I
DONALD F. OTHMER and DAVID ZUDKEVITCH Polytechnic Institute, Brooklyn 1 , N. Y.
Correlating latent Heats and Entropies of Vaporization with Temperature N e w equations and nomograms come from the Clapeyron relation of latent heat and vapor pressure. Reference substance plots and thermodynamic corrections for nonideal gas behavior and liquid compressibility give accurate values immediately over wide temperature and pressure ranges LATENT HEATS of vaporization or liquids are essential in calculations, derivations, and other theoretical functions, also in engineering design problems in heat transfer, evaporation, distillation, and adsorption. They have been calorimetrically measured (7) for very few liquids, usually only a t the normal boiling point. Similarly, entropies of vaporization could be utilized if values were known. They also are not available; and the general practice is to evaluate them from correlated heats of vaporization. Engineering processes are important at pressures other than atmospheric. Hence, latent heat values at temperatures other than the normal boiling point are necessary. Latent heats always decrease with rising temperature and disappear a t the critical point. Vapor pressure data are relatively easily and precisely determined and are usually used in evaluating latent heats. They are interrelated by the Clapeyron equation :
-dP= -
dT
L TAV
The Clausius modification, corrected for nonideality of the gas, and the compressibility of the liquid, is: = Va
-
RT VL = P ( Z Q - Z L ) (2)
Reprints of this article, with the nomograms on stiff paper, are available from the Special Publications Dept., American Chemical Society, 1155 Sixteenth Street, N.W., Washington 6, D. C. Single copies, $1.00; in lots of 10 or more, $0.50. Copies of nomogram and tables of previous article (11) may also be secured at the same rate.
m
Combining these two equations : dT L dlnP== R T 2 ( Z Q- ZL)
(3)
A corresponding equation can be written a t the same temperature for another substance used as a reference. dlnP‘=-
dT L‘ R T 2 (2;- 2;)
(4)
Equation 3 divided by Equation 4 eliminates the T function:
Integration of Equation 5 results in: log P =
L L’
2; - Z’ & l o gP’ Za - Z L ~
+c
(6)
A plot (8)on standard logarithmic paper has the vapor pressure of the reference substance on the X axis. I t is calibrated with even values of temperature corresponding tovapor pressures from standard tables; ordinates are erected a t these temperature calibrations. Pressure values corresponding to the Y axis are plotted as points on these new, temperature ordinates. These points fall on a straight line, the slope of which, m, is constant and the equation of which is: log P = m log P’
+C
(7)
Previously, m values were noted for some 600 organic compounds (70). For every liquid m is constant through a wide range of pressures, u p to nearly the critical region of the liquid or the reference substance. However, from Equation 6 : L Z& -
m=--
z; = a constant
L’ Za - Z L
=
L / L ’ or L = mL‘
This relation holds exactly for only a limited range. Latent heats calculated therefrom deviate increasingly from experimental values a t higher pressures. Equations 6 and 8 correct both for nonideal gas behavior and compressibility of the liquid, and have demonstrated the advantage for using a reference substance. By substituting known values for the reference substance in Equation 8, accurate values of the molal heat can thus be obtained. Hougen and others (5, 7) have given a table of values of the difference of gas and liquid compressibility factors. An equation has now been developed to express these data more conveniently than the table. I t holds for any substance in the commonly used pressure ranges of PR from 0.01 to
0.5 20 - Z L
1.003 - 0.66 pRo”‘
(9)
When the quantity 1.003 - (2, - 2,) is plotted us. P R on logarithmic paper, a straight line is obtained, which is exact and convenient in use. The nomograms (Figures 1 and 2) allow direct reading of Z a - Z L from values of PR. Equation 8 may be rearranged for engineering use in pounds or grams. If water, molecular weight M’ = 18, is used as a reference substance:
I and I‘ are the latent heats per unit weight a t the same temperature. Thus,
(8)
At low pressures, the difference of gas and liquid compressibility factors approaches unity with decreasing pressure. The ratio of this difference to that for the reference substance a t the same temperature approaches unity even more closely. I n this range:
m’l‘
1.003 - 0.66 1.003 - 0.66
PAO.74
(11)
Equation 11 calculates latent heat as the product of three functions: The m‘ Values. Table I lists these values for more than 500 substances obtained (70) directly from m and the respective molecular weights. VOL. 51, NO. 6
JUNE 1959
791
Table I. Values of m’ for Some Organic Compounds Name
nzI = (18,kf/m)
Hydrocarbons Isoprene +Pentane Benzene Cyclohexane n-Hexane Toluene 2-Heptene Methylcyclohexane n-Heptane Ethylbenzene o-Xylene m-Xylene p-Xylene Ethylcyclohexane 2-Methyl-2-heptene n-Octane or-Methylstyrene 0-Methylstyrene 4-M ethylstyrene Cumene Nonane Naphthalene Tetralin Butylbenzene Isobutylbenzene p-Cymene &Limonene Dipentene Myrcene a-Pinene @-Pinene Terpenolene &-Decalin trans-Decalin Decane 4-Isopropylstyrene 3-Ethyl cumene 4-Ethyl cumene 1,2-Diisopropylbenzene Triisobutylene Dodecane Heptylbenzene Tetradecane Pentadecane Hexadecane Halogenated Hydr.ocarbons Tribromomethane Chloroform Dibromomethane Methylene chloride Carbon tetrachloride Tetrachloroethylene Trichloroethylene 1,l-Ethylidene chloride
l,I,l,Z-Tetrabromoethane 1,1,2,2-Tetrabromoethane 1,1,1,2-Tetrachloroethane 1,1,2-Trichloroethane 1-Bromo-2-chloroethane 1,2-Dibromoethane 1,2 -Dichloroethane Ethyl bromide Ethyl iodide 2,3-Dibromopropene Methyl dichloroacetate 1,2,3-Tribromopropane Epichlorohydrin 1,2,3-Trichloropropane l,l,l-Trichloropropane 1,2-Dibromopropane 1,3-Dibromopropane Propylene dichloride I-Bromopropane 2-Bromopropane 1-Chloropropane 2-Chloropropane 1-Iodopropane 2-Iodopropane 1,2,2,-Tribromobutane 1,2,3,-Trichlorobutane 1,2-Dibromobutane 1,4-Dibromobutane
792
0.1630 0.1565 0.1771 0.1608 0. I516 0.1669 0.1489 0.1465 0.1479 0.1591 0.1627 0.1611 0. I601 0.1444 0.1496 0.1445 0.1675 0.1536 0.1643 0.1484 0.1377 0.1706 0.1589 0.1499 0.1280 0.1487 0.1414 0.1420 0.1419 0.1303 0.1352 0.1626 0.1421 0.1296 0.1382 0.1459 0.1384 0.1402 0.1339 0.1133 0.1344 0.1459 0.1271 0.1215 0.1199 0.0627 0.1085 0.0862 0.1415 0.0882 0.0967 0.1068 0.1293 0.0789 0.0703 0.1046 0.1184 0.1076 0.0880 0.1474 0.1059 0.0899 0.0875 0.1340 0.0795 0.1841 0.1301 0.1035 0.0925 0.0925 0.1271 0.1066 0.1016 0.1545 0.1384 0.0855 0.0817 0.0731 0.1042 0.0853 0.0974
Halogenated Hydrocarbons 1,2-Dichlorobutane 2,3-Dichlorobutane Dichloroethyl ether 1-Bromobutane n-Butyl chloride tert-Butyl chloride 1-Bromo-3-methylbutane Pentachlorobenzene 1,2,3,4-Tetrachlorobenzene
Aldehydes, Ketones, and Ethiers
1,2,4-Trichlorobenzene 1,4-Bromochlorobenzene 1,4-Dibromobenzene 1,4-Dichlorobenzene o-Dichlorobenzene Bromobenzene Chlorobenzene Fluorobenzene Iodobenzene Dichloroisopropyl ether 2-Bromotoluene 3-Bromotoluene 4-Bromotoluene 3-Chlorotoluene o-Chlorotoluene p-Chlorotoluene 3-Fluorotoluene 4-Fluorotoluene 2-Iodotoluene I-Chloro-3-ethylbenzene 2-Bromoethylcyclohexane I-Iodo-octane Iodononane I-Bromonaphthalene I-Chloronaphthalene
0.1190 0.1213 0.1446 0.1094 0.1468 0.1262 0.1054 0.1162 0.1052 0.1175 0.1164 0.1104 0.0936 0.1335 0.1357 0.1138 0.1485 0.1472 0.0961 0.1264 0.1120 0.1201 0.1152 0.1439 0.1452 0.1439 0.1453 0.1453 0.0961 0.1397 0.1082 0.0902 0.1103 0.1205 0.1479
Alcohols and Polyols Methanol 2-Chloroethanol Ethyl alcohol Ethylene glycol 2,3-Dibromo-l-propanol Isopropyl alcohol n-Propyl alcohol Propylene glycol I-Bromo-2-butanol Isobutyl alcohol tert-Butyl alcohol %-Butyl alcohol sec-Butyl alcohol 1,3-Butanediol Furfuryl alcohol n-Amyl alcohol 2-Pentanol tert-Amyl alcohol Isoamyl alcohol 3-Pentanol Cyclohexanol 2-Hexyl alcohol n-Hexyl alcohol 2-Methyl-1-pentanol 2-Methyl-4-pentanol Dipropylene glycol Heptyl alcohol Methylphenylcarbinol Cyclohexylethanol Tetraethylene glycol Cinnamyl alcohol Nonanol Tripropylene glycol a-Terpineol Citronellol I-Menthol Decyl alcohol
0.4970 0.2379 0.3832 0.4280 0.1128 0.3829 0.4035 0.3955 0.1585 0.2653 0.2491 0.2770 0.2787 0.2590 0.2438 0.2399 0.2305 0.2187 0.2383 0.2406 0.2164 0.2047 0.2202 0.2099 0.2088 0.2081 0.2057 0.2039 0.1901 0.2146 0.1893 0.1787 0.1527 0.1544 0.1722 0.1609 0.1667
1,2,4,5-Tetrachlorobenzene
Aldehydes, Ketones, and Ethers Bromal 0.0727 Chloral 0.0979 Chloral hydrate 0.1279 2-Propenal 0.2202 Acetone 0.2259 Propylene oxide 0.2024 2-Methylpropionyl bromide 0.1318 1-Bromo-2-butanone 0.1293 Di-2-bromoethyl ether 0.0985 Methyl ethyl ketone 0.1882 Dichloroethyl ether 0.1445
INDUSTRIAL A N D ENGINEERING CHEMISTRY
1,4-Dioxane Methyl propyl ether Diethyl ether 2-Furf urylaldehyde Tiglaldehyde Levulinaldehyde 3-Pentanone 2-Pentanone 3-Methyl-2-butanone 2-Chloroethyl-2-chloroisopropyl ether 2-Chloroethyl-2-chloropropyl ether 4-Hydroxy-3-methyl-2-butanone Ethyl propyl ether Cyclohexanone Mesityl oxide Dichlorodiisopropyl ether Allyl propyl ether Allyl isopropyl ether Paraformaldehyde Dipropyl ether Diisopropyl ether Acetal Diethyl Cellosolve Benzaldehyde Salicylaldehyde 4-Bromoanisole Anisole 4-Heptanone Methyl-n-amyl ketone Acetophenone Anisaldehyde 2-Octanone Caprylaldehyde 1,2-Dipropoxyethane Diethylene glycol butyl ether Cinnamyl aldehyde Benzyl ethyl ether Phorone Isophorone Azelaldehyde 2-Monanone Eugenol Isoeugenol Cineole Capraldehyde 2-Decanone Diisoamyl ether Dipropylene glycol monobutyl ether Diphenyl ether I-Acetonaphthone 2-Dodecanone Lauraldehyde Tripropylene glycol monoisopropyl ether Benzyl phenyl ether
0.1175 0.1481
Acids Trichloroacetic acid Dichloroacetic acid Bromoacetic acid Chloroacetic acid Acetic acid Acrylic acid Propionic acid Methoxyacetic acid Succinyl chloride Succinic anhydride Chloroacetic anhydride Methacrylic acid Acetic anhydride Butyric acid Tiglic acid Valeric acid Isovaleric acid Benzenesulfonyl chloride Propionic anhydride Benzoyl chloride Phenylacetyl chloride
0.1563 0.1862 0.1819 0.2575 0.2880 0.2788 0.2683 0.2841 0.1490 0.2687 0.1682 0,2634 0.1855 0.2170 0.2207 0.2162 0.2011 0.1345 0.1620 0.1484 0.1481
Esters Methyl formate Methyl dichloroacetate Ethyl formate Methyl acetate Methyl glycolate Ethyl chloroglyoxylate
0.2002 0.1340 0.1745 0.1800 0.1956 0.1318
0.1728 0.1610 0.1602 0.1216 0.1887 0.2121 0.2240 0.2238 0.2203 0.1318 0.1318 0.2455 0.1342 0.1814 0.1796 0.1266 0.1454 0.1484 0.1367 0.1371 0. I245 0.1423 0.1159 0.1884 0.1767 0.1196 0.1728 0.1839 0.1802 0.1787 0.1902 0.1644 0.3076 0.0759 0.1475 0.2016 0.1515 0.1679 0.1508 0.1560 0.1487 0.1598 0.1641 0.1232 0.1571 0.1433 0.1272 0.1434 0.1383 0.1822 0.1441 0.1419
Esters Ethyl trichloroacetate Ethyl dichloroacetate 2-Chloroethyl chloroacetate Methyl acrylate Dimethyl oxalate Ethyl acetate Methyl propionate Propyl formate Ethyl glycolate Ethyl acrylate Isopropyl chloroacetate Ethyl propionate Methyl butyrate Methyl isobutyrate Isopropyl acetate %-Propyl acetate Isobutyl formate Butyl formate sec-Butyl formate Diethyl carbonate Dimethyl maleate Isobutyl dichloroacetate Ethyl acetoacetate Glycol diacetate Diethyl oxalate Dimethyl-Z-maleate Dimethyl-d-tartrate Methyl isovalerate Ethyl butyrate Ethyl isobutyrate Propyl propionate Isobutyl acetate Isoamyl formate sec-Butyl glycolate Dimethyl citraconate trans-Dimethyl mesaconate Dimethyl itaconate Butyl acrylate Diethyl malonate Methyl caproate Ethyl isovalerate Propyl butyrate Propyl isobutyrate Isoamyl acetate Isopropyl isobutyrate Isobutyl propionate Triethyl orthoformate Phenyl acetate Methyl benzoate Methyl salicylate Diethyl maleate Dipropyl oxalate Diethyl fumarate Diisopropyl oxalate Ethyl isocaproate Propyl isovalerate Isobutyl isobutyrate Isobutyl butyrate Amyl isopropionate Benzyl acetate Ethyl benzoate Ethyl salicylate Diethyl itaconate Diethyl glutarate Methyl caprylate Isobutyl isovalerate Isoamyl butyrate Isoamyl isobutyrate Methyl cinnamate Dimethyl phthalate Propyl benzoate Diethyl adipate Diisobutyl oxalate Isoamyl isovalerate Isobutyl benzoate Bornyl formate Geranyl formate Neryl formate Menthyl formate 2-Ethylhexyl acrylate Octyl acrylate Isoamyl benzoate Geranyl acetate Linalyl acetate Citroneryl acetate Menthyl acetate Dimethyl sebacate
0.1098 0.1252 0.1481 0.1664 0.1792 0.1673 0.1681 0.1659 0.1968 0.1550 0.1426 0.1556 0.1572 0.1509 0.1461 0.1563 0.1498 0.1570 0.1510 0.1507 0.1258 0.1159 0.1664 0.1560 0.1767 0.1643 0.1816 0.1423 0.1401 0.1383 0.1467 0.1401 0.1461 0.1653 0.1528 0.1491 0.1866 0.1124 0.1227 0.1203 0.1106 0.1126 0.1064 0.1159 0.1030 0.1143 0.1081 0.1654 0.1580 0.1466 0.1387 0.1360 0.1387 0.1359 0.1349 0.1337 0.1315 0.1298 0.1332 0.1514 0.1481 0.1471 0.1210 0.1351 0.1385 0.1251 0.1274 0.1235 0.1540 0.1479 0.1402 0.1476 0.1241 0.1192 0. I394 0.0950 0.1012 0.0992 0.0887 0.0919 0.0976 0.0969 0.1008 0.0887 0.1103 0.0885 0.1043
Esters Diisoamyl oxalate Bornyl propionate Bornyl butyrate Bornyl isobutyrate Geranyl butyrate Geranyl isobutyrate ~
0.0903 0.0889 0.0879 0.0863 0.1043 0.1015
Nitrogen Compounds Formamide Nitromethane Tetranitromethane Acetonitrile Methyl thiocyanate Methyl isothiocyanate Acetamide Acetaldoxime Ethylamine Nitroethane 1,2-Ethanediamine Acrylonitrile Propionitrile Ethyl isothiocyanate 2-Bromo-2-nitrosopropane Ethyl carbamate 1-Nitropropane 2-Nitropropane Propylamine 3-Butene nitrile cis-Crotonitrile trans-Crotonitrile Methacrylonitrile Allyl isothiocyanate Diacetamide Ethyl methyl carbamate Propyl carbamate Diethylamine 3-Bromopyridine 2-Chloropyridine Tiglonitrile Angelonitrile Ethyl cyanoacetate Piperidine Isobutyl carbamate Isoamyl nitrate 2,4,6-Trichloroaniline Nitrobenzene 2-Chloroaniline 3-Chloroaniline 4-Chloroaniline &Nitroaniline Aniline 2-Picoline 1,3-Phenylenediamine Phenylhydrazine Benzonitrile Phenyl isocyanide Phenyl isocyanate Phenyl isothiocyanate Benzylamine 2-Toluidine 2-Tolunitrile Phenyl acetonitrile 2-Tolyl isocyanide 4-Ethylaniline 2,4-Xylidine 2,6-Xylidine Tetramethylpiperizine Diisobutylamine Quinoline Isoquinoline 4-Cumidine Triisobutylamine
0.6700 0.2570 0.0815 0.3323 0.2250 0.2164 0.5526 0.4179 0.2373 0.2088 0.3052 0.2480 0.2712 0.2036 0.1069 0.2725 0.1943 0.1853 0.2131 0.2416 0.2299 0.2412 0.2042 0.1873 0.2654 0.2154 0.2532 0.1755 0.1187 0.1706 0.1842 0.2110 0.2651 0.1814 0.2248 0.1440 0.2202 0.1775 0.1734 0.1874 0.1889 0.2156 0.2267 0.1898 0.2983 0.2731 0.2023 0.1861 0.1585 0.1667 0.1981 0.2068 0.1814 0.1209 0.1750 0.1959 0.2011 0.1817 0.1353 0.1375 0.1803 0.1879 0.1801 0.1222
Phenols
2,3,4,6-Tetrachlorophenol 2,5,4-TrichlorophenoI 2,4,6-Trichlorophenol 2,4-Dichlorophenol 2,6-Dichlorophenol 2-Chlorophenol 3-Chlorophenol 2-Nitrophenol Phenol Pyrocatechol 2-Methoxyphenol 2-Ethylphenol
0.1274 0.1248 0.1349 0.1503 0.1550 0.1452 0.1714 0.1669 0.2406 0.2375 0.2026 0.1911
Phenols 3-Ethylphenol 4-Ethylphenol 4,6-Dimethylresorcinol 2-Isopropylphenol 3-Isopropylphenol 4-Isopropylphenol Thymol 1-Naphthol 2-Naphthol 4-Isobutylphenol 4-sec-Butylphenol 2-sec-Butylphenol
0.2125 0.2085 0.1685 0.1833 0.1856 0.1939 0.1692 0.1896 0.1900 0.1788 0.1717 0.1591
Metal Organic Compounds Carbon disulfide Methyl trichlorosilane Methyl dichlorosilane 2-Methyldisilazine Trichloroethoxysilane Trichloroethylsilane Dimethyldichlorosilane Dimethylantimony 2-Ethyldisilazane Allyltrichlorosilane
Trichloroisopropylsilane Dichloroethoxymethylsilane Trimethylchlorosilane Trimethyldiborane Trimethylgalium Trimethyl phosphate Selenophene Diethylzinc Dichlorodiethylsilane Diethyldifluorosilane Diethyl sulfate Diethyl sulfide Tetramethyllead Allyldichloroethylsilane Trifluorophenylsilane Trichlorophenylsilane Benzenethiol Hexamethyldisiloxane Diallyldichlorosilane Diallyl sqIfi.de Chlorotriethylsilane Triethyl phosphate Diethox ydimethylsilane Trimethylpropylsilane
Hexamethylcyclotrisiloxane Benzyldichlorosilane
Dichloromethylphenylsilane Triethoxymethylsilane Butyltrimethylsilane Triethylmethylsilane
Dichloroethylphenylsilane Chlorodime thylphenylsilane Dime th ylph enylsilane Dibutyl sulfide Tetraethoxysilane Tetraethyllead Amyltrimethylsilane Tetraethylsilane Tetraeth ylbistibine
0.1491 0.0827 0.0970 0.1272 0.0835 0.0984 0.1028 0.0962 0.1147 0.0919 0.0870 0.0944 0.1141 0.1693 0.1168 0.1469 0.1009 0.1291 0.1109 0.1084 0.1492 0.1647 0.1657 0.1032 0.0964 0.0983 0.1856 0.1041 0.1028 0.1480 0.1147 0.1229 0.0694 0.1153 0.0805 0.1257 0.1103 0.1018 0.1211 0.1235 0.1059 0.1207 0.1337 0.1461 0.0927 0.0719 0.1170 0.1218 0.0687
1,3-Diethoxytetramethyldisiloxane
0,0939
1,7-Dichloro-octamethyltetrasiloxane Octamethoxytrisiloxane
Octamethylcyclotetrasiloxane Chloroethoxymethylphenylsilane Hexyltrimethylsilane Triethylpropylsilane Diisoamyl sulfide Heptyltrimethylsilane Butyltriethylsilane
Decamethylcyclopentasiloxane Decamethyltetrasiloxane
Diethoxymethylphenylsilane Trimethyloctylsilane Amyltriethylsilane Difluorodiphenylsilane Triethoxyphenylsilane Triethylhexylsilane
0.0669 0.0654 0.0702 0.1110 0.1158 0.1226 0.1278 0.1156 0.1179 0.0613 0.0694 0.1167 0.1220 0.1184 0.1104 0.1094 0.1130
1,7-Diethoxyoctamethyltetrasiloxane
Dodecamethylpentasiloxane Dodecamethylcyclohexasiloxane
VOL. 51, NO. 6
JUNE 1959
0.0700
0.0629 0.0563
793
Figure 1.
Nomogram for latent heats
The value of m' for the substance i s found from Table I; and the value of the reduced pressure, PR, or o ZG ZL is calculated. The point on the temperature scale is then connected with the point on the reducedf pressure (ZC - ZL)scale; and the line is extended to the pivot line. From this point a line through the corresponding m' value gives the latent heat on the left scale. For compounds not in Table I i f values are known of any two of these properties: latent heat at one temperature, m' (or twa points of vapor pressure from which i t can be calculated), reduced pressure (or critical pressure) the other i s fixed; and the nomogram may b e used far all temperatures. The double calibration of the reduced pressure scale shows values of ZC- Z L at each value of P R far any substance.
350
-
250
Properties of Reference Substance-e.g., water. T h e value, l ' / Z & - ZL, was
r o
403 $
0.60 -
200
350
50
10
8o 70
20
calculated at temperatures from 0 to 300 O C. using Equation 9 a n d the steam tables. Because it has a single value for each temperature, values of it may be used to calibrate a temperature scale. The Quantity ZQ - ZL. This is defined for all substances by Equation 9. It can be determined by a logarithmic straight line plot of 1.003 - (Z, - Z L ) LIS. PR.
0.45
0.50 30 0.45
300
IO0
-
IS0
-F -
0.40
50
0.35 7 -
0.35 60
-
. -
2oo
5!
L
-$
Fo0 -$
-
0.40
40 I10
ta7'
Entropies of Vaporization
I50 I60
Entropies of vaporization may likewise be related on a molal basis:
70
0.30
:;ii 80
I90
0.25
90
LT(Z& - 2;) -
200
-
-
210
-iF
100
TL'(ZG
-
ZL)
(d:
90
I5O
on grams or pound basis
80
=
'
!W2
0.15 0.0
-
0.09
-
_. E
-
CL
- 0.10
c -
0.09
Or: AS =
0
0.0 5
UY W
0.04
m'
: 40 70
-
-
r
-
60
-I
__
0.07
z;)( Z G
-
-
ZL) =
1.003 - 0.66 E'$.i4
0.02
a >
Nomograms
0.0 I
Nomograms allow ready determination of latent heats and entropy changes. From Equation 10
F 0.08
-:
AS'
(Zd
m'AS' 1.003 - 0.66PA"'* (14)
0.03
3 -I
0
-
AS (20- ZL) ( Z C - Z,)
as'
500%
-
',%:
log I = log m'
+
270 EXAMPLE
r
0.06
-
Also from Equation 14:
70.05
log A S = log m' log
-
f
z;- z;+ AS'
~
log ( Z a - Z L ) (16)
-
0.04
-
794
INDUSTRIAL AND ENGINEERING CHEMISTRY
Equations 15 and 16 are illustrated by Figures 1 and 2, respectively. The m ' scale in each, also the latent heat scale
:If
60
b
LATENT H E A T S A N D . V A P O R PRESSURE
20
50 90 IO0
40 I10
50
60 I50 70 I70
Figure 2. Nomogram for entropy of vaporization
80
I80
0.50 f0"o
I90
90 200
The value of m' for the substance is found from Table I; and the value o f the reduced pressure, PR, or o f Zc ZL i s colculated The paint on the temperature scale i s then connected with the point on the reduced pressure (ZG ZL)scale; and the line is extended t o the pivot line. From this point, a line through the corresponding m' value gives the entropy o f vaporization on the left scale.
-
-
For compounds not in Table I, i f values are known of any two of the three properties: entropy of vaporization (or latent heat) at one temperature, m' (or two points b f vapor pressure from which it can be calculated), reduced pressure (or critical pressure), the other is tlxed; and the nomogram may b e used for all temperatures. The double calibration of the reduced pressure scale shows values of Zc ZL at each value of PR for any substance.
-
350% 360
I80
I90
200
400
4 3200 - 3
220
44 0 EXAMPLE
260
530
540
560
VOL. 51, NO. 6
JUNE 1959
795
and the entropy scale, respectively, are standard logarithmic scales with values increasing upwardly. The pivot line on the right is spaced appropriately. T o draw the temperature scale in Figure 1, values of log I’/Z& - ZA were first regularly spaced on the line. Even values of temperature were then calibrated a t corresponding values of this relation. Similarly, in Figure 2 the function log AS‘/Zh - ZA, using values from the steam tables, was used to calibrate the points on the temperature scale. In both nomograms the (ZG - ZL) scale was first calibrated with uniform divisions. Then values of PR were calibrated on the other side of this line using values calculated from Equation 9. For the determination of latent heats and other uses of the important term 2, - ZL values may be immediately read for any substance at any temperature from the value of PR on the reduced pressure scale. Nomographic Method of Solution
To use the nomograms, the reduced pressure at the given temperature of the saturated vapor is required. This saturated reduced pressure is defined as PR = Po/Pc. The vapor pressure, P o , if otherwise unknown, can be determined from the nomogram and constants previously published for some 600 compounds (70). The critical pressures can be read from a handbook ( I ) or can be calculated by various methods. A straight line connects the point on the temperature or middle scale with the corresponding Pa to locate the intersection on the pivot line (on right side of nomogram). This point and the m’ value determine a straight line which yields the values of latent heat and entropy on the respective scales. At temperatures below the normal boiling point of the substance, an average value of ZQ - Z L = 0.98 may be assumed; and the nomograms can be used directly without the vapor pressure and critical pressure data, because either PE or Z, - Z L calibrations on the same scale determine the point to be used. Examples 1. Determine the latent heat of carbon tetrachloride at 180’ C. From tabulated values P, = 45 atm. and vapor pressure at 380’ C. = 10.39 atm.
Vapor
Temp., O
c.
40 80.1 120 160
Pressure,
Mm. Hg, 185.0 760.0 2238.1 5281.9
From Table I, m’ = 0.0882. The reduced pressure PE = 0.231. The latent heat I = 35.1 as read from Figure 1. This compares with I = 35 as given in the International Critical Tables. 2. Determine the latent heat and entropy of vaporization for benzene (at temperatures shown in table below). Nomogram UseUnlisted Substances
For those substances not listed in Table I, the nomograms may be used by either one of two methods, depending on available data. I. If there is known or can be determined the critical conditions, and the latent heat and vapor pressure at any temperature-e.g., the boiling point and critical pressures are given or can be found-with no vapor pressure data, m’ can be calculated by using the appropriate values for water and Equation 11. I t can also be established by connecting the boiling temperature with PR = l / P o and by using the intersection with the pivot line to establish m ’. The m’ point, once it has been established, can be used for determining latent heats at other temperatures. 2 . If vapor pressures at tu70 temperatures and the critical pressure are known, the value of m (on a molal basis) can be evaluated as the slope of a line drawn through these two points on a logarithmic paper against the vapor pressure of water (8). Values of m can also be established by the use of Equation 5 or by the vapor pressure nomogram ( I O ) . This m value is used for two purposes: 1 , To calculate (or determine graphically) the vapor pressure of the component at a certain temperature (70). From this value is established the reduced pressure at that temperature. 2. For evaluating m’ by the use of Equation 10. From these two values (PEand m’) the latent heat and entropy of vaporization are determined a t the desired temperature as above. Some deviations were found between the tabulated values and the values which were read from the correlation nomograms. In acetic and propionic acids, the deviations were of the order of magnitude of more than 30%. These and a few other examples can be explained by phenomena of association in the vapor phase which gives a much
Properties of Pure Benzene Pe = 47.9 atm.; m ’ = 0.1771 Latent Heat, Cal./G. From PR Figure 1 ICT 0.00507 0.00209 0.06137 0.14488 ~
796
INDUSTRIAL AND ENGINEERING CHEMISTRY
100.3 94.8 86.7 77.9
100.7 94.14 86.5 78.5
larger molecular weight than the monomer. Association to dimers occurs in vapor phase with certain organic compounds-e.g., acids. Nomenclature constant (intercept of line on log plot with P = 1) molar latent heat molecular weight vapor pressure critical pressure reduced pressure entropy change molar volume of gas or vapor molar volume of liquid compressibility factor of gas compressibility factor of liquid latent heat per unit weight latent heat per unit weight of reference substance slope of logarithmic plot of vapor pressure ( M ‘ / M ) m = (18, M ) m with capital letters = reference substance Acknowledgment The authors thank the Esso Education Foundation for its support of this project and Michael W. Maresca for help in calculating and plotting the nomograms. References (1) Chemical Engineering Handbook (J. H. Perry, editor), 3rd ed., McGrawHill, New York, 1950. (2) Davis, D. S.,“Nomography,” Reinhold, New York, 1955. (3) Gambil, W. R., Chem. Eng. 65, No. 5 (M’arch1958). (4) Hildebrand, J. H., Scott, R. L . , “The Solubility of Non-Elcctrolytcs,” Reinhold, New York, 1950. (5) Hougen, 0. A , , Watson, K. M., Ragatz, R. A., “Chemical Process Principles,” p. 275, Wiley, New York, 1954. ( 6 ) Lange, N. A., “Handbook of Chemistry,” Handbook Publishers, Sandusky, Ohio, 1949. (7) Lydersen, A. L., Greenkorn, R. R., Hougen, 0. A., “Generalized Thermodynamic Properties of Pure Fluids,” Coll. Eng. Univ. Wisconsin, Eng. Exptl. Sta. Kept. 4 (October 1955). (8) Othmer, D. F., I N D . ENG.CHEIII. 32, 841 (1940). (9) Othmer, D. F., &d., 34, 1072 (1942). (IO) Othmer, D. F., Maurer, P. W., Molinary, C. F., Kowalski, R. C., Zbid., 49, 125 (1957). (11) Othmer, D. F., Ten Eyck, E. H., Zbid., 41, 2897 (1949). (12) Reid, R. C., Sherwood, T. K., “The Properties of Gases and Liquids,” McGraw-Hill, New York, 1958. (13) Stull, D. R., IND.ENG.CHEM.39, 517 (1947).
RECEIVED for review November 17, 1958 ACCEPTED March 11, 1959
A S from
Figure 2 0.321 0.268 0.22 0.179
Previous articles in this series have appeared in IND.END.CHEM.during 1940, 1942-46, 1948-51, 1953,1955,1957; Chem. Eng. Data, 1956; Chem. Met. Eng., 1940: Chimie @ Industrie (Paris), 1948 ; Euclides (Madrid), 1948 ; Sugar, 1948; Petroleum Refiner, 1951-53; World Petroleum Congr. Proc., 3d Congr., Hague, 1951; Proc. Intern. Congr. Pure and Appl. Chem., 11th Congr., London, 1947.
