coefficient, p is the density, and $ is the fluidity) and the dimensionless temperature T = (Tc - T ) / T ; where TC is the gas-liquid critical point temperature and T is the temperature at which D, p , and $ values have been obtained. If we compare T with P ' of this work, we note that T* additionally to critical state involves reference to the normal boiling point temperature; i.e., T* makes references to two points, first to the critical state and then to the normal boiling point.
Liquid Metals For liquid metals (Na, K, Rb, Cs, Hg, and Te), eq 4 becomes s* =' (T* + T* 0 0 2 0 9 5 7)/(I + T* -0.11467 ) (5) where n = 0.20957 and m = -0.17467 (Figure 1, Tables I, 11, and VI). The error mean square for the input of 51 liquid metal experimental data point pairs, s2 = 0.0046983, while the standard deviation (Halm, 1972) for coefficient "n" is 0.026308 and for coefficient "m" is 0.016356. Note that for use in this work, we have defined error mean square as
where y, is the input data point, 7 is the calculated point, n is the number of data pairs, p is the number of program variables with weight such that the curve was forced through the point S* = 1.0, and T* Z= 1.0. Equation 5, although derived mainly from the contributions of the alkali metal group (Na, K, Rb, Cs), nevertheless predicts the latent heat of vaporization for liquid metals of different liquid electronic structure patterns (compare with the following: Egelstaff (1973); Cusack (1973); Grosse (1966); and Tourand et al. (1972)) for the proposed reduced
Figure 1. S*-T* relation (eq 5 ) for liquid metals.
75
1w
Figure 2. S*-T* relation (eq 9) for quantal liquids.
S * - P coordinate system over the entire liquid range from the melting point to the critical point. Consequently, all the correlated metals may be considered to be in the corresponding state with respect to the introduced S*-T* coordinates. Indeed, eq 5 supports Grosse's (1966) findings that properties of sodium, potassium, and lithium will correspond to cesium and rubidium and that, therefore, the physical properties of the liquid higher alkali metals can now be estimated with more accuracy a t high temperatures.
If we recall that our reduced coordinates S* and T* (eq 2 and 3) include critical and normal boiling points by definition, it may well be expected that our proposed empirical correlation (eq 4 and 5 ) may be generalized to include all metals. This is brought out indirectly by comparing tellurium behavior in our coordinate (S*-T*) system to that shown in Grosse's (1966) reduced coordinate (AS\ = A H V / T and Tred = T / T c ) system. For instance, using Baker's (1967) data, tellurium at 1000°K has latent heat of vaporization of 26.9 kcal/mol, while at 1400°K the heat Ind. Eng. Chem., Fundam., Vol.
14, No. 3, 1975
249
s* ' I
~
i
,
I
3
T*
Figure 3. S*-T* relation (eq 10) for inorganic and organic liquids.
of vaporization is 23.3 kcal/mol. If the critical temperature of tellurium is 2329°K (Horvath, 1973), then, using Grosse's (1966) coordinates, we have that for Tred = 0.43, the value of 3s" = 26.9 cal/g-atom OK and for Tred = 0.60, AS" = 16.6 cal/g-atom O K . When these values are compared with Grosse's (1966) correlation for alkali metals, then it is found that these experimental tellurium data points fall distinctly outside the range of Grosse's correlation, and so tellurium does not conform to Grosse's (1966) S vs. Tred curve obtained for Hg, Rb, and Cs. However, tellurium easily corresponds to alkali metal group and mercury behavior in our proposed S * - P reduced coordinate system (Figure 1; Tables I, 11, and VI) Grosse (1966) has also suggested that heat of vaporization values for liquid metals may be found from the knowledge of the melting behavior a t their normal melting points by means of correlation AHv = AH,(l
-
( 7)
where S H Y is the heat of vaporization in cal/mol a t the given temperature, T , O K , AHw is the heat of melting, cal/mol a t the normal melting point, T M , OK, and a is constant needed for each metal separately, while 0 is defined as Comparing our proposed correlation (eq 4 and 5 ) with Grosse's relation (eq 7 ) , we see that our proposed correlations refer directly to the thermodynamic state changes a t the normal boiling point including a t the same time reference to the critical point (definition of P )while Grosse (1966) utilizes the normal melting point and critical point relations to indirectly estimate the latent heat of vaporization values. In view of the presented evidence (Figure 1, Tables I and 11) it is felt that eq 5 is sufficiently general and accurate (Table 11) to be used for predicting the latent heat of vaporization values over the entire liquid metal range from the melting to the critical point. 250
Ind. Eng. Chem.,
Fundam.,Vol. 14, No. 3,
1975
Quantal Liquids In this work we consider the following substances for which we have adequate data as quantal fluids (see de Boer and Kranendonk (1948) and Lennard-Jones and Devonshire (1937)): He, He3; equilibrium Hz; para-HZ and Dz. For these quantal liquids, eq 4 becomes S* = ( T $
+ T*
0.14513)/(1
+ T*
o . ~ z ~ ~ @ ) (9)
Equation 9, as proposed, is valid very close to the critical point, up to ( T c - T ) / T c 5 1% of the critical point temperature value with the error mean square s2 = 0.14369 for 36 data point pairs. The standard deviation for constant n is 0.12818, while for constant m it is 0.0040486. Figure 2 compares the calculated S* and T* values (eq 9) with available experimental data, Table I, while Table I11 shows a comparison between the results obtained in this work and by other estimation methods. In general, the S * - P relation behaves smoothly (Figure 2); nevertheless this S*-T* curve can be divided into three large regions characterized by the number of data point pairs found in the given region and the value of T*. Taking the closeness of experimental data to the calculated data point values (or the goodness of the curve fit) as the criterion for the region formation, then the first region in T* can be visualized as 0.0 5 T* 5 20. In this AT* we find nearly all data point pairs. This region includes both the critical and the normal boiling point. The second region is for the P range of 20 5 P 5 373. In this region we find only ten data point pairs, all for liquid helium. For this region T g >> T; that is the given state temperature T has become very small. Consequently, this region is of practical significance only for those liquids which have exceedingly low normal boiling points and which are very close to absolute zero (0°K). The third region is for the P range of 375 5 T* 5 a . For this region we have no available experimental data point pairs, and the expected S * - P curve behavior for this P range appears to be largely hypothetical.
Table I. Summary of Data Compound class 1. Liquid metals 2. Quantal liquids 3. Inorganic and
organic liquids a. Paraffins
-
By definition (eq 2 and 3) when T O"K, then both coordinates, S* and T* approach infinity separately; Le., References S* m and T* m . However, when these separate coordinates (S* and T*) are connected by a mathematical 1,7, F8,69,53,100,12:: relation, such as eq 8, we have a more complex situation. 55,79,158,189,190 Indeed, if we substitute 7' = 0°K in eq 9, then S* appears to become indeterminate since both the denominator and numerator, separately, although a t different rates, ap36,42,46,4%48,50,51,52,53, proach the value of infinity. Whether the lifting of the in56,57,G1,62,73,80,81,91, determinancy would in the limit yield finite value for s* 96,107,112.11 4,119,131, when T O"K is not clear, even if this possibility seems 159,164,191 ,1 93, 135,15 if to be substantiated by the behavior of the calculated (eq 199,201,202 9) S* - T* curve as shown by Figure 2. 5,9,21,33,41,35,19,59,97,98,Since the third region represents the S*-T* relation a t 11 G,155,167,180,182,198, extremely low temperatures, the experimental verification 21 2 of this relation a t these temperatures is very difficult, to 15,26,75,86,126,148,1i3, say the least.
-
-
-
b. Unsaturated
hydrocarbons c . Aromatics d. Alcohols
e . Ethers f . Aldehydes
g. E s t e r s h. Miscellaneous oxygenated organics i. Hydrocarbon halides j . Aromatic halides k. Nitrogen organics
1. Sulfur organics
m. Organometallics n. Rare gases 0. Halogens p. Inorganic oxides q. Inorganic sulfides r . Inorganic halides
s. Miscellaneous inorganic s
181 , I 88 11,12,13,29,33,40,75,1OS, 109,L10,125,157,1SG,znG, 207 34,67,S5,87, 105,113,141, 142,144,152,l 86,187,195 24,32 208 27,203 8,l F , 19,20,22,65,66,93,140, 149,183,207,209 3,30,31,70,i1,163,1 75,179 4,F,l4,35,82,89,9O,117,122, 138,161,171,17-1,205 10,72,94,95,129,130,131, 132,133,136,137,139,145, 146,151,153,162,165,l 6 6 , 168,169,170,l 72,l 76,177, 200 2,78,178 18,127,185,211 21 0
28,58,99,106 25 37,38,39,101,102,1O3,101, 120,121,124 60,150
Inorganic and Organic Liquids In this general liquid group we have included a series of selected experimental data for the following main compound classes: paraffins, unsaturated hydrocarbons, aromatics, alcohols, ethers, ketones, aldehydes, esters, miscellaneous oxygenated compounds, hydrocarbon halides, aromatic halides, nitrogen-organic compounds, sulfur-organic compounds, organometallic compounds, rare gases, halogens, inorganic oxides, inorganic sulfides, inorganic halides, and miscellaneeous inorganic compounds. This collection of 297 compounds representing 1360 data point pairs does not include the six metals (Na, K , Rb, Cs, Hg, and Te) used to establish eq 5 and the quantal fluids, He, He3; equilibrium-H2; para-H2 and Dz. For the listed inorganic and organic liquid compound classes, eq 4 becomes The error mean square for the input 1360 inorganic and organic liquid data point pairs is s2 = 0.0010144, while the standard deviation for constant n value is 0.0050142 and that for constant m is 0.0046815. Figure 3 compares the calculated S* and T* values (eq 10) with the available experimental data listed by references in Table I. The evaluation of the goodness and over-all generality of the proposed correlation (eq 4 and 10) for the given sets of inorganic and organic liquids is found in Tables IV, V, and VI and through comparison of the results of this work with
Table 11. Comparisona between This Work (Eq 5), Experimental Data,* Watson I, and Watson I1 Methods for Liauid Metals Watson I (Re-f. 204)
This work Compound
Temp range, "K
NC N
Sodium Potassium Rubidium Cesium Mercury Tellurium
426.79-1 8 56.33 1088.70-1644.30 310.80-1367.00 302.00-1366.00 234.00- 750.40 1100 .OO-1400.00
9 8 10 10 9 5
Total
51
Av %
Max
3 4 2 0 0 3
2.73 1.52 1.33 0.77 0.74 1.99
10.07 2.93 3.90 1.95 1.71 -2.76
12
1.51
...
>
2%'
F
A'
Av
Max
%
%
4 5 6 4 7 0
3.36 2.19 2.40 1.77 3.95 0.25
12.64 3.97 5.19 -3.74 -7.52 -0.38
26
2.32
.. .
> 2Qc
Watson I1 ( R e f . E l i ) Av
Max
96
9%
4 3 4 2 7 0
3.04 1.35 1.80 1.14 4.06 0.51
11.87 2.60 4.23 -2.47 -7.73 1.16
20
1.98
...
s, 2 ;
If we compare the results of this table with results obtained by using Grosse's relation (Grosse, 1966) with a = 0.38, we find that Grosse's relation then yields the following 70deviation: for Na, 97.66%; K, 97.47%; Rb, 97.36%; Cs, 97.22%; Hg. 96.487~,and for Te, 71.0070. This indicates that the used a = 0.38 value is incorrect and that the associated assumption t h a t liquid metals might behave as van der Waals fluids is not justified. * See Table I. c N = number of data points. N > 2% = number of data points exceeding 2% average error. a
Ind. Eng. Chem., Fundam., Vol. 14, No. 3, 1975
251
Table 111. Comparisona between This Work (Eq 9), Experimental Data,* Watson I, and Watson I1 Methods for Quantal Liquids This work
Compound
Nc N
Helium Helium 3 Equilibrium-H, P-Hz
> 2%
Watson I (Ref. 203)
Watson I1 ( R e f . 197)
Av
Max
Av
Max
Av
Max
%
%
"2%
%
%
N>2%
%
%
9 10 5 7 5
9 9 5 7 5
10.44 8.08 6.93 20.61 36.94
43.37 -18.63 10.36 -72.51 -111.25
8 9 5 7 4
14.97 53.86 7.95 9.61 4.11
-37.42 -214.23 -15.25 -16.49 -9.23
8 8 5 7 4
10.79 29.69 6.37 6.22 6.85
-24.97 -137.09 -12.70 -13.85 -19.07
Total: 36
35
16.60
...
33
18.10
...
32
11.98
...
-
DZ
I t should be noted that cases with (Tc.- r ) / T r 5 1.0% have not been included in this correlation. As the critical point is approached, the intermolecular force effects become excessively large. See Table I. N = number of data points.
Table IV. Comparisona between This Work (Eq lo), Experimental Data,b Watson I, and Watson I1 Methods for 22 Inorganic and Organic Liquids This work
N> Compound Methane n-Nonane Ethylene Acetylene Benzene Vinyl benzene Propanol t- Butanol Ethylene glycol Diethyl ether 2-Me-tetrahydrofuran Acetaldehyde Methyl formate Perfluorocyclobutane Methyl chloride Fluorobenzene Methylamine Carbon disulfide Fluorine Water Titanium tetrachloride Mercuric chloride
Temprange,'K Nc 90.66-190.00 298.16-590.00 104.01-277.60 192.40-307.80 266.50-560.90 273.20-403.20 298.16-533.16 298.16-505.38 313.20453.20 273.00453.00 222.00-373.00 344.30-422.00 255.40-333.20 233.20-377.60 244.27413.00 318.40-382.37 223.00-380.40 273.20-522.40 53.48-140.00 273.00-644.30 273.20-373.20 700.00-970.00 Total
a
9 11 8
N>
2%d Av % 3 6 2 4 1 0 3 9 5
12
5 8 3 5 2 0 1 4 2 1
7
7
9
9
2.36 11.37 2.94 4.53 0.86 0.94 3.74 5.38 3.70 0.72 7.78 19.61 1.78 2.00 1.22 0.15 1.09 4.14 1.55 1.82 8.81 13.42
188 81
4.54
7 11
6 9 14 8 9 5 8 8 8 14 3 8 7 7
1
Watson I ( R e f . 204)
Watson II (Ref. 197)
N >
Max%
2%
Av %
Max%
2%
Av%
Max%
-9.71 -60.17 -16.73 -15.24 2.89 -1.38 -20.36 43.60 7.96 2.19 -11.55 -36.12 4.21 3.02 -7.03 0.29 2.03 -13.47 3.53 -13.07 -12.84 -29.21
6 9
4.00 -71.79 -14.43 -8.76 6.39 -2.04 -25.28 41.65 8.67 -5.56 -12.97 40.68 3.59 4.88 -3.11 0.19 -1.71 -11.51 4.76 2.92 -14.18 -14.62
3 9 2 6 2 2 4 10 8 4 5 8 2 3 2 0 1 3 5 8
9
2.55 16.30 2.21 4.04 1.15 1.38 6.03 7.28 6.38 1.72 8.18 21.92 1.60 1.87 0.85 0.09 1.03 3.23 3.10 1.52 9.30 10.64
9
2.05 17.15 3.07 3.99 1.20 1.43 3.45 4.87 5.54 1.99 8.34 22.48 1.53 2.08 0.85 0.13 1.26 3.62 1.98 3.62 9.21 6.87
-7.32 -75.72 -18.72 -8.58 5.08 -2.11 -11.47 -22.35 8.48 -6.64 -13.21 41.77 3.45 -5.60 -3.27 0.24 2.17 -12.96 -2.88 10.26 -14.05 -8.70
100
5.11
...
103
4.85
...
1
6 1 2 4 11 8 4 5 8 2
3 1 0 0 3 6 4 7
7
Results for other methods used for this set of compounds, are given in Table V. bd See Table I.
those obtained by means of other well-known, latent heat of vaporization calculation methods (Table V). As the results of comparison indicate, the proposed method, this work, is indeed of general validity.
Discussion The proposed empirical relations (eq 4, 5, 9, and 10) connect the reduced evaporation entropy (eq 2) with the reduced fractional temperature (eq 3) along the entire liquid range from the melting point up to the vapor-liquid critical point. This unique dimensionless S*-'P coordinate system predicts the latent heat of vaporization values without direct reference (Tables 11, 111, IV, V, and VI) to the nature of vaporization process, the magnitude and type of intermolecular forces, and the molecular structure, but requires the knowledge of vapor-liquid critical point 252
Ind. Eng. Chern., Fundarn., Vol. 14, No. 3, 1975
and normal boiling point temperatures and latent heat of vaporization value a t the normal boiling point. Indirectly, however, the physical and structural property influence on the evaporation process of substance is noted through the introduction of the characteristic group constants rn and n, for the three proposed general liquid classes (Table VI, eq 5, 6, and 10): the inorganic and organic liquids, the liquid metals and the quantal liquids. In effect the 'P-S* coordinates bring together the widely varied inorganic and organic compounds (297 substances, 1360 data point pairs) and at the same time distinguish between the liquid metals (6 substances, 51 data point pairs) and quantal liquids (5 substances, 36 data point pairs). As a matter of fact, Figures 1, 2, and 3 show that the inorganic compound group (Figure 3) is found between liquid metals (Figure 1) and quantal liquids (Figure 2). It is characteristic that liquid metals (Figure 1) form nearly
Table V. Validity of the Proposed Correlation (Eq 5, 9, and 10) Correlation and reference This work
Watson I (204)
Watson I1 (197)
Watson I11 (159)
Klein (77)
Haggenmacher ( 76,159)
F ishtine (77)
P i t z e r q hen (159)
Viswanath and Kuloor (196)
Narsimhan (143)
Chen (23)
Pitzer (23)
Liquid class Metal liquids, eq 5 Quantal liquids, eq 9 Inorg. and org. liquids, eq 10 22-tested compounds (Table IV) Metal liquids, eq 5 Quantal liquids, eq 9 Inorg. and org. liquids, eq 10 22-tested compounds (Table IV) Metal liquids, eq 5 Quantal liquids, eq 9 Inorg. and org. liquids, eq 10 22-tested compounds (Table IV) Metal liquids, eq 5 Quantal liquids, eq 9 Inorg. and org. liquids, eq 10 22-tested compounds (Table N ) Metal liquids, eq 5 Quantal liquids, eq 9 Inorg. and org. liquids, eq 10 22-tested compounds (Table IV) Metal liquids, eq 5 Quantal liquids, eq 9 Inorg. and org. liquids, eq 10 22-tested compounds (Table IV)
A'
AT
51 36 1360
188 51 36
1360 188 51 36 1360 188
Metal liquids, eq 5 Quantal liquids, eq 9 Inorg. and org. liquids, eq 10 22-tested compounds (Table IV) Metal liquids, eq 5 Quantal liquids, eq 9 Inorg. and org. liquids, eq 10 22-tested compounds (Table N ) Metal liquids, eq 5 Quantal liquids, eq 9 Inorg. and org. liquids, eq 10 22-tested compounds (Table IV) Metal liquids, eq 5 Quantal liquids, eq 9 Inorg. and org. liquids, eq 10 22-tested compounds (Table IV) Metal liquids, eq 5 Quantal liquids, eq 9 Inorg. and org. liquids, eq 10 22-tested compounds (Table IV) Metal liquids, eq 5 Quantal liquids, eq 9 Inorg. and org. liquids, eq 10 22-tested compounds (Table IV)
a straight line S*-T* relation very close to the critical point, the liquid metal S*-T* relation experiences a very sharp change of curvature since the S*-T* curve becomes force-fitted (thermodynamic critical point requirement) to go through the fixed critical point coordinates (S* = 0; T* = 0). Whether the values of the axial S* intercept obtained when the linear part of the liquid metal S*-T* curve is extended to cut the S* coordinate would represent the residual metallic bonding forces found a t the critical point remains a question worthy of further study. Tables 11, III, IV, and V show the predictive accuracy of
> 2%
Av %
12 35 504 81 26 33 465 100 20 32 469 103
1.51 16.60 2.56 4.54 2.32 18.10 2.72 5.11 1.98 11.98 2.63 4.85
Parameters needed
Remarks General range from melting to critical point
TC,
TB
TC,
TB
General, constant power = 0.38
TC,
TB
General, power is function of
AHVB
TB
TC,
TB,
General, power is function of T B , "%B
188
109
6.81 Tc, pc TB, P ( T )
188
143
12.47
T c , P,, P(T)
188
109
17.60
Antoine constants B and C Tc,Pc, P(T ) T B , AHVB
188
110
25.63
P(T)
131
Range T 5 T , for B a n d C need accurate P(T)
Range T 5 T, for C need accurate P(T)
Antoine constant C Tc, pc
188
Limited, correction factor needed
Version of Chen method, general
8.28 Tc, pc
0.3 5 T , 5 0.95 0.01 5 P , 5 0.4
P(T)
188
162
13.47 T c , PCC
General TB
188
146
10.74 Tc, pc P(T)
188
134
8.40 TC> TB,
Pc or W
188
95
Gener a1, accurate P ( T ) data required 0.56 5 T , 5 1.0 accurate w r e quired
7.72
the proposed S*-T* correlation for separate compound classes, while Table V sums up all obtained results and compares the proposed method with other selected methods for calculating the latent heat of vaporization. For liquid metals, Tables I1 and V compare the results of this work with Watson I (Watson, 1943) and Watson I1 (Viswanath and Kuloor, 1967) methods. For the total of 51 data point pairs (6 compounds, Figure 1) our proposed method (average error 7'0 = 1.51) appears to compare very well indeed to either of the considered Watson methods. In case of quantal liquids (5 substances, 36 data point Ind. Eng. Chem., Fundam.. Vol. 14. No. 3, 1975
253
Table VI. Summary of the Values of Constants n and m (Eq 5 , 9 , and 10)
Liquid class and equation Liquid metals Eq 5 Quantal liquids Eq 9 Inorganic and organic liquids Eq 10
Calcd value
Calcd value
)z
III
0.20957
-0.17467
0.14543 0.35298
SimpliCorrelafied conChange in tion e r r o r Simplified Simplified stant e r r o r e r r o r mean mean square 12 value i i i value mean square square, % 0.3909896
0.210
-0.175
0.52740
0.14369
0.145
0.527
0.14373
0.028
0.13856
0.0010144
0.353
0.139
0.0010146
0.020
pairs), using restrictive critical region data ( ( T c - T ) / T 5 1%); our proposed method (average error 70 = 16.60) appears to be second to the Watson 11 method (Tables I11 and V). This comparison perhaps may not be complete since in this liquid group we had only 36 data point pairs (Figure 2) with very few data point pairs available in the 20 5 T* 5 373 range, and none available a t the very low temperatures approaching the 0°K limit. It is expected that the accuracy of the proposed S*-T* relation for quantal liquids (eq 9) will be increased if more experimental data become available. The proposed correlation for inorganic and organic liquids (eq 9, Figure 3) representing 297 compounds and 1360 data point pairs competes with any of the other methods selected for comparison. Thus Table IV shows results obtained for 22 test compounds (188 data point pairs) between this work, Watson I (Watson, 1943) and Watson I1 (Viswanath and Kuloor, 1969) methods. The proposed method, this work, yields an average error percent of 4.54%. On the other hand, Table V additionally to summing u p and comparing the results for all compound classes, also gives separately a comparison between our work and 11 other well known prediction methods for all 297 inorganic and organic compounds. This comparison confirms (average error % = 2.56) the overall general validity of our proposed method. Comparing the amount of nature of the needed input parameters (Table V), we note that the proposed method needs relatively few input data-the critical point and the normal boiling point temperatures (Tc, T B ) and the heat of vaporization a t the normal boiling point (AHve).Other methods may need additional parameters, or again may be restricted in the range of usage. Table VI lists the values of constants m and n to be used in eq 5, 6, and 10. We have given two sets of m and n values. First, we have listed the calculated m values (obtained using nonlinear least squares regression methods, (Halm, 1972), and second, we have listed the proposed simplified m and n values. As shown in Table VI, the subsequently introduced change in error mean square 70 value between the first and the second set of the m and n values is negligible for quantal and inorganic and organic liquids (0.03%), while much larger for liquid metals (>20%). Hence, for quantal and inorganic and organic liquids one may use the simplified m and n values while for liquid metals one should use the calculated m and n values (Table VI). Literature Cited ( 7 ) Achener, P Y.. Mackewicz. W. V.. Fisher. D. L.. Camp, D . C., Report NO. AGN-8195. Vol. 1, Nuclear Divlsion, Aerojet-General Corp , Apr 1968. 12) Anderson, R . D . . Taylor. H. A,. J. Phys. Chem . 56, 161 (1952). ( 3 ) Andon, R L . J , Counsell. J . F., Hales, J. L . . Lees, E B , Martin, J F , J Chem SOC.A . 2357 (1968) ( 4 ) Andon, R. J L , Cox, J . D . , Herington, E. F. G.. Martin, J. F.. Trans faraday Soc.. 53. 1074 (1957).
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Received f o r review D e c e m b e r 13, 1974 Accepted April 11,1975 T h e f i n a n c i a l s u p p o r t o f t h e N a t i o n a l Research C o u n c i l of C a n a da i s g r a t e f u l l y acknowledged.
