Ind. Eng. Chem. Res. 1996, 35, 343-348
343
CORRELATIONS Prediction of Heat Capacities of Solid Inorganic Salts from Group Contributions A. T. M. Golam Mostafa, James M. Eakman,* and Mark M. Montoya Department of Chemical Engineering, New Mexico State University, Box 30001, Department 3805, Las Cruces, New Mexico 88003
Stephen L. Yarbro Los Alamos National Laboratory, Los Alamos, New Mexico 87545
A group contribution technique is proposed to predict the coefficients in the heat capacity correlation, Cp ) a + bT + c/T2 + dT2, for solid inorganic salts. The results from this work are compared with experimental data from the literature. The technique is shown to give good predictions for both simple and complex solid inorganic salts. Literature heat capacities for a large number (664) of solid inorganic salts covering a broad range of cations (129), anions (17), and ligands (2) have been used in regressions to obtain group contributions for the parameters in the heat capacity temperature function. A mean error of 3.18% is found when predicted values are compared with literature values for heat capacity at 298 K. Introduction Group contribution techniques are well studied and implemented to predict thermophysical properties such as heats of formation, ∆H298 f , free energies of formation, ∆G298 , heat capacities, Cp, and liquid molar volf ume at normal boiling point, Vb, for organic compounds. Mostafa (1995) has previously described a group contribution method for predicting ∆H298 and ∆G298 of f f solid inorganic salts. Reid et al. (1987) describe a large number of the group contribution methods available in the literature for predicting heat capacity correlation coefficients for organic compounds. Joback (1984) used the following functional form to predict Cp of organic ideal gases:
∑j nj∆a,j - 37.93) + (∑j nj∆b,j + 0.210)T + (∑nj∆c,j - 3.91 × 10-4)T2 + (∑nj∆d,j + 2.06 × j j
Cp ) (
10-7)T3 (1) where nj is the number of groups of the jth type and ∆k,j is the contributions for the kth (k ) a, or b, or c, or d) coefficient and the jth atomic or molecular group where T is in kelvin (K). Yoneda (1979) uses the same concept but a different equation for ideal gases:
Cp )
( ) T
∑j ∆a,j + (∑j nj∆b,j) 1000
( ) T
∑j nj∆c,j) 1000
+(
2
(2)
where nj is the number of groups of the jth type and ∆k,j is the contributions for the kth (k ) a, or b, or c, or d) coefficient and the jth atomic or molecular group * To whom correspondence should be addressed. E-mail:
[email protected].
0888-5885/96/2635-0343$12.00/0
where T is in kelvin (K). Thinh et al. (1976) and Benson et al. (1969) also used the same idea but different functional forms of eqs 1 and 2 to predict Cp for organic substances. The following functional forms have been used by Robie et al. (1979) to describe heat capacity of different inorganic solids:
c e + dT2 + 2 T xT
(3)
c + dT2 T2
(4)
Cp ) a + bT +
c T2
(5)
Cp ) a + bT +
e xT
(6)
Cp ) a + bT +
Cp ) a + bT +
Cp ) a + bT
(7)
ASPEN PLUS (1990), the interactive flowsheet simulator for process modeling, developed by Aspen Technology, Inc., Cambridge, MA, uses the following functional form for the heat capacities of solids:
Cp ) a + bT +
g c f + dT2 + + 2 T T xT3
(8)
where a, b, c, d, f, and g are constants. Knacke and Kubaschewski (1990) used the functional form of eq 4 for solid inorganic salts which is shown to fit experimental data well. The inverse square of the absolute temperature functionality which appears in many of these relationship comes from Einstein’s (McQuarrie, 1976) theory of the specific heat of crystals, which is © 1996 American Chemical Society
344
Ind. Eng. Chem. Res., Vol. 35, No. 1, 1996
Table 1. Estimation of Cp from Eq 10 step 1
CaAl2O4 group
step 2
Ca2+ Al3+ O2-
step 3
∑jnj∆i,j
step 4
∆aj
∆bj
1(20.4698) 2(10.3059) 4(28.1522)
Cp )
1(-6.2249) 2(4.5183) 4(12.0434)
153.6904
∑n ∆ j
+(
a,j
j
∑n ∆ j
b,j
× 10-3)T + (
j
50.9853
∑n ∆ j
c,j
j
1
× 106) + ( T2
∑n ∆ j
d,j
∆cj
∆dj
1(-0.02629) 2(-0.62271) 4(-0.74718)
1(-3.21927) 2(-3.7007) 4(-4.02248)
-4.26043
-26.71059
× 10-6)T2
j
) 153.6904 + (50.9853)(10-3)(298) + (-4.26043)(106)/(298)2 + (-26.71059)(10-6)(298)2 ) 118.54 J/(g-mol K)
( )
Cv ) 3Nk
hνE kT
2
e-hνE/kT (1 - e-hνE/kT)2
(9)
where Cv is the heat capacity at constant volume, N is the number of atoms, k is the Boltzmann constant, h is Planck’s constant, νE is the Einstein frequency, and T is the absolute temperature. The inverse square term appears as a dominant temperature functionality in the Taylor series expansion of eq 9. Development of Method Heat capacity correlation coefficients of solid inorganic salts are determined using an ion-based group contribution technique. A multiple linear regression is used to determine values for group contributions which give “best fit” prediction of the coefficients of the heat capacity temperature function for solid inorganic salts. These predictions employ the functional form of eq 4 which is shown to fit experimental data of solid inorganic salts well by Knacke and Kubaschewski (1990) and written as follows:
Cp )
∑j nj∆a,j + (∑j nj∆b,j × 10-3)T + ∑j nj∆c,j × 106)
(
1 2
T
∑j nj∆d,j × 10-6)T2
+(
(10)
where nj is the number of groups of the jth type, is the contributions for the kth (k ) a, or b, or c, or d) coefficient and the jth atomic or molecular group where T is in kelvin (K). The values of the group contributions are obtained by regressing on Cp data from a number (664) of solid inorganic salts. The parameters are predicted using the multiple linear regression relationship
b ) [XTX]-1XTy
(11)
where b is a vector the jth element of which contains ∆k,j, the calculated prediction for the kth coefficient (k ) a, or b, or c, or d) the jth group contributions, X is a matrix in which the element in the ith row and jth column contains (ni,j - n j j), where ni,j the number of occurrences of group j in compound i and n j j is the mean number of occurrences of group j in all salts in the regression, and y is a vector the ith element of which contains the experimental values found for Cp, heat capacity for compound i. The reader is referred to Himmelblau (1970) for a discussion of regression and statistical parameter estimation. The data for most inorganic solid salt heat capacities used in the regression are obtained from Knacke and Kubaschewski (1990). Data for neptunium and the
hydrates are obtained from Barin (1989) at different temperatures and fitted in the functional form of eq 4 to get the coefficients a, b, c, and d. Table 2 gives all the groups and their contributions, ∆k,j, estimated by the regression for a, b, c, and d. The columns in this table list, in order, the group being estimated, the number of occurrences of that group in the regression, the contribution of that group to ∆a,j, ∆b,j, ∆c,j, and ∆d,j. Procedure for Prediction by the Proposed Method A stepwise procedure for the prediction of Cp by the proposed method is described below. Table 1 gives an illustrative example of this sequence of calculations. The successive steps are as follows: Step 1: Write the molecular structural formula for the solid inorganic salt. Step 2: Break the molecular structural formula into appropriate cationic, anionic, or ligand molecule structural groups as given in Table 2. Then calculate the numerical contributions of each group by picking the numerical value of that specific group from Table 2 and multiplying it by the number of occurrences of the same group in the molecular structural formula. Step 3: Sum of the numerical values of the various groups to yield prediction for ∑jnj∆a,j, ∑jnj∆b,j, ∑jnj∆c,j, and ∑jnj∆d,j for eq 10. Step 4: Calculated Cp by substituting ∑jnj∆a,j, ∑jnj∆b,j, ∑jnj∆c,j, and ∑jnj∆d,j in eq 10 and using any absolute temperature T (K) as required. Dean (1979) discusses the historically simple Kopp’s rule for estimating heat capacity of a liquid or solid, which states that the heat capacity of a solid inorganic compound is approximately equal to the sum of the heat capacities of the constituent elements and that an approximate value expressed in gram calories per gram formula weight can be calculated by assigning atomic heat capacities to the elements. Table 3 gives a comparison of predictions of heat capacities (C298 p ) for different solid inorganic salts using Kopp’s rule and by the proposed method with the corresponding fits to experimental values. The columns in this table list, in order, the solid inorganic compound being considered, experimental values of heat capacity 298 (Cp,literature ) at 298 K found in the literature, predicted 298 heat capacity (Cp,predicted ) by the proposed method at 298 K, percent error associated with the prediction by the proposed method as compared to the experimental 298 values, predicted heat capacity (Cp,predicted ) by Kopp’s rule, and percent error associated with the prediction by Kopp’s rule. It can be seen that the proposed method gives better prediction. Heat capacities at the maximum temperature for which each correlation is valid (Knacke and Kubaschewski, 1990; Barin, 1989) are also
Ind. Eng. Chem. Res., Vol. 35, No. 1, 1996 345 Table 2. Solid Inorganic Salt Groups: Cationic, Anionic, and Ligand Molecule Groups, Their Contributions, Number of Occurrences in the Regression, and Standard Deviation (s∆,j) Associated with Them for a, b, c, and d group
no. of occurrences
∆a,j
∆b,j
∆c,j
∆d,j
group
Ag2+ Al3+ As3+ As5+ B3+ Ba2+ Be2+ Bi3+ Ca2+ Cd2+ Ce3+ Ce4+ Cm3+ Co2+ Co3+ Cr2+ Cr3+ Cr6+ Cs+ Cu+ Cu2+ Dy3+ Er3+ Eu2+ Eu3+ Fe2+ Fe3+ Ga3+ Gd3+ Ge2+ Ge4+ H+ Hf2+ Hf4+ Hg+ Hg2+ Ho3+ In+ In2+ In3+ Ir3+ Ir4+ K+ La3+ Li+ Lu3+ Mg2+ Mn2+ Mn3+ Mo2+ Mo3+ Mo4+ Mo5+ Mo6+ Na+ Nb3+ Nb4+ Nb5+ Nd3+ NH4+ Ni2+ Np3+ Np4+ Np6+ P5+
8 81 6 4 59 23 16 9 64 14 9 1 1 17 1 4 22 4 8 7 11 4 2 2 6 15 26 9 7 2 2 12 1 9 8 6 6 3 1 5 2 1 48 8 47 3 32 15 4 3 4 3 2 11 91 1 3 5 9 4 14 2 3 2 12
11.398 10.306 -18.849 -16.941 -13.188 26.676 -5.164 -3.363 20.470 21.910 13.011 8.506 -5.346 26.552 34.158 16.779 21.086 -23.112 20.779 11.289 19.179 21.060 17.635 22.957 35.717 20.486 16.618 16.577 20.474 17.815 -7.106 -5.217 8.841 20.580 23.601 8.228 33.919 15.693 5.362 13.433 13.108 5.576 25.309 15.803 15.639 18.605 14.639 21.419 8.913 22.549 -28.076 19.349 -48.550 -7.473 14.186 20.593 12.016 19.843 15.042 4.205 22.497 23.683 -12.661 -19.545 -39.486
24.063 4.518 69.121 25.671 16.765 -15.773 22.314 14.344 -6.225 -23.495 -7.089 -6.387 10.658 -19.272 -42.075 -8.351 -16.392 27.266 -0.058 12.683 3.018 -15.320 -17.241 -9.764 -36.906 -5.415 3.328 -11.146 -20.495 -5.864 41.113 25.754 -27.084 -22.025 0.155 19.740 -49.884 13.602 29.459 -4.548 -18.124 -3.667 -2.284 -11.296 4.124 -16.153 -0.637 -15.908 3.070 -9.633 84.735 -24.670 202.730 -7.423 9.665 -14.387 -14.823 -33.720 -4.286 116.120 -6.671 -10.822 62.760 -43.129 50.248
0.434 -0.623 0.995 1.183 0.273 0.054 -0.002 1.658 -0.026 0.561 0.762 0.734 0.076 0.327 -0.149 0.152 0.067 1.662 0.520 0.270 -0.002 0.476 1.542 0.500 0.298 0.055 0.082 -0.034 0.163 0.556 0.226 0.232 0.280 -0.050 0.078 0.585 0.133 0.212 0.502 0.310 0.469 0.394 0.218 0.458 -0.007 0.172 -0.074 0.265 0.454 0.289 1.976 0.62 1.237 1.279 0.529 0.043 0.494 0.517 0.450 1.206 -0.022 0.742 1.250 1.616 1.686
Cations 2.106 Pb2+ -3.701 Pb4+ 6.034 Pd2+ 5.796 Pr3+ -0.219 Pr4+ 7.523 Pt2+ 6.544 Pt3+ 2.171 Pt4+ -3.219 Pu3+ 6.178 Pu4+ 4.687 Pu6+ 8.045 Rb+ -3.166 Re4+ 5.245 Re5+ -4.614 Re7+ 0.073 Rh3+ 5.298 Ru3+ 8.719 Ru4+ 11.137 Sb3+ 4.246 Sc3+ -1.275 Se4+ 2.925 Se5+ -3.292 Se6+ 3.050 Si4+ 18.834 Sm2+ 5.393 Sm3+ 5.002 Sn2+ 5.207 Sn4+ 3.088 Sr2+ 3.312 Sr4+ 6.623 Ta3+ 2.576 Ta4+ 2.601 Ta5+ 2.633 Tb3+ 0.036 Tb4+ 2.319 Tc4+ 19.321 Tc6+ 0.561 Te4+ -1.314 Th4+ 2.953 Ti2+ 0.573 Ti3+ 8.045 Ti4+ 5.174 Tl+ 3.344 Tl3+ 7.973 Tm3+ 3.904 U3+ -0.609 U4+ 5.652 U5+ 5.448 U6+ 1.122 V2+ 0.108 V3+ 0.855 V4+ -5.487 V5+ 13.430 W2+ 4.851 W4+ -1.971 W5+ 2.709 W6+ 1.065 Y3+ 3.311 Yb2+ 2.166 Yb3+ 6.234 Zn2+ -3.274 Zr2+ -48.715 Zr3+ 356.929 Zr4+ 2.325
BrClClO3ClO4CO32FHCO3IN3-
166 295 2 4 15 247 2 132 2
26.093 26.609 56.325 113.901 47.278 22.041 26.758 26.999 24.083
14.767 10.376 104.045 46.380 86.757 15.652 138.905 13.542 31.806
-0.201 -0.251 -1.043 -3.511 -0.887 -0.244 -0.373 -0.182 -0.764
Anions -1.039 NO20.657 NO3-5.013 O-6.944 O2-5.133 OH1.538 PO43-5.013 SO32-1.300 SO423.644
18
36.332
20.257
-0.265
Ligand Molecules -1.780 H 2O
CO
no. of occurrences
∆a,j
∆b,j
∆c,j
∆d,j
14.632 0.7457 15.727 20.343 21.266 14.033 28.058 43.127 31.509 35.028 -17.410 19.747 11.056 3.204 -7.326 10.964 35.233 20.326 -11.375 12.644 -14.644 0.120 -63.818 -2.308 24.182 22.693 16.189 10.166 15.410 17.666 16.373 27.033 12.959 18.899 8.506 12.316 23.493 20.470 20.291 21.321 0.689 10.043 15.850 25.172 23.189 17.864 26.669 13.890 -14.769 20.124 20.671 2.121 -1.279 18.062 6.405 -6.784 -23.788 22.197 14.772 23.895 12.599 30.330 26.864 17.188
7.448 4.913 -2.973 -11.940 0.933 4.772 -25.164 -50.286 -15.262 -39.982 43.930 5.160 -11.407 -19.960 6.207 3.8319 -31.127 -13.287 36.678 -15.208 16.581 -8.773 110.803 4.382 -4.011 -10.606 5.634 -7.447 -0.275 -6.047 -14.807 -41.503 -20.800 -10.177 -6.387 -12.577 -36.130 -25.515 -28.458 -9.603 23.784 -7.562 20.050 -13.439 -19.727 -10.375 -29.923 -31.178 50.875 -11.173 -27.881 2.604 -5.243 1.149 3.023 42.333 35.451 -22.411 0.169 -21.427 -0.744 -18.041 -34.755 -23.478
0.642 1.076 0.496 0.213 0.419 0.453 0.679 0.905 0.127 -0.268 0.385 0.526 0.201 0.492 1.683 0.338 0.754 -0.920 1.780 0.330 1.731 0.697 2.335 -0.041 0.502 0.327 0.378 -0.180 0.381 1.494 0.043 -0.208 0.439 0.342 0.733 0.038 2.242 0.999 0.351 0.209 0.839 0.299 0.507 -0.203 0.482 0.780 -0.066 0.627 1.654 0.087 0.199 1.214 0.697 0.502 0.790 1.062 1.836 0.002 0.502 0.346 0.377 0.077 -0.144 -0.063
4.828 8.045 1.895 2.084 8.045 0.382 0.573 0.764 3.352 -1.420 -1.932 7.198 8.045 12.067 15.007 4.214 -1.971 8.045 0.090 2.953 5.589 10.056 10.990 -3.301 -1.314 3.826 1.586 8.045 4.159 8.045 -1.971 -2.628 4.567 3.065 8.045 8.045 12.067 2.709 -1.633 1.847 3.471 11.414 0.635 5.087 3.370 0.108 2.171 -2.126 -14.399 1.847 0.867 1.988 13.608 -1.314 2.709 0.956 9.893 3.065 -1.314 2.925 4.203 0.072 0.108 7.098
1 120 36 1155 16 6 1 49
18.307 49.766 34.706 28.152 28.917 95.827 78.739 85.866
126.388 83.928 3.050 12.043 30.730 56.863 24.184 52.357
0.596 -0.478 -1.052 -0.747 -0.628 -2.462 -1.058 -1.925
-4.044 -7.040 -6.604 -4.023 3.257 -7.691 -9.702 -0.047
39
15.458
66.593
0.470
-40.518
26 1 3 6 1 2 2 2 7 2 2 16 1 1 2 4 1 1 6 5 6 2 2 79 1 8 6 1 14 1 1 1 9 5 1 1 1 2 11 4 7 34 7 3 4 4 7 4 11 4 4 3 20 1 2 2 20 5 1 4 18 4 4 13
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Table 3. Literature Values, Predicted Values, and Percent Error: Comparison of Heat Capacities by the Proposed Method, Kopp’s Rule with the Corresponding Experimental Values, and Percentage of Errors (% Error) Associated with Some Selected Solid Inorganic Salts proposed method 298 , Cp,literature
298 Cp,predicted ,
compound
J/(g-mol K)
J/g-mol K
% error
Ag2CO3 Ag2SO4 AgNO2 AgNO3 Al(OH)3 Al2(SO4)3 AlOCl AlPO4 Ba(NO3)2 Ba2TiO4 Ba3Al2O6 BaCO3 BaHfO3 BaSO4 BaTiO3 BaV2O6 BaZrO3 Be(OH)2 BeAl2O4 BeAl6O10 BeSO4 Bi2(SO4)3 BiOCl Ca(NO3)2 Ca(OH)2 Ca2B2O5 Ca2P2O7 Ca2V2O7 Ca3Al2O6 Ca3B2O6 Ca3P2O8 Ca3WO6 CaAl2O4 CaAl4O7 CaB2O4 CaB4O7 CaCO3‚MgCO3 CaCr2O4 CaFe2O5 CaMgO2 CaMoO4 CaO‚HfO2 CaUO4
111.60 131.44 81.34 93.01 93.05 260.15 56.83 92.98 151.32 149.13 215.35 87.97 108.25 101.67 102.38 172.78 105.39 65.55 104.88 261.81 85.60 278.82 70.16 149.29 87.56 147.05 187.70 213.88 209.65 187.77 236.03 202.24 120.55 193.83 103.83 157.74 155.18 146.59 194.05 79.61 117.06 105.71 130.44
109.98 127.08 85.97 92.41 98.19 248.00 54.21 88.67 160.79 150.56 216.20 85.94 105.86 103.04 104.34 173.51 102.27 64.63 102.55 257.61 81.84 278.93 69.67 155.57 80.62 140.63 186.21 209.29 200.54 181.63 227.22 200.25 118.54 196.07 99.63 158.25 156.98 144.77 188.88 77.54 115.82 100.64 127.65
1.45 3.31 5.70 0.64 5.53 4.67 4.61 4.63 6.25 0.95 0.39 2.31 2.21 1.35 1.91 0.42 2.95 1.40 2.22 1.60 4.40 0.04 0.70 4.20 7.92 4.37 0.79 2.15 4.34 3.27 3.73 0.99 1.67 1.16 4.05 0.33 1.16 1.24 2.66 2.60 1.06 4.80 2.13
Kopp’s rule 298 Cp,predicted ,
proposed method 298 Cp,literature ,
298 Cp,predicted ,
J/g-mol K
% error
compound
J/(g-mol K)
J/g-mol K
% error
109.62 141.42 85.35 102.09 105.02 320.49 68.62 115.48 178.24 144.77 230.12 83.68 102.09 115.48 102.09 178.24 102.09 78.66 144.77 348.95 115.48 320.49 68.62 178.24 78.66 158.16 214.22 220.92 230.12 200.83 256.90 204.18 144.77 246.86 115.48 188.28 167.36 144.77 187.44 85.35 118.83 102.09 118.83
1.77 7.59 4.94 9.76 12.87 23.20 20.74 24.19 17.79 2.93 6.86 4.88 5.69 13.58 0.29 3.16 3.13 20.01 38.03 33.28 34.90 14.95 2.20 19.39 10.16 7.55 14.13 3.29 9.76 6.96 8.84 0.96 20.08 27.36 11.22 19.36 7.85 1.24 3.40 7.22 1.51 3.42 8.90
CaV2O6 Li2B4O7 Li2BeF4 Li2CO3 Li2HfO3 Li2Nb2O6 Li2SO4 Li2Ta2O6 Li2WO4 Li2ZrO3 Li3AlF6 LiBeF3 LiClO4 LiNO3 LiOH Mg(NO3)2 Mg(OH)2 Mg2V2O7 Mg3P2O8 MgAl2O4 MgCO3 MgFe2O4 MgMoO4 MgSO4 MgTi2O5 MgTiO3 MgV2O6 MgWO4 UBr3 UBr4 UCl3 UCl4 UCl5 UCl6 UF3 UF4 UF5 UF6 UI3 UI4 UO2 UO2SO4 UOBr2
170.74 170.34 134.97 97.96 111.87 198.55 117.14 200.14 133.34 110.20 202.30 92.61 104.94 89.11 49.75 141.90 77.49 202.26 213.09 115.94 75.40 137.76 110.88 95.57 146.60 91.17 159.87 109.83 108.73 128.11 102.52 120.77 144.57 175.53 95.08 116.01 132.30 167.43 112.02 134.43 63.57 144.88 97.88
168.29 175.21 133.42 97.68 117.59 204.22 114.78 197.02 135.20 114.01 201.38 91.83 105.06 86.26 48.79 151.10 76.15 200.35 213.82 114.07 76.26 143.41 111.35 93.36 152.76 94.65 163.82 113.78 107.96 129.74 104.35 124.93 146.12 179.32 95.86 113.60 131.96 162.33 110.18 132.69 63.14 143.47 96.44
1.44 2.86 1.15 0.30 5.11 2.86 2.01 1.56 1.39 3.45 0.45 0.84 0.12 3.20 1.93 6.48 1.73 0.94 0.34 1.61 1.13 4.10 0.42 2.31 4.21 3.82 2.47 3.59 0.71 1.27 1.79 3.44 1.08 2.16 0.82 2.08 0.26 3.05 1.65 1.30 0.68 0.97 1.47
Table 4. Literature and Predicted Values of Some Compounds Not Used in the Regression and Percentage of Errors Associated with Them compound
298 , Cp,literature J/g mol K
298 Cp,predicted , J/g mol K
% error
Ca3(PO4)2 CaCO3 CaSO4 CaSO4‚0.5H2O CaSO4‚2H2O Mg3(PO4)2 MnCO3 Zn3(PO4)2 ZnSO4 ZnSO4‚7H2O
227.780 81.250 99.650 112.380 186.020 213.110 81.500 234.050 99.060 379.150
222.890 80.740 97.850 116.350 171.860 209.500 82.870 219.780 96.820 355.820
2.140 0.630 1.800 3.540 7.610 1.690 1.680 6.100 2.270 6.150
predicted. A mean error of 8.17% is calculated for 649 of the 664 salts. The predictions for the reminder of the 15 compounds are poor. The errors for these poor predictions at their maximum temperature are as follows: VO (43.53% at 1973 K), UO2F2 (49.2% at 1500 K), UO2 (50.3% at 2000 K), TiO2 (46.3% at 2130 K), TiF3 (47.4% at 1310 K), ThO2 (94.5% at 2500 K), ScF3 (55.5% at 1825 K), NpO3‚H2O (100.4% at 800 K), MoF3 (93.1% at 1237 K), MoCl3 (52.1% at 926 K), MoBr3 (60.4% at 1082 K), MgO (43.8% at 3105 K), GeO2 (68.0% at 1409
Kopp’s rule 298 Cp,predicted ,
J/g-mol K
% error
178.24 214.22 161.50 109.62 128.03 204.18 141.42 204.18 144.77 128.03 229.28 114.64 118.83 102.09 52.30 178.24 78.66 220.92 256.90 144.77 83.68 144.77 118.83 115.48 161.50 102.09 178.24 118.83 103.76 129.70 103.76 129.70 155.64 181.59 88.70 109.62 130.54 151.46 103.76 129.70 59.41 148.95 94.56
4.39 25.76 19.66 11.90 14.44 2.83 20.73 2.02 8.57 16.18 13.34 23.79 13.24 14.56 5.13 25.61 1.50 9.23 20.56 24.87 10.98 5.09 7.16 20.83 10.17 11.98 11.49 8.19 4.57 1.24 1.21 7.40 7.66 3.45 6.71 5.51 1.33 9.54 7.37 3.52 6.54 2.81 3.40
K), FeF3 (46.5% at 1132 K), and Be2SiO4 (63.5% at 1806 K). Essentially all of these higher error levels are for oxides and halides of single cations. Summary and Conclusion As mentioned earlier, Cp values predicted by the proposed method are in good agreement with the literature values at low and moderate temperatures. In Figures 1 and 2, the literature and predicted heat capacities of CaTiO3 and Fe2(SO4)3 have been plotted as a function of absolute temperatures. The temperature range used in each of the figures (Figures 1-4) is the range given by Robie et al. (1979) over which their fit of experimental heat capacity data is said to be valid. For CaTiO3, the predictions are close to experimental values up to 1200 K and then the proposed method deviates by predicting higher values. For Fe2(SO4)3, predictions are good from 300 to 600 K and then deviate. One of the poorest predictions obtained is shown in Figure 3. Here, for FeTiO3, the predictions begin to deviate at a much lower temperature. Figure 4 illustrates one of the cases found for underprediction at high temperature. Here, for Mg(OH)2, predictions and the experimental values are almost the same from 298
Ind. Eng. Chem. Res., Vol. 35, No. 1, 1996 347
Figure 1. Heat capacity (Cp, J/(g-mol K)) vs temperature (K) of CaTiO3 for literature (from Robie et al., 1979) and preducted values by the proposed method.
Figure 2. Heat capacity (Cp, J/(g-mol K)) vs temperature (K) of Fe2(SO4)3 for literature (from Robie et al., 1979) and predicted values by the proposed method.
Figure 4. Heat capacity (Cp, J/(g-mol K)) vs temperature (K) of Mg(OH)2 for literature (from Robie et al., 1979) and predicted values by the proposed method.
frequency, νE. The group contribution model outlined in this paper assumes that there is a contribution to the Einstein vibrational frequency characteristic to each anionic or cationic group in a lattice and that these contributions can be combined in a linear fashion to estimate heat capacity data. Thus, the group contribution for the Einstein term, and probably the other terms, is sensitive to lattice structure and may explain why structural deviations in the heat capacity from literature values may occur. Carbides, sulfides, and nitrides such as SiC, CdS, and GaN have a coordination number of 4, which means that they are less closely packed in the crystal lattice as compared to most of the salts having coordination numbers of 6, 8, and 12. This could be one of the reasons that the proposed method does not work for carbides, sulfides, and nitrides. This method is also not applicable for alloys such as Al2Se3, ScAs, NiSe, NiTe, and PbTe because of the formation of two or more dissimilar domains in the crystal structure, each having a different heat capacity. One can estimate the heat capacity of each one of these domains if the exact composition is known. Acknowledgment This work has been supported in part by Subcontract No. 9-XX3-1442E-1 from U.S. Department of Energy, Los Alamos National Laboratory. Nomenclature
Figure 3. Heat capacity (Cp, J/(g-mol K)) vs temperature (K) of FeTiO3 for literature (from Robie et al., 1979) and predicted values by the proposed method.
to 600 K and then the predictions deviate. In general, it can be said that the proposed method gives estimates of heat capacity within an acceptable range of accuracy at most temperatures. In the regression of heat capacity 664 solid inorganic salts have been used and their corresponding C298 p values are predicted. A mean error of 3.18% and a maximum error of 13.63% are found for these salts. The Einstein model for heat capacity (eq 9) predicts that there is a law of corresponding states for any crystal lattice based on a characteristic vibrational
b ) vector the jth element of which contains ∆i,j, the calculated estimates for the jth group contributions C298 ) heat capacity at 298 K, J/(g-mol K) p Cp ) heat capacity, J/(g-mol K) Cv ) heat capacity at constant volume, J/(g-mol K) h ) Planck’s constant k ) Boltzmann constant N ) number of atoms ni,j ) number of occurrences of group j in compound i n j j ) mean number of occurrences of group j in all salts in the regression nj ) number of groups of the jth type Vb ) liquid molar volume at normal boiling point X ) matrix the i,jth element of which contains (ni,j - n j j) y ) vector the ith element of which contains the experimental values found for Cp for compound i νE ) Einstein frequency ) heat of formation, kJ/(g-mol) ∆H298 f ) free energy of formation, kJ/(g-mol) ∆G298 f
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∆k,j ) group contribution for k of jth ionic, atomic, or ligand molecule group ∆a,j ) group contribution for a of jth ionic, atomic, or ligand molecule group, J/(g-mol K) ∆b,j ) group contribution for b of jth ionic, atomic, or ligand molecule group, J/(g-mol (K)2) ∆c,j ) group contribution for c of jth ionic, atomic, or ligand molecule group, (J K)/(g-mol) ∆d,j ) group contribution for d of jth ionic, atomic, or ligand molecule group, J/(g-mol (K)3)
Supporting Information Available: List of all the solid inorganic salts used in the regression, literature reference, corresponding literature value, prediction by the proposed method, percentage of error for the proposed method, prediction by Kopp’s rule, and percentage of error for Kopp’s rule for heat capacity at 298 K (17 pages). Ordering information is given on any current masthead page. Literature Cited ASPEN PLUS. User Guide Appendices, 2nd ed.; Aspen Technology, Inc.: Cambridge, MA, 1990. Barin, Ihsan. Thermochemical Data of Pure Substances; VCH Verlagsgesellschaft mbH: D6940 Weinheim, Germany, 1989. Benson, S. W.; Cruickshank, F. R.; Golden, D. M.; Haugen, G. R.; O’Neal, H. E.; Rodgers, A. S.; Shaw, R.; Walsh, R. Additive Rules for the Estimation of Thermochemical Properties. Chem. Rev. 1969, 69, 279-324. Dean, J. A. Lange’s Handbook of Chemistry; McGraw-Hill Book Company: New York, 1979.
Himmelblau, D. M. Process Analysis by Statistical Methods; John Wiley & Sons Inc.: New York, 1970. Joback, K. G. Thesis, Massachusetts Institute of Technology, Cambridge, Massachusetts, June, 1984. Knacke, O.; Kubaschewski, O. Thermochemical Properties of Inorganic Substances, 2nd ed.; Springer-Verlag: Berlin, Germany, 1990. McQuarrie, D. A. Statistical Mechanics; Harper & Row Publishers: New York, 1976. Mostafa, A. T. M. G. Predictions of Thermophysical Properties of Solids from Group Contributions. Ph.D. Dissertation, New Mexico State University, Las Cruces, NM, 1995. Reid, R. C.; Prausnitz, J. M.; Poling, B. E. The Properties of Gases and Liquids, 4th ed.; McGraw-Hill, Inc.: New York, 1987. Robie, R. A.; Hemingway, B. S.; Fisher, J. M. Thermodynamic Properties of Minerals and Related Substances at 298.15 K and 1 Bar (105 Pascal) Pressure and at higher Temperatures; United States Government Printing Office: Washington, DC, 1979. Thinh, T. P.; Duran, Jean-Louis; Ramalho, R. S. Estimation of Ideal Gas Heat Capacities of Hydrocarbons from Group Contribution Techniques. Ind. Eng. Chem. Process Des. Dev. 1976, 10, 576-582. Yoneda, Y. Bull. Chem. Soc. Jpn. 1979, 52, 1297.
Received for review February 28, 1995 Revised manuscript received August 22, 1995 Accepted September 25, 1995X IE9501485
X Abstract published in Advance ACS Abstracts, December 1, 1995.
