J. B. OPFELL and B.
H. SAGE
California Institute of Technology, Pasadena, Calif.
K. S. PITZER University of California, Berkeley, Calif.
Application of Benedict Equation to Theorem of Corresponding States
THERE
has been interest in reduced equations of state for the purpose of generalizing the volumetric behavior of pure substances. Van der Waals was one of the first ro recognize the utility of such reduced analytical expressions (4852). With increasing application of the thermodynamic properties of pure sub-
Table 1.
Information Used in Evaluation of Coefficients
source N ~ of
of Compound Data States Methane (28) 67
Ethane
Propane
n-Butane
n-Hexane
Pressure Range, Lb./Sq. Inch Lower Upper
Specific Volume Range, Temperature Range CU. F t . / l h . 'R.a Lower Upper Lower Upper
1051.42 (1.563)b 293.92 (0.437) 269.25 (0.400)
10010.67 (14.879)b 8817.61 (13.105) 5647.75 (8.394)
0.054142 (0.5474)b 0.042746 (0.4322) 0.099390 (1.0048)
0.339795 (3.435)b 1.28836 (13.025) 1.16890 (11.818)
559.69 (1.631)b 365.69 (1.065) 491.69 (1,433)
919.69 (2.680)b 851.69 (2.482) 761.69 (2.219)
309.83 (0.438) 163.27 (0.231)
9966.00 (14.074) 5157.56 (7.284)
0.033166 (0.4202) 0.0533128 (0.6755)
0.72348 (9.166) 1.06626 (13.510)
559.69 (1.018) 536.69 (0.976)
919.69 (1.673) 986.69 (1.795)
202.90 (0.329) 343.30 (0.556)
9945.80 (16.117) 4493.74 (7.282)
0.027811 (0.3818) 0.03632 (0.499)
0.112550 (1.545) 0.36323 (4.987)
559.69 (0.840) 665.69 (1.000)
919.69 (1.381) 986.69 (1.482)
125
(26)
118
(57)
169
(1)) (6)
178
(58)
153
(3)
111
(29)
113
98.44 (0.179)
10010.72 (18.188)
0.0261844 (0.3726)
0.~3924787 559.69 (1.3160) (0.731)
919.69 (1.202)
(6)
103
215.74 (0.392) 14.696 (0.0267)b
5268.96 (9.573) 400.08 (0.727)b
0.032449 (0.4617) 0.20613 (2.9332)b
0.551630 (7,8500) 6.,36003 (9:'.618)b
761.69 (0.995) 559.69 (0.731)b
1031.69 (1.348) 739.69 (0.967)b
14.696 (0.0300) 105.75 (0.216) 485.85 (0.993)
10000.00 0.029038 (20.437) (0.4205) 9990.40 0.024012 (20.418) (0.3477) 490.53 0.055467 (1.003) (0.8032)
9.245142 (E3.87) 0.121194 (1.7549) 0.083724 (1.2123)
559.69 (0.662) 559.69 (0.662) 844.85 (0.999)
919.69 (1.088) 919.69 (1.088) 845.75 (1.000)
96.31 (0.219) 82.30 (0.187)
9754.67 (22.195) 4582.40 (10.426)
0.022884 (0.3329) 0.025449 (0.3703)
0. 104268 (1.5171) 0.041858 (0.6090)
559.69 (0.612) 671.69 (0.735)
919.69 (1.006) 896.69 (0.981)
104.60 (0.264) 395.62 (0.997)
5165.60 (13.018) 399.00 (1.006)
0.022638 (0.3321) 0.053454 (0.7842)
0.039998 (0.5868) 0.074822 (1.0975)
545.69 (0.561) 972.02 (1.000)
941.69 (0.968) 972.38 (1,000)
66
(48)
75
(40)
275
(4)
72
(47)
99
(20)
44
n-Heptane (46) (2)
n-Nonane
,
(28)
(41)
n-Pentane
stances at eliwated pressures, there has been a revival of interest in generalizing methods for predicting their properties. The early work of Lewis (74, 23, 24) and Brown (44) is illustrative of the application of reduced variables in generalizing the properties of pure substances. Keyes (27) set forth some
46 37
(12)
299
0.179 (0.001)
10000.00 (29.860)
0.021287 (0.3082)
O.Cl30955 (0.4482)
559.69 (0.522)
919.69 (0.857)
%-Decane (43)
161
- 'F. +
0.073 (0.000)
10000.00 (32.113)
0.021015 (0.3043)
0.C129730 (0.4305)
559.69 (0.502)
919.69 (0.824)
459.69.
a
R.
Corresponding values of reduced pressure, reduced specific vcdume, and reduced temperature shown in parentheses.
of the relationships of a reduced equation of state and made available appropriate coefficients for a number of pure substances. Newton (27) summarized the behavior of pure substances in regard to their agreement with the theorem of corresponding states. Cook and Rowlinson (73) more recently carried out an extensive evaluation of the deviations of pure substances from this theorem. I t appears that the theorem of corresponding states as originally proposed by van der Waals yields uncertainties of several per cent even for gases (27) that follow his molecular model. For this reason the theorem was used with reservation in most quantitative applications. Meissner and Seferian (25) made a significant contribution to the correlation of the volumetric behavior of gases by the use of the compressibility factor at the critical state as a correlating parameter. Suggestions for application of the technique to mixtures involving simplifications of their behaviors expressed in terms of the properties of the components as described by Weber (53) were included in the discussion (25). Meissner and Seferian obtained significant improvement in the accuracy of predicting the behavior of a pure substance in terms of a universal function of the reduced pressure, temperature, and compressibility factor at the critical state. More recently Pitzer and others (33, 35, 36) considered other means of describing deviations from the theorem of corresponding states. The molecular conditions for strict conformity with this theorem have been fairly well established (77, 78, 34). Pitzer (35) defined a correlation parameter in the following way:
I n Equation 1 the primed quantities are associated with the behavior of a reference material defined by Pitzer as a VOL. 48, NO. 11
NOVEMBER 1956
2069
Methane
Table II. Ethane
Coefficients of Benedict Equation for Normal Paraffin Hydrocarbons n-Heptane n-Nonane n-Hexane Propane n-Butane n-Pentane
n-Decane
y = 0.0
0.559246 -1.613218 0.I37968 0.172102 0.219985 0.023133 -0.226843 0.0095 0.0324
0.831612 - 2.099713 0.318368 0.006218 0.580253 0.021982 -0.401203 0.0208 0.0259
0,123444 -0.770738 - 0,315539 0.501635 -0.316385 0.025059 - 0.007315 0.0229 0.0305
- 1.035290 1.206060 - 1.162693
0.301572 0.956373 - 0.420659 0.389337 -0.341071 0.030779 0.267622 0.0095 0.0146
- 0.116748
0.290710 0.771582 - 0.742404 0.725684 - 1.027600 0.049442 0.775558
0.140509 0,755950 -0.325152 0.553083 -0.410366 0,027373 0.000029 0.0349 0.0561
1.013099
- 1.1631 19
0,025369 0.338461 0.0293 0.0486
-0.516270 0.848586 - 1.223454 0.604884 -0.755194 0.026945 0.280479 0.0397 0.0620
-2.055331 2.310343 -0.777473 1.577199 - 1.895046 0.032552 0.200290 0.0535 0.0310
-0.119645 -0.092125 -0.648697 0 906909 -0,957254 0.027916 0.153279 0.0235 0.0287
-4.197646 4.865633 0 333655 2.801485 -3.689610 0 508355 -0.147241 0.0370 0 0564
-10.0756Otb 11.392926 - 0.Of 0933 5.141723 -6.434534 0.058320 0.000594 0.0832 0.1224
0.915605 -1.361521 0.414615 0.705475 -0.941378 0.042664 0.530994 0.0479 0.0353
- 3.262720
-1.745214 1.533241 - 0.286506 I.. 978974 - 2.486962 0.047532 0.473849 0.0375 0.0540
- 1.530988
I
e
y = 0.4
-
0.039594 - 1.187327 0.824959 -1.359562 0.059092 1.074925 0.0062 0.0274
-
.
I
.
0.0248
-
0.854850
- 1.363102
- 0.697514 0.668222 -1.041141 0.053057 0.842154 0.0245 0.0419
y =
0.246511 0.10632 1 1.390419 0.386825 0.229974 -1.195672 -.0.969841 - 0.741052 - 0A76932 - I, .797676 -0.184682 - 0.264500 - 0.328950 -0.304547 -0.467039 0.301339 0.423661 0.511666 0.538202 0.091932 - 0.OS2126 - 0.214187 -0.333118 - 0.389776 0.204892 0.027707 0.022238 0.021689 0.025029 0.027234 0.0104 0.0219 0.0229 0.0408 0.0331 0.0343 0.0306 0.0684 0.0256 0.0536 1.619 2 677 1.541 2.000 2.432 Defined in nomenclature and by Equation 5. Rounding error greater than magnitude of standard error of estimate. Defined in nomenclature. a
Table 111.
- 0.227474 -0,049335 -0.213968 0.930080 - 1.061676 0.032761 0.0552 0.0324 3.152
Critical Properties of the Normal Paraffin Hydrocarbons
Compound
Critical Temp., a E.
Methane" Ethane" Propanea n-Butane" n-Pentanea n-Hexane' a-Heptanea n-Nonaned %-Decaned
343.19b 549.77 665.95 765.31 845.60 914.15 972.31 1072.73 1115.03
Critical Critical AIolal Volume, Pressure, Cu. Ft./ Lb./Sq. Inch Lb. Mole 672.8 708.1 617.1 550.4 489.3 439.5 396.8 334.9 311.4
1.587 2.373 3.212 4.084 4.982 5.923O 6.830 8.857 9.908
17101.
Weight
W
16.042 30.068 44.094 58.120 72,146 86.172 100.198 128.250 142.276
0.013 0.099 0.152 0.201 0.252 0.298 0.350 0.439 0.466
See (39).
R. =
F. + 459.69.
Estimated. See (16).
Table IV.
0.000000 1
- 0.259081
K2
- 0-615277
Ka
0.054406 0.960514 - 0.786484 0.0259926 - 0.086763 0,2406
K4
K 5
RB K7 (SPIN" a
0 400000 0.005116 - 0,689777 - 0 269733 0.680066 -0.494678 0.0226202 -0.001734 0.2472 e
e
Defined in nomenclature and by Equation 5.
2070
6
0.'&65
2.578214
- 3.079385 0.056667 1.048294 0.0778 0.1325
1.234592
- 1.903184 - 0.185931 0.501643 -0.404139 0.028634 0.0259 0.0334 2 000 a
- 5.539851 6.246882 - 0.135555
-10.070938 11.388478 - 0.008955 5.140679 3.168425 - 3 995188 - 6,433958 0.058335 0.047817 0.0832 0.0394 0.1157 0.5861 4.545 4.977
simple fluid (33, 35). The vapor pressure corresponding to P, was arbitrarily taken by Pitzer at a reduced temperature of 0.7. This choice was made in the interest of simplicity because for a fluid (33) such as methane, argon, krypton, or xenon, the ratio P,'/P,' is almost exactly 0.1 a t a reduced temperature of' 0.7. Under these circumstances Equation l reduces to w =
log
(2) -
1.000
(2)
Pitzer called his correlation parameter the "acentric factor." He reports (36) average deviations from the modified theorem of corresponding states of about 0.5% over a wide range of pressures and temperatures. On the basis of Pitzer's modified theorem of corresponding states it follows that:
Reduced Composite Coefficients for Benedict Equation of State
Y
'I
-
0.616692
- 0.284445
m
-
*
-
2.365850 -0.363439 3.537565 4.395480 0.078460 1 384489
INDUSTRIAL AND ENGINEERING CHEMISTRY
m
- 0,308105 - 0.344831 - 0.248072 0.855236
- 0.704492 0.0237546
...
0.2461
In Equation 3 function Z is general, and Pitaer found for most substances, particularly hydrocarbons, that Equation 3 may be rewritten in the following simpler form :
z
=
z q r , , P,) 4- w Z ( ' ) ( T r ,P,)
(4) Equation 4 indicates that the compressibility factor, Z, is the sum of a universal function of the reduced temperature and pressure and the product of the acentric factor and another uni-
Table V.
Standard Error of Estimate in Compressibility Factor with Pressure as Dependent Variable
Tablea Methane
Ethane
Propane
n-Butane
I1 IV VI
0.0095 0.0893 0.067
0.0208 0.0570 0.049
0.0229 0.0858 0.045
0.0293 0.1395 0.073
I1 IV VI
0.0095 0.0752 0.048
0.0062 0.0437 0.037
0.0827 0.046
0.0349 0.1372 0,056
I1 IV VI
0.0104 0.0752 0.060 1.541
0.0219 0.0480 0.048 1.619
0.0229 0.0795 0.059 2.000
0.0331 0.1372 0.063 2.432
n-Decane
n-Heptane
n-Nonane
0.0235 0.2977 0,190
0.03700 0.2740 0.274
0.0832 0.4690
0.0479 0.2780 0.147
O.;b'SS 0.135
0.03748 0.2800 0.280
0.0778 0.4730 0.378
0.0552 0.2805 0 * 200 3.152
0.0259 0.3110 0.173 2.000
0.0394 0.273 0.273 4.545
0.0832 0.467 0.467 4.977
n-Hexane
n-Pentane y = 0.0 0.0397 0.2090 0.067
0.05348 0.2777 0.200
~
...
b
y = 0.4 1..
0.0245 0.2065 0.065 y =
Zmc
C
m
0.0408 0.2082 0.067 2,677
Coefficients taken from designated table. Rounding error greater than magnitude of standard error of estimate. Defined in nomenclature.
Table VI. versa1 function of the reduced temperature and pressure. Equation 4 is applicable only to compounds of low polarity which form normal liquids. Tables of the two functions have been made available (36). The present discussion describes an investigation of the usefulness of the reduced form of the Benedict equation as the basis for an analytical statement of the modified theorem of corresponding states. The Benedict equation (9) was chosen for this investigation as a result of its specific development and application to hydrocarbons (7, 8, 70). Recently a generalized equation of state was developed for gases and liquids, which might prove more suitable for describing the volumetric behavior of fluids in relation to the theorem of corresponding states (79).
Compound
Optimum Adjusting Parameters for Critical Properties for Use with Composite Coefficients y = 0.0 Standard Adjusting error of parameter estimate
Methane Ethane Propane n-Butane n-Pentane n-Hexane n-Heptane n-Nonane n-Decane
1. 06n 1.02 0.985 0.980 0.970 0.980 0.980 1.00 1.03
0.067 0.049 0.045 0.073 0.067 0.200 0.190 0.274
...
b
y = 0.4
y = m
Adusting parameter
Standard error of estimate
Adjusting paramatsr
Standard error of estimate
1.06 1.01 0.985 0.975 0.970 0.980 0.980 1.00 1.01
0.048 0.037 0.046 0.056 0.065 0.147 0.135 0.280 0.378
1.06 1.01 0.985 0.980 0.970 0.980 0.980 1 .oo 1.00
0.060 0.048 0.059 0.063 0.067 0.200 0.173 0.273 0.467
Values in this table multiplied by the reciprocal molal volume at the critical state may be used to reduce the reciprocal molal volume for use in application of the theorem of corresponding states. The adjusting parameters for temperature are all unity except for n-hexane, where it is 1.01 when y = 0.4. Rounding error greater than magnitude of standard error of estimate.
Reduced Form of Benedict Equation I n order to evaluate the influence of various modifications to the theorem of corresponding states expressed in terms of the Benedict equation, several steps were employed. In the first step coefficients for the Benedici equation were obtained from the sets of data for each of nine different normal paraffin hydrocarbons from methane through decane. No experimental information appeared to be available for n-octane and for this reason it was omitted from the study. The second step involved evaluation by least squares methods (77) of reduced coefficients upon the assumption that the theorem of corresponding states applied to all nine paraffin hydrocarbons and yielded a single set of COefficients. The third step involved a n adjustment in the reducing parameters for temperature and specific weight for each hydrocarbon, in order to improve accuracy of prediction with the reduced
Coefficients for Modified Reduced Benedict Equation of State
Table VII. Coeflcienta
y = 0.4
y = 0.0
- 0.31882075
m
0.23142374 -1.13726353 0.11730587 0.61433275 -0.57936412 0.02961377
0.08120322 -0.73581982 0.22746193 0.43752897 -0.20878237 0.01962536 -0.58504384 -0.39810914 -0.14320654 -1,17935710 2.68606163 -3.76329981 0.05946795 2.91104674 0.08563
0.57979775 -1.48997825 0.55981371 -0.75513314 0.02078748 0.53935321 -2.93095975 2.02476666 2.34162042 2.62356106 -2.59662226 0.02792576 -1.00758631 0.085 18 a b
y =
...
- 0.35025387 1.09281975 -0.64291419 1.24740049 -1.31393463 0.00425465
...
0.09085
Defined in nomenclature. Defined in nomenclature and by Equation 5. ~
~~~
VOL. 48, NO. 11
b
NOVEMBER 1956
2071
Table VHI.
Coefficients for Benedict Equation Obtained from Modified Corresponding States Theorem Ethane
Methane
Propane
n-Butane
n-Pentane
n-Hexane
n-Heptane
n-;\;onane
n- Decaize
- 1.605512
- 1.684648
y = 0.0
- 0.356923 0.606120 1.459537 0.593920 - 0.788889 0.0211505 0.526255 0.0471
-
- 0.608986 - 0.764327 0.780250 0.887562 - 1,258158 - 1. I34052 0.819546 0.958595 - 1.012199 - 1.149820
- 1.057423
- 0.907944
0.986776 -1.019313 1.087 150 - 1. 277054 0.0250322 0.0264006 0.336828 0.386200 0.0757 0.0488
0.0235521 0,439602 0,0479
- 1,192247 - 1.344657 1.090039 1.183178 1.288466 - 0.899890 - 0.792175 -- 0,670411 1.220951 1.341635 1,478060 - 1.409482 - 1.528927 - 1.663951 0.0278248 0.0291094 0.0305615 0.285442 0.239093 0.186698 0.1072 0.1776 0 0502
Y = 0.4 0.041790 0.020691 0.076028 0.001183 -0.019120 0.749997 0.757587 - 0,764604 -0.771908 - 0,737682 0.212130 0.110706 0.048200 - 0.009589 - 0.069736 0.703449 0.472448 0.845810 1.114417 0.977427 -0.581349 - 0.780804 - 0,965206 1.157134 -0.257705 0.0203984 0.0286645 0.0255127 0.0315784 0.0346113 0.148540 - 0.547200 - 0 296850 - 0.142565 0.0000766 0.0521 0,0703 0.0615 0.0781 0.1115
-
R 1
Kz K3 K4
-
a
y =
0.226870 - 1.123057 0.108948 0.630549 - 0.596445 0.0296691 0.0545 1.541 a b
Coeflcientb
AD
Bo
co x
10-6
U
b
0.178185 - 0.971155 0.019583 0.803938 -0.779082 0.0302605 0.0592 2,000
-
1.523339
- 0.398783 1.782393
- 1.965159 0.0338009 0.069818 0.1s99
-0.104316 -0.802554 -0.290276 - 0.322119 1.689234 1.616710 - 1,860871 - 1,962480 0 0457318 0.0473374 0.692906 0.771504 0.1852 0.1500 -0,093567
- 0 + 798687
m
0.161023 0.143160 0.127048 0. 108835 -0.917607 -0.861873 - 0.811603 -0.754777 -0.011920 - 0.044709 -0.074283 -0.107714 0.865060 0.928678 0.986058 1.050923 -0.910476 -0.843465 -0.970917 - 1.039241 0.0304690 0.0306859 0.031 1029 0.0308817 0,0990 0.1163 0.1650 0.0625 2.432 2.677 3.152 2.000
0.077662 -0.657516 - 0.164934 1.161942 - 1.156181 0.0314816 0.1950 4.545
0.068205 -0.628010 -0.182292 1.195621 .- 1.191658 0.0315964 0.1780 4.977
Defined in nomenclature and by Equation 5. Defined in nomenclature.
Table IX.
c
0.196749 - 1.029074 0.053657 0.737825 - 0.709444 0.0300350 0.0387 1.619
-0.037433 - 0.058135 -0,778495 0.785942 - 0.123987 -0.185313 1.237975 1.377650 - 1,330246 - 1,525937 0.0373468 0.0404391 0.282448 0.433823 0.1636 0.0577
1.468670 -0.462007 1.711557 - 1.895050 0.0330469 0.097023 0.I720
x
10-
CY
Y
Conventional Coefficients for Benedict Equation Obtained from Modified Corresponding States Theorem"
Methane
Ethane
- 3541.99
- 10924.67
Propane
n-Butane
n-Pentane
Y = 0.0 -49284.86 -5.268610 29093.287 317525.3 30.31049 45979.89 2.441822 0.000000
-20370.77 -33100.37 - 0.566330 - 1.445246 2.454636 - 3,708316 5324.401 I1543.177 1004.562 20026.355 7314.8 33634.9 84751.3 174961.0 4.61575 1.49526 18,13530 9.88669 4414.99 12624.89 574.73 27028.12 0.107100 0.310051 0.721096 1.408499 0* 000000 0.000000 0.000000 0.000000
-
n-Hexane
n-Heptane
- 68743.53 - 7.06 1083
- 91825.65
38462.647 526106.1 47.05919 68752.55 3.955140 0.000000
n-lvonane
n-Decane
-1149747.82 -180600.75 -9.184142 -14.220020 -16.691155 45169.179 58780.537 54208.115 I711428.6 2308330.3 809951.3 174.96774 134.26549 68.95195 85914.82 101962.72 100826.81 5.852 148 12.116289 16.728690 0 .oooooo 0.000000 0 .oooooo
y = 0.4
Ao Bo
co x
10-6
a b
c
x
10-6
oi
Y
10501.08 43 10.80 0.099176 0.120634 146.004 - 468.495 19317.3 2389.5 1,18944 3.96188 -597.590 -2981.31 0,316197 0.586573 2.25283 1 1.007048
17387.66 0.066449 -490.610 57551.8 8.72346 - 4660.28 1.215979 4.125492
25647.84 0.004832 188.388 132236.6 16.30496 6.15 2.229069 6.672604
6562.82 0.359975 -74.986 5530.4 I, 58748
22289.35 0.572241 -.199.328 57424.9 8.29160
30780.17 0.657666 234.187 115557.7 14.43051
0.'5'6&64
1.286519
2.461180
m
m
-
34900.93 -0.095265 2254.556 260676.9 27.66574 23927.31 3.699781 9.930124 y =
A0 Bo
co x
10-
a b C CY
Y
...
0.198709 03
14408.58 0.466925 -227.073 23573.7 4.15549
...
... c5
45231.14
6019.937 457739.7 43.42322 81219.72 5.832269 14,030400
81435.33 -0.828723 34058.590 12485.525 742771.3 1680501.5 126.82507 64.26781 199636.75 720072.73 8.443965 17.075039 31.378560 18.660120
38968.60 0.713295 1445.418 205110.2 23.05471
47154.74 0.752442 3606.658 334094.0 34 58697
53791.01 0.743354 7257.275 505865.2 49.02587
67041.33 0.687852 19351.949 1044115.7 91,15014
74454.26 0.675761 26869.813 1399754 . O 117.36758
4.168818
6.607460
9,535987
18.918645
25.788026
- 0.221697
56012.05
- 0 397068 e
95147.38
- 1,033542 47480.192 2305183.4 165.82278 1126710.00 23.460157 39.265800
m
... c3
I
..
03
... m
... m
. . a
m
a These values may be used in the Benedict equation when pressure is expressed in pounds per square inch, temperature in R., andvolume in cubic feet per pound mole. b Defined in nomenclature.
coefficients obtained in the second step. I n the fourth step of the problem, values of the acentric factor proposed by Pitzer (33) were employed and the
2072
optimum values, in the least squares sense, of the 14 coefficients associated with the two functions shown in Equation 4 were evaluated. This set of cal-
INDUSTRIALAND ENGINEERING CHEMISTRY
culations made it possible to compare the effectiveness of the Benedict equation as applied to individual compounds and as used in several different ways in connec-
Table X.
Comparison of Root Mean Square Deviations in Compressibility Factor with Pressure as Dependent Variable for Several Different Modifications of the Theorem of Corresponding States
Tablea
Methane
Ethane
Propane
11 IV VI VI1
0.0324 0.156 0.183 0.0471
0.02592 0.0836 1 0.08243 0.04793
0.03054 0.160 (0.080) 0.0488
0.04858 0.2848 0.1400 0.0757
I1
0.0146 0.118 0.185 0.0522
0.02 737 0.07051 0.06695 0.0703
0,0248 0.158 0.0460 0.0615
0.05461 0.3274 0.10056 0.0781
n-Butane
n-Pentane
n-Hexane
n-Heptane
n-Nonane
n-Decane
0.0620 0.5197 0. I310 0.1072
0.0310 0.5086 0.2115 0.1776
0.0287 0.469 0.162 0.0502
0.0564 2.772 2.772 0.172
0.1224 0.482 1.111 0.160
= 0.4 0.0419 0.5185 0.1357 0.115
0.0353 0.5106 0.2083 0.1636
0.0365 0.486 0.168 0.0577
0.540 1.656 1.656 0.185
0.1325 0,469 0.574 0.150
0.0324 0.5187 0.2173 0.1650
0.0334 0.493 0.174 0.0625
0.0586 0.363 0.363 0.195
0.1159 0.458 0.458 0.178
y = 0.0
y
IV VI VI1
y = m
I1 IV VI VI1
0.0343 0.02555 0.03065 0.118 0.06743 0.153 0.180 0.06104 (0.078) 0.0545 0.0387 0.0592 Coefficients taken from designated table,
a
0.05365 0.3279 0. 1392 0.0990
0.0684 0.5237 0.1433 0.1163
Table XI. Comparison of Predictions Using Reduced Benedict Coefficients with Those of Tabular Correlation (36) Involving Acentric Factor Molal Volume, Cu. Ft.1 Lb. Mole
Exptl. Pressure, Lb./Sq. Inch
z, - z
z
z, - Z n
ZD
- Zb
METHANE
T = 365.69' R. = -94' 4.695654 1.347255 0.906437 0.771043 0.706327 0.668566 0.632392
587.84 1028.72 2057.44 4408.80 7348.01 10287.21 14696.01
9.131745 4.284176 2.307642 1.201321 0.935842 0.796982 0.748311
587.84 1175.68 2057.44 4408.80 7348.01 11756.81 14696.01
0.7034 0.3532 0.4752 0.8662 1.3235 1.7526 2.3682
F.
-0.005 -0.003 0.002 -0.001 a
... ...
e..
0.023 12 0.0881 0,0018 -0.0072 - 0.0049 -0.0012 0.0117
0.00 0.00 -0.01
-0.04
... ... ...
T = 536.69' R. = 77' F. 0.9320 0.8745 0.8244 0.9196 1.1380 1.6269 1.9094
0.000 -0.003 -0.000 0,004
... ...
...
0.9970 1.022 1.067 1.1981 1.4368 1.6046
440.88 2939.20 4408.80 7348.01 11756.81 14696.01
0.000 0.000 0.005
... ... ...
-0.0466
-0.0401 -0,0212 0.0008 0.0147
0.003 0.000 0.006 0.024
...
... ...
0.0111 0.0458 0.0458 0.0397 0.0309 0.306
0.003 0.010 0.013
... ... ...
n-PENTANE
T 1.866273 1.864108 1.851916 1.806897 1.776667 1.747304 1.726814
961.80 1522.70 1992.10 4064.90 6053.00 8139.30 9965.40
2.276206 2.225488 2.183860 2.071528 1.991157 1.944551 1.903933
1002.40 1502.30 2005.00 4043.00 6129.90 8004.70 9987.50
= 559.69O
0.29884 0.47258 0.61422 1.22285 1.79048 2.36782 2.86505
R. = 100' F.
... ... ...
... ... ...
...
- 0.0008 -0.0013 -0.0029 - 0.0046 -0.0023 - 0 0003 0.0044
... ... ... ... ... ... ...
0.0032 0.0025 0.0024 0.0054 0.0073 0.0149 0.0249
... ... ... ... ... ... ...
6
T = 739.69' R. = 280'F.
a
0.2874 0.4212 0.5516 1.0551 1.2206 1.5566 1.9016
0.001 0.002 0 * 000
0.009
...
... ...
Individual Compounds
By following methods which have been described (7 7) and by employing the experimental backgrounds used in earlier studies for methane, ethane, nbutane, and n-pentane (32) and for nhexane, n-heptane, n-nonane, and ndecane (37), as well as for propane (45), the coefficients were evaluated for each compound. These calculations were carried out for values of the ratio
- 0,0169 -0.0295
T = 851.69' R. = 392' F. 20.667861 3.180647 2.212833 1.490238 1.117004 0.997925
tion with the theorem of corresponding states.
Molal volume is dependent variable, coe5cients for y = 0 are from Table VI. Subscript D means 2 obtained from Figure V-2, p. 161 (15). Pitzer's table (36)applies only to reduced pressures less than 9.
(5)
of 0, 0.4, and infinity. This ratio is indicated in the tables and elsewhere in the text by y. The experimental background employed for these calculations is summarized in Table I. References to the sources of the experimental data are included in this table. The results of these calculations are summarized in Table I1 for the nine normal paraffin hydrocarbons involved. I n each case the standard error of estimate with pressure as the dependent variable was included for each of the points and for a selected sample of 20 points. The standard error of estimate is a measure of the degree to which two functions are equivalent and is evaluated from all the experimental points in the range of the independent variable used in evaluating the coefficients. I t was described for present purposes as
I n Equation 5, i corresponds to each of the n experimental points. The standard error of estimate based upon the sample of 20 states may be called the root-mean-square deviation. Occasionally, the sample of 20 states was not particularly representative of the entire data background, with the result VOL. 48, NO. 11
0
NOVEMBER 1956
2073
~~
~
Table XI. Comparison of Predictions Using Reduced Benedict Coefficients with Those of Tabular Correlation (36) Involving Acentric Factor (Confd.) Molal Volume, Cu. Ft./ Lb. Mole
Lb./Sq. Inch
3.374845 3,077099 2,831226 2.435793 2.232919 2.163803 2.095192
1190.30 1489.80 1986.00 4170.00 6130.50 8131.80 9883.70
Exptl. Pressure,
Z T
Z p
z,- Z"
-z
= 919.69' R. = 460' F. 0.40701 0.009 0.4645 0.002 0.5623 0.005 -0.011 1.0291 1.3870 1.7828 2.0708
ZD
0.0208 0.0068 0.0028 0.0080 -0.0130 0.0141 0.0192
... ...
...
- Z*
-0.012 -0.004 0.008 -0.007
... ... ...
DECANE
T = 559.69' R. 0.07
3.17 3.15 3.11 3.04 3.02 2.99
2000.00 6000.00 8000.00 10000.00
3.58 3.54 3.50 3.45 3.31 3.25 3.20
2.08 400.00 1000.00 2000.00 6000.00 8000.00 10000.00
400.00
=
0.000004 0.2100 1.0362 3.0387 4.0156 4.9796
100' F.
... ... ...
0.0000
... ... e . .
- 0.0021 - 0.0055 - 0.0024
... ...
... ... ...
0.0142 0.0201
. . a
T = 739.69' R. = 280' F.
... ... ... ... ...
0.0023 0.1783 0.4404 0.8682 2.4995 3.2763 4.0324
... ... ... ...
0.0002
- 0.0009
9 . .
a
64.72 400.00 1000.00
2000.00 6000.00 8000.00 10000.00
0.0277 0.1680 0.4091 0.7911 2 2006 2.8620 3.5086
0.00033 0.00022 - 0.00078 - 0.00049 - 0.00231 0.01031 0.00872
0.001 -0.007 -0.006 -0.001
... ... I..
I
Reduced Equation of State
The Benedict equation of state (9) may be written in terms of reduced properties in the following form :
Table
(8)
(3)
a = -
Y =
e . .
...
(11)
RTc3Tc'K,
... ... . . I
(10)
b == Ye=Ka
. a .
XII.
VcKz
...
(12)
Ks
Yc3 K6
(13) (14)
Yc2r
Coefficients for each compound may be obtained from the data of Tables H I and IV and the foregoing relations. The standard error of estimate in compressibility factor encountered with each
Predicted Critical State Properties Volume,
Temp., Compound
Y
Methane
0.0 0.4
Ethane
0.0
m
0.4 m
Propane
0.0 0.4 m
n-Butane
0.0 0.4
%-Pentane
0.0
m
0.4 m
n-Hexane
0.0
0.4 63
n-Heptane
0.0
0.4 m
n-Nonane
0.0 0.4 m
' R. 2094
a =
(7)
- RTc3ycK3 - RTcVczK~
BO
c
Molal volume is dependent variable, coefficients for y = 0 are from Table VI. Subscript D means 2 obtained from Figure V-2, p. 161 ( 1 5 ) .
that the root-mean-square error is large in comparison with the standard error of estimate. These coefficients represent nothing that is particularly new. The matrices of the coeffjcients of the normal equations used (77) in evaluating those of Table I1 form the basis for subsequent calculations associated with the theorem o i corresponding states. The coefficientsgiven for n-hexane, n-heptane, n-nonane, and n-decane have been published (37). These coefficients do not necessarily yield the minimum value of the standard error of estimate, since only three values of the exponential constant, y, have been considered. The coefficients of Table I1 correspond to the least squares values for the three values of y .
- RTYV,Kz
=
Co =
... ...
T = 919.69O R. = 460' F, 4.23 4.15 4.04 3.90 3.62 3.53 3.46
Ao
. . a
-0.0015 -0.0047 - 0.0027 -0.0078 -0.0102
.*.
For the sake of completeness there are recorded in Table 111 the critical temperature, pressure, and molal volume for each of the normal paraffin hydrocarbons involved in this study. I n most cases these values were based upon a critical survey by Rossini (39). By employing the same matrices used in the evaluation of the data of Table 11, a composite set of coefficients for the reduced form of the Benedict equation shown in Equation 6 was obtained and is recorded in Table IV. These reduced coefficients were obtained for each of the same three values of employed in the calculations associated with Table 11. The reduced coefficients of Equation 6 ma)' be used to evaluate the individual coefficients in the usual form of the Benedict equation. The reIationships between the reduced coefficients and the conventional coefficients (7-10) are given in the following equations:
INDUSTRIAL AND ENGINEERING CHEMISTRY
=
O
F.
+ 459.69.
Pressuie, Lb./Sq. Inch
Cu. Ft./ Lh. X o l e
Compressihiltty Factor
363.38 358.08 363 e 70 581.50 549.48 582.26 707.32 679.36 707.29 812.77 819.40 819.73 886.32 854.89 884.87 974. IO 926.39 960.63 1019.51
522.61 793.23 857.78 925.89 663.49 922.37 840.42 651.05 840.23 756.03 785.63 782.23 734.16 528.06 753.46 821.77 575.44 744.63 572.65
1.628 1.696 1.668 2.474 2 792 2.482 3.354 3.655 3.354 4.281 4.218 4.229 4.983 5.620 4.915 5.435 5.894 6.115 6.865
0.21818 0.35010 0.36657 0.36706 0.31415 0.36638 0.37135 0.32640 0.37128 0.37107 0.37685 0.37605 0.38462 0.32348 0.38998 0.42726 0.34116 0.44169 0.35932
1014.26 1207.35 1108.25 1167.97
618.37 1245.02 678.1 887.68
6.832 6.856 7.333 8.363
0.38814 0.65880 0.41793 0.59442
R."
...
. . e
...
...
compound is set forth in a part of Table V. For the most part the agreement is not nearly so good as that obtained with the individual coefficients recorded in Table 11. Such behavior is to be expected, as it is well known (27) that the normal paraffin hydrocarbons deviate significantly and in a rather systematic manner from the theorem of corresponding states.
Adjusted Critical Parameters By following in some respects the methods of Meissner and Seferian (25), the critical volume and critical temperature were adjusted by an iterative least squares procedure so as to yield the minimum standard error of estimate for each compound based upon the reduced composite coefficients of Table IV. Values of the adjusted reducing parameters for temperature and molal volume are recorded for each of the nine compounds in Table VI. In addition, the standard errors of estimate obtained for each compound with the adjusted reducing temperature and volume parameters are included. A comparison with similar information in Table V for the theorem of corresponding states indicates a distinct improvement in the accuracy of description of the volumetric behavior of these compounds by the use of adjusted reducing parameters which differ from the critical temperature and volume. Such an improvement is not unexpected, since in effect it adds two more coefficients to the functional relations describing the volumetric behavior of each pure substance. Adjustment of the two reducing parameters did not bring the results into good agreement, as the standard error of estimate was still several times that obtained with the coefficients presented in Table 11.
Acentric Factors The semitheoretical work now available concerning the use of an acentric factor (36) as a correlating device to obtain a modified theorem of corresponding states affords another approach. If the Benedict equation as described in Equation 6 is employed to approximate the two functions shown in Equation 4, there are a total of 14 universal coefficients to describe the volumetric behavior of pure substances. I n addition, a knowIedge of the reduced state and the acentric factor, w , is required. By utilizing the background of experimental information for the nine compounds as described, it was poi+ sible to evaluate the 14 generalized coefficients. Table I11 records values of the acentric factor for each of the nine compounds. In the evaluation of the 14 coefficients it was convenient to express the coefficients in the following form:
K I = Kpi
+ wKpe
(15)
K 2
=
Kp2
+
WKpS
(16)
The last analogous equation of this group is of the form,
K, =
Kp7
WKpl4
(17)
For the purposes of evaluating the coefficients, which are interrelated by seven expressions analogous to Equations 15, 16, and 17, it is convenient to rewrite the Benedict equation in the following expanded form:
Values of these 14 coefficients as established from Equation 18 are set forth in Table VII. In addition there are presented in Table VIII, coefficients associated with the acentric factor established from the set of relations typified by Equations 15 through 17 for the nine hydrocarbons. As a matter of interest the values of the specific coefficients for each of the paraffin hydrocarbons were computed by application of the data of Table VII. The values of these specific coefficients recorded in Table IX may be compared with the corresponding coefficients recorded in Table 11. A summary of the root-mean-square deviations in compressibility factor for the several different methods of modifying the theorem of corresponding states constitutes Table X. This table shows a progressive improvement in the accuracy of representation in passing from the simple theorem through modification in the reducing parameters for the temperature and volume to the use of the acentric factor just described. The latter correlation is comparable in its accuracy of prediction to that obtained with the coefficients set forth in Table 11. I n the case of methane, the root-mean-square deviation for the modified reducing parameters (Table VI) is greater than for the composite coefficients (Table IV). This situation reflects the effect of sample size in the computation of the root-mean-square deviation. The compressibility factor predicted from the reduced Benedict coefficients was compared with values interpolated in the tabular correlation (36)with pressure as the dependent variable. Results of this comparison are shown in Table XI. The tabular correlation yields somewhat smaller deviations when compared with experiment than was realized with the equation of state approach. Such behavior is not unexpected, as the tabular presentation of the first function of Equation 4 can be made
to describe the actual behavior with greater accuracy than the analytical representation given by the Benedict equation of state. I n addition, scme values of the deviation obtained with the graphical correlation of Dodge (75) were included in this table. A detailed record of pressures predicted by each of the four methods investigated here is available (30) for each of the nine hydrocarbons included in this study. The extent to which the acentric factor (36) can be applied to generalizations of the behavior of pure substances by the use of equations of state has not been explored fully. The present investigation indicates that the acentric factor modification of the theorem of corresponding states yields results of only slightly greater uncertainty than are obtained by the use of specific coefficients for the Benedict equation which apply to each compound individually. Significant differences between the accuracy of description of the volumetric behavior of a pure substance by means of the Benedict equation of state and by tabular methods are a n indication of the need for more precise analytical representation of the influence of volume and temperature upon the pressure of a pure substance when a t thermodynamic equilibrium.
Prediction of Critical States It was considered of interest to explore the operations inverse to those employed in the first portion of this discussion as a part of a general evaluation of the effectiveness of the Benedict equation to describe the behavior of pure substances. As an approximation of the thermodynamic criteria used for the identification of the critical state it was assumed that the following equations described the necessary and sufficient conditions :
Zimm (54) indicated that the higher derivatives of pressure with respect to volume a t constant temperature are also zero. Imposition of these additional restraints did not appear worth while, considering how poorly the equation describes the details of the v o h metric behavior in the vicinity of the critical state. By utilizing Equations 19 and 20 as auxiliary restraints, the predicted critical temperature and molal volume were calculated for each of the nine paraffin hydrocarbons using the individual coefficients listed in Table 11. The results of these calculations are set forth in Table X I I . I t was not possible to obtain satisfactory solutions of the VOL. 48, NO. 11
NOVEMBER 1956
2075
Benedict equation for n-decane, which accounts for its omission from Table XII. Results for n-heptane when y was 0.4 were omitted for the same reason. The agreement is not particularly satisfactory and indicates a limitation of the Benedict equation as a means of predicting the critical state of pure substances. Acknowledgment
This paper is a contribution from American Petroleum Institute Research Project 37 located a t the California Institute of Technology and Project 50 at the University of California. C. J. Pings, Jr., and W. G. Schlinger contributed to the methods of analysis and June Gray assisted with the calculations. W. N. Lacey reviewed the manuscript and Elizabeth McLaughlin assisted in its preparation.
2,
compressibility factor predicted by equation of state Z, = maximum compressibility factor ZP = compressibility factor obtained from tabular correlation (36) using acentric factor Z(O),Z(1) = Pitzer’s (36) universal functions of reduced temperature and reduced pressure Z ( T,, P7,u) = function of reduced temperature, reduced pressure, and acentric factor
5
=
= summation operator
i=l
?r
w
a
reduced reciprocal molal volume = acentric factor = partial differential operator =
Aa, BO,CO,a, b , c, a , y = coefficients of Benedict equation of state exp( ) = exponential function t = individual points used in analysis Ki-K,,y = constants computed from coefficients of Benedict equation of state, dimensionless &I, Kp2,Kp7,Kp8,Kpg,I$M= coefficients of functions Z(0)and Z(1) log () = common logarithm AT = total number of states P = observed pressure, lb. per sq. inch pc = critical pressure, lb. per sq. inch p,’ = critical pressure of simple fluid, lb. per sq. inch pe = pressure predicted by equation of state, Ib. per sq. inch P, = reduced pressure PS = vapor pressure a t T, = 0.7, Ib. per sq. inch p,’ = vapor pressure of simple fluid a t T, = 0.7, Ib. per sq, inch R = uniGersa1 gas constant; 10.73147 (Ib./sq. inch) (cu.ft.9 Ob. mole) io R.) = standard ‘error of estimate SI5 of predicted compressibility factors from experimental values with pressure as dependent variable T = absolute temperature, O R. = critical temperature, R. TC = reduced temperature T, V = specific volume, cu. ft. per Ib. = molal volume, cu. ft. per mole = critical molal volume, cu. ft. per lb. mole Z = compressibility factor = compressibility factor obZD tained from graphical correlation (75) I
v
2076
-
I
Nehton,‘ R. H., IND.ENG. CHEM. 27,302 (1935).
Olds, R. H., Reamer, H. H., Sage, B. H., Lacey, W. N., Ibid., 35, 922 (1943). Ibid., 36,282 (1944). Opfell, J. B., Pitzer, K. S., Sage, B. H., Am. Doc. Inst., Doc. 4985 (19561, Opfell, ’J. B., Sage, B. H., Chem. Eng. Data Series 1, 62. (1956). Opfell, J, B., Schlinger, W. G., Sage, B. H., IND. EXG. CHEM. 46,1286 (1954). Pitzer, K. S., J. Am. Chem. SOC. 77, 3427 (1955). Pitzer, K. S., J . Chem. Phjls. 7, 583 (1939). Pitzer, K. S., Lippmann, D. Z . ,
“Extended Theory o f Corresponding States,” Report of R.esearch Project 50, American Petroleum Institute, Univ. of California, Berke-
lev. 1954. ~-~ . (36) Pitzer, K. S., Lippmann, D. Z . , Curl, R. F.:Jr., Huggins, C. M., ~
References Nomenclature
(26) Michels, A., Nederbragt, G. W., Physica 2, 1000 (1935); 3, 569 (1936’1.
(1) Beattie, J. A , , Hadlock, C., Poffenberger, N., J. Chem. Phys. 3, 93 (1935). ( 2 ) Beattie, J. A., Kay, W. C., J , Am. Chem. SOC.59, 2586 (1937). (3) Beattie. J. A , Kay, W.C., Kaminsky, J., Zbid., 59, 1589 (1937). ( 4 ) Beattie, J.-A.,’Levink,S. IV., Douslin, D. R., Zbid., 73, 4431 (1951). \ - - - - , -
( 5 ) Beattie, J. A., Simard, G. L., Su, G.-J., Zbid., 61, 26 (1939). ( 6 ) Zbid., p. 924. (7) Benedict, M., Webb, G. B., Rubin, L. C.? Chem. Eng. Progr. 47, 419 (1951). ( 8 ) Zbid., p. 449. ( 9 ) Benedict, M., Webb, G. B., Rubin, L. C., J . Chem. Phys. 8, 334 (1940). (10) . , Benedict. M., ”ebb. G. B.. Rubin.
L. C.,’Friend, L., &hem. Eng. Progr:
47, 609 (1951). (11) Brough, H. W., Schlinger, W. G., Sage, B. H., IND.ENG.CHEM.43, 2442 (1951). (12) Carmichael, L. T., Sage, B. H., Lacey, W. N., Zbid., 45, 2697 (1953). (13) Cook, D., Rowlinson, J. S., Proc. Roy. SOC. LondonA219.405 (1953). (14) Cope, J. Q., Lewis, Mi. K,,‘Weber, H, C. IND.ENG. CHEM.23, 887 (1931). (15) Dodge, B. F., “Chemical Engineering
Thermodynamics,” McGraw-Hill, New York, 1944. (16) Doss, M. P., compiler, “Physical Constants of the Hydrocarbons,” Texas Co., New York, 1939. (17) Guggenheim, E. A., J . Chem. Phys. 13,253 (1945). (18) Guggenheim, E. A., Reus. Pure and Afipl. Chem., Australia 3, 1 (1953). (19) Hirschfelder, J. O., Buehler, R. J.,
McGee, H. A . , Jr., Sutton, J. R., “Generalized Equation of State for Both Gases and Liquids,” Naval Research Laboratory, University of Washington, Tech. Rept. Wis0 0 R - 1 2 (March 22, 1956). (20) Kelso, E. A., Felsing, W. A , , J . Am. Chem. SOC. 62,3132 (1940). (21) Keyes, F. G., Zbid., 60, 1761 (1938). (22) Kvalnes, H. M., Gaddy, V. L.,
Ibid., 53, 394 (1931). (23) Lewis, W. K., Luke, C . D., Trans. A m . SOC.Mech. Engrs. 54 [PME 541, 55 (1932). (24) Lewis, W. K., Luke, C. D., IND. ENG.CHEW25,725 (1933). (25) Meissner, H. P., Seferian, R., Chem. Eng. Progr. 47, 579 (1951).
INDUSTRIAL AND ENGINEERING CHEMISTRY
/
1
Petersen, D. E., J . A m . Chem. SOC.
77, 3433 (1955).
(37) Reamer, H. H., Olds, R . H., Sage, B. H., Lacey, W. N., IND. ENG. CHEM.36,956 (1944). (38) Reamer, H. H., Sage, B. H., Lacey, W. N., Ibid., 41, 482 (1949). (39) Rossini, F. D., others, “Selected Values of Physical and Thermodynamic Properties of Hydro-
carbons and R.elated Compounds,” Carnegie Press, Pittsburgh, 1953. (40) Sage, B. H., Lacey, W. N., IND. ENG. CHEW34,730 (1942). (41) Sage, B. H., Lacey, W. N., “Thermodynamic Properties of the Lighter Paraffin Hydrocarbons and Nitrogen,” p. 47, American Petroleum Institute, New York, 1950. (42) Zbid., p. 71. (43) Zbid., p. 83. (44) Selheimer, C. W., Souders, M., Jr., Smith, R. L.? Brown, G. G., I&. ENG.CHEM.24, 515 (1932). (45) Selleck, F. T., Opfell, 5. B., Sage, B. H., Ibid., 45, 1350 (1953). (46) Smith, L. B., Beattie, J. A , , Kay, W. C., J . Am. Chem. SOC.59, 1387 (1937). (47) Stewart, D. E., Sage, B. H., Lacey, W. N., IKD. ENG. CHEM. 46, 2529 (1954). (48) Waals, J. D. van der, Arch. Neerland sci. ( 2 ) 4, 231 (1901). (49) Waals, J. D. van der, dissertation, Leiden, 1873. (50) Waals, J. D. van der, Proc. Acad. Sci. Amsterdam 3, 515, 571, 643 (1901). (51) Waals, J. D. van der, Verslug. Akad. Wetenschappen Amsterdam 9, 586, 614,701 (1901). (52) Waals, J. D. van der, 2.physik. Chem. 38,257 (1901). (53) Weber, H. C., (‘Thermodynamics for
Chemical Engineers,” Chap. VIII, Wiley, New York, 1939. (54) Zimm, B. H., J . Chem. Phys. 19, 1019 (1 951).
RECEIVED for review April 2, 1956 ACCEPTED May 24, 1956 Material supplementary to this article has been deposited as Document No. 4985 with the AD1 Auxiliary Publications Project Photoduplication Service, Library of Congress, Washington 25, D . C . A copy may be secured by citing the document number and by remitting $3.75 for photoprints or $2 .00 for 35-mm. microfilm. Advance payment is required. Make checks or money orders payable to Chief, Photoduplication Service, Library of Congress ~
