Abstract
Twenty-seven selected equations were tested on 162 compounds with 1958 calorimetric data for their abilities to reflect the temperature influence on the heat of vaporization of pure compounds. A new equation is recommended (overall percent deviation 0.27% and percent deviation 0.59% above 0.9T r ).
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Y. C. Ahn, J.M. Cho, H. S. Shin, Y. J. Hwang, C.G. Lee, J.K. Lee, H.U. Lee and T.W. Kang, Korean J. Chem. Eng., 20, 873 (2003).
R. F. Dye, Korean J. Chem. Eng., 18, 571 (2001).
S. P. Serbanovic, I.R. Grguric, M. Lj. Kijevcanin, A. Z. Tasic and B. D. Djordjevic, 21, 858 (2004).
V. Majer and V. Svoboda, Enthalpies of vaporization of organic compounds, A Critical Review and Data Compilation, IUPAC Chemical Data Series No.32, Blackwell, Oxford (1985).
J. S. Chickos and W. E. Jr. Acree, J. Phys. Chem. Ref. Data, 32, 519 (2003).
R. Mollier, Z. ges. Kalte.-Ind., 2, 66 (1895).
M. Thiesen, Verh. Physik. Ges. (Berlin), 16, 80 (1897).
O. Pilling, Physik. Z., 10, 162 (1909).
D. Tyrer, J. Phys. Chem., 17, 717 (1913).
J. Kendall, J. Am. Chem. Soc., 36, 1620 (1914).
N. S. Osborne and M. S. van Dusen, J. Am. Chem. Soc., 40, 14 (1918).
F.G. Keyes, R. S. Taylor and L. B. Smith, J. Math. Phys. (MIT), 1, 211 (1922).
R.M. Winter, J. Phys. Chem., 32, 576 (1928).
P. G. Nutting, Ind. Eng. Chem., 22, 771 (1930).
J. Eggert, Lehrbuch der physikalischen Chemie, S. Hirzel, Leipzig (1931).
W. J. Jones and S. T. Bowden, Phil. Mag., 37, 480 (1946).
P.M. Silverberg and L. A. Wenzel, J. Chem. Eng. Data, 10, 363 (1965).
D. J. Graue, V. Berry and B.H. Sage, Hydrocarbon Proc., 45(6), 191 (1966).
D.C.-K. Lin, I.H. Silberberg and J. J. McKetta, J. Chem. Eng. Data, 15, 483 (1970).
C. F. Chueh and A. C. Swanson, Chem. Eng. Prog., 69(7), 83 (1973).
C. F. Chueh and A. C. Swanson, Can. J. Chem. Eng., 51, 596 (1973).
M.-H. Guermouche and J.-M. Vergnaud, J. Chim. Phys., 71, 1049 (1974).
S. S. Todd, I.A. Hossenlopp and D.W. Scott, J. Chem. Thermodynamics, 10, 641 (1978).
M. Radosz and A. Lydersen, Chem.-Ing.-Tech., 52, 756 (1980).
V. Tekač, V. Majer, V. Svoboda and V. Hynek, J. Chem. Thermodynamics, 13, 659 (1981).
S. Torquato and G. Stell, J. Phys. Chem., 85, 3029 (1981).
B. Armstrong, J. Chem. Eng. Data, 26, 168 (1981).
A. Vetere, Chem. Eng. J., 32, 77 (1986).
G. R. Somayajulu, Int. J. Thermodynamics, 9, 567 (1988).
V. Svoboda and P. Basařová, Fluid Phase Equilibria, 93, 167 (1994).
H.W. Xiang, Fluid Phase Equilibria, 137, 53 (1997).
R. J. L. Andon, J. F. Counsell, D.A. Lee and J. F. Martin, J. Chem. Soc. Farad. Trans. I, 69, 1721 (1973).
J. Laynez and I. Wadsö, Acta Chem. Scand., 26, 3148 (1972).
V. Majer, L. Šváb and V. Svoboda, J. Chem. Thermodynamics, 12, 843 (1980).
M. Månsson, B. Ringnér and S. Sunner, J. Chem. Thermodynamics, 3, 547 (1971).
T. R. Rubin, B.H. Levedahl and D.M. Yost, J. Am. Chem. Soc., 66, 279 (1944).
V. Majer, V. Svoboda, A. Pošta and J. Pick, Coll. Czech. Chem. Comm., 45, 3063 (1980).
J. H. Mathews, J. Am. Chem. Soc., 48, 562 (1926).
Y. J. Rao and D. S. Viswanath, J. Chem. Eng. Data, 22, 36 (1977).
S. Sundaram and D. S. Viswanath, J. Chem. Eng. Data, 23, 62 (1978).
K. D. Williamson and R. H. Harrison, J. Chem. Phys., 26, 1409 (1957).
V. Majer, V. Svoboda, A. Pošta and J. Pick, Coll. Czech. Chem. Comm., 46, 817 (1981).
V. Svoboda, V. Kubeš and P. Basařová, J. Chem. Thermodynamics, 24, 555 (1992).
I. Wadsö, Acta Chem. Scand., 22, 2438 (1968).
Н. Л. Ярым-Агаев, Н. Н. Феодосьев and К. Г. Скорнков, ж. Фuз. Хuм., 23, 1257 (1949).
Y. J. Rao and D. S. Viswanath, J. Chem. Eng. Data, 18, 49 (1973).
S. Sundaram and D. S. Viswanath, J. Chem. Eng. Data, 21, 448 (1976).
I.A. Hossenlopp and D.G. Archer, J. Chem. Thermodynamics, 20, 1061 (1988).
N. S. Osborne and D. C. Ginnings, J. Res. NBS, 39, 453 (1947).
I.A. Hossenlopp and D.W. Scott, J. Chem. Thermodynamics, 13, 423 (1981).
G. Natarajan and D. S. Viswanath, J. Chem. Eng. Data, 30, 137 (1985).
J. H. Mathews and P. R. Fehlandt, J. Am. Chem. Soc., 53, 3212 (1931).
V. Svoboda, V. Majer, F. Veselý and J. Pick, Coll. Czech. Chem. Comm., 42, 1755 (1977).
I. Wadsö, Acta Chem. Scand., 20, 544 (1966).
J. F. Counsell, J. L. Hales and J. F. Martin, Trans. Farad. Soc., 61, 1869 (1965).
R. Fuchs and L. A. Peacock, Can. J. Chem., 58, 2796 (1980).
J. Polák and G.C. Benson, J. Chem. Thermodynamics, 3, 235 (1971).
V. Svoboda, F. Veselý, R. Holub and J. Pick, Coll. Czech. Chem. Comm., 38, 3539 (1973).
P. Ulbig, I.M. Kluppe and S. Schulz, Thermochim. Acta, 271, 9 (1996).
J.G. Aston, H. L. Fink, A. B. Bestul, E. L. Pace and G. J. Szasz, J. Am. Chem. Soc., 68, 52 (1946).
W. Kozicki, R. F. Cuffel and B. H. Sage, J. Chem. Eng. Data, 7, 173 (1962).
J.G. Aston, S.V. R. Mastrangelo and G.W. Moessen, J. Am. Chem. Soc., 72, 5287 (1950).
D. Kulikov, S. P. Verevkin and A. Heintz, Fluid Phase Equilibria, 192, 187 (2001).
C. Svensson, J. Chem. Thermodynamics, 11, 593 (1979).
M. Månsson, P. Sellers, G. Stridh and S. Sunner, J. Chem. Thermodynamics, 9, 91 (1977).
V. Majer, V. Svoboda, V. Uchytilová and M. Finke, Fluid Phase Equilibria, 20, 111 (1985).
J. C. Brown, J. Chem. Soc., 83, 987 (1903).
V. Svoboda, V. Charvátová, V. Majer and V. Hynek, Coll. Czech. Chem. Comm., 47, 543 (1982).
J. F. Counsell, J.O. Fenwick and E. B. Lees, J. Chem. Thermodynamics, 2, 367 (1970).
V. Majer, V. Svoboda, V. Hynek and J. Pick, Coll. Czech. Chem. Comm., 43, 1313 (1978).
J. F. Mathews and J. J. McKetta, J. Phys. Chem., 65, 758 (1961).
F. Veselý, L. Šváb, R. Provazník and V. Svoboda, J. Chem. Thermodynamics, 20, 981 (1988).
V. Svoboda, V. Hynek and B. Koutek, J. Chem. Thermodynamics, 30, 1411 (1998).
V. Majer, V. Svoboda, S. Hála and J. Pick, Coll. Czech. Chem. Comm., 44, 637 (1979).
K. S. Pitzer, J. Am. Chem. Soc., 62, 1224 (1940).
D. Kulikov, S. P. Verevkin and A. Heintz, J. Chem. Eng. Data, 46, 1593 (2001).
V. Svoboda, V. Kubeš and P. Basařová, J. Chem. Thermodynamics, 24, 333 (1992).
J.G. Aston and G.H. Messerly, J. Am. Chem. Soc., 58, 2354 (1936).
I.A. Hossenlopp and D.W. Scott, J. Chem. Thermodynamics, 13, 415 (1981).
G.M. Barrow, J. Am. Chem. Soc., 73, 1824 (1951).
J. F. Lemons and W.A. Felsing, J. Am. Chem. Soc., 65, 46 (1943).
G. Waddington, J.C. Smith, D.W. Scott and H. M. Huffman, J. Am. Chem. Soc., 71, 3902 (1949).
V. Majer, V. Svoboda and M. Lencka, J. Chem. Thermodynamics, 17, 365 (1985).
H. L. Finke, J. F. Messerly, S.H. Lee, A.G. Osborn and D. R. Douslin, J. Chem. Thermodynamics, 9, 937 (1977).
N. S. Berman and J. J. McKetta, J. Phys. Chem., 66, 1444 (1962).
G. Della Gatta, L. Stradella and P. Venturello, J. Solution Chem., 10, 209 (1981).
G. Geiseler, K. Quitzsch, H.-P. Hofmann and R. Pfestorf, Z. Phys. Chem. Leipzig, 252, 170 (1973).
J.K. Nickerson, K.A. Kobe and J. J. McKetta, J. Phys. Chem., 65, 1037 (1961).
S. Sunner, C. Svensson and A. S. Zelepuga, J. Chem. Thermodynamics, 11, 491 (1979).
V. Uchytilová, V. Majer, V. Svoboda and V. Hynek, J. Chem. Thermodynamics, 15, 853 (1983).
W.N. Hubbard, W.D. Good and G. Waddington, J. Phys. Chem., 62, 614 (1958).
J. P. McCullough, H. L. Finke, D.W. Scott, R. E. Pennington, M. E. Gross, J. F. Messerly and G. Waddington, J. Am. Chem. Soc., 80, 4786 (1958).
L. Guttman and K. S. Pitzer, J. Am. Chem. Soc., 67, 324 (1945).
J. Huisman and B.H. Sage, J. Chem. Eng. Data, 9, 536 (1964).
V. Uchytilová, V. Majer, V. Svoboda and I. Hemer, J. Chem. Thermodynamics, 16, 475 (1984).
V. Majer, V. Svoboda and V. Hynek, J. Chem. Thermodynamics, 16, 1059 (1984).
E.M. Arnett and D. Oancea, J. Chem. Educ., 52, 269 (1975).
E. T. Jr. Beynon and J. J. McKetta, J. Phys. Chem., 67, 2761 (1963).
S.C. Schumann, J.G. Aston and M. Sagenkahn, J. Am. Chem. Soc., 64, 1039 (1942).
D.W. Scott, J. P. McCullough, K. D. Williamson and G. Waddington, J. Am. Chem. Soc., 73, 1707 (1951).
V. Majer, V. Svoboda and M. Lencka, J. Chem. Thermodynamics, 16, 1019 (1984).
D.W. Scott, W.N. Hubbard, J. F. Messerly, S. S. Todd, I.A. Hossenlopp, W.D. Good, D. R. Douslin and J. P. McCullough, J. Phys. Chem., 67, 680 (1963).
N.S. Berman, C.W. Larkam and J. J. McKetta, J. Chem. Eng. Data, 9, 218 (1964).
J. L. Hales, J. D. Cox and E. B. Lees, Trans. Farad. Soc., 59, 1544 (1963).
D. M. T. Newsham and E. J. Mendez-Lecanda, J. Chem. Thermodynamics, 14, 291 (1982).
D. Harrop, A. J. Head and G.B. Lewis, J. Chem. Thermodynamics, 2, 203 (1970).
J. L. Hales, E. B. Lees and D. J. Ruxton, Trans. Farad. Soc., 63, 1876 (1967).
K.G. McCurdy and K. J. Laidler, Can. J. Chem., 41, 1867 (1963).
E. Morawetz, Acta Chem. Scand., 22, 1509 (1968).
I. Wadsö, Acta Chem. Scand., 20, 536 (1966).
D.W. Scott, W.D. Good, G. B. Guthrie, S. S. Todd, I.A. Hossenlopp, A.G. Osborn and J. P. McCullough, J. Phys. Chem., 67, 685 (1963).
L.A. Peacock and R. Fuchs, J. Am. Chem. Soc., 99, 5524 (1977).
I.A. Hossenlopp and D.W. Scott, J. Chem. Thermodynamics, 13, 405 (1981).
B. T. Collins, C. F. Coleman and T. De Vries, J. Am. Chem. Soc., 71, 2929 (1949).
R. E. Pennington and K. A. Kobe, J. Am. Chem. Soc., 79, 300 (1957).
T. K. Yarlett and C. J. Wormald, J. Chem. Thermodynamics, 18, 371 (1986).
C. F. Coleman and T. De Vries, J. Am. Chem. Soc., 71, 2839 (1949).
R.H. Davies, A. Finch and K. Salem, Thermochim. Acta, 170, 213 (1990).
L. Kahlenberg, J. Phys. Chem., 5, 215 (1901).
J. Konicek, Acta Chem. Scand., 27, 1496 (1973).
G. Natarajan and D. S. Viswanath, Rev. Sci. Instrum., 54, 1175 (1983).
T. S. Storvick and J. M. Smith, J. Chem. Eng. Data, 5, 133 (1960).
L. Šváb, L. Petroš, V. Hynek and V. Svoboda, J. Chem. Thermodynamics, 20, 545 (1988).
G. Waddington and D. R. Douslin, J. Am. Chem. Soc., 69, 2275 (1947).
E. Morawetz and S. Sunner, Acta Chem. Scand., 17, 473 (1963).
D.W. Scott, J. P. McCullough, W. N. Hubbard, J. F. Messerly, I. A. Hossenlopp, F. R. Frow and G. Waddington, J. Am. Chem. Soc., 78, 5463 (1956).
K. Kusano, Thermochim. Acta, 88, 109 (1985).
V. Majer, V. Svoboda, J. Koubek and J. Pick, Coll. Czech. Chem. Comm., 44, 3521 (1979).
V. Svoboda, V. Uchytilová, V. Majer and J. Pick, Coll. Czech. Chem. Comm., 45, 3233 (1980).
L. Petroš, V. Majer, J. Koubek, V. Svoboda and J. Pick, Coll. Czech. Chem. Comm., 44, 3533 (1979).
J. Cihlář, V. Hynek, V. Svoboda and R. Holub, Coll. Czech. Chem. Comm., 41, 1 (1976).
J.G. Aston, G. J. Szasz and H. L. Fink, J. Am. Chem. Soc., 65, 1135 (1943).
A. J. B. Cruickshank and A. J. B. Cutler, J. Chem. Eng. Data, 12, 326 (1967).
J.O. Fenwick, D. Harrop and A. J. Head, J. Chem. Thermodynamics, 7, 943 (1975).
J. Furukawa, M. Sakiyama, S. Seki, Y. Saito and K. Kusano, Bull. Chem. Soc. Japan, 55, 3329 (1982).
R. Jochems, H. Dekker, C. Mosselman and G. Somsen, J. Chem. Thermodynamics, 14, 395 (1982).
W. Kozicki and B. H. Sage, Chem. Eng. Sci., 15, 270 (1961).
J. P. McCullough, W. B. Person and R. Spitzer, J. Am. Chem. Soc., 73, 4069 (1951).
R. Sabbah, R. Chastel and M. Laffitte, Thermochim. Acta, 5, 117 (1972).
R. Spitzer and K. S. Pitzer, J. Am. Chem. Soc., 68, 2537 (1946).
D. J. Subach and B. J. Zwolinski, J. Chem. Eng. Data, 20, 232 (1975).
I. Wadsö, Acta Chem. Scand., 14, 566 (1960).
P. Sellers and S. Sunner, Acta Chem. Scand., 16, 46 (1962).
J.G. Aston, H. L. Fink and S. C. Schumann, J. Am. Chem. Soc., 65, 341 (1943).
J. P. McCullough, R. E. Pennington, J. C. Smith, I.A. Hossenlopp and G. Waddington, J. Am. Chem. Soc., 81, 5880 (1959).
H. T. Couch, W. Kozicki and B.H. Sage, J. Chem. Eng. Data, 8, 346 (1963).
R. J. Irving, J. Chem. Thermodynamics, 4, 793 (1972).
S. C. Mraw and C. F. Keweshan, J. Chem. Thermodynamics, 16, 873 (1984).
J. T. Clarke, E. B. Rifkin and H. L. Johnston, J. Am. Chem. Soc., 75, 781 (1953).
L. J. Paridon, G. E. MacWood and J.-H. Hu, J. Phys. Chem., 63, 1998 (1959).
V. Majer, Z. Wagner, V. Svoboda and V. Čadek, J. Chem. Thermodynamics, 12, 387 (1980).
I. Wadsö, Acta Chem. Scand., 23, 2061 (1969).
J. F. Counsell, D.A. Lee and J. F. Martin, J. Chem. Soc. (A), 313 (1971).
R. L. Montgomery, F.D. Rossini and M. Månsson, J. Chem. Eng. Data, 23, 125 (1978).
G. H. Montillon, K. L. Rohrbach and W. L. Badger, Ind. Eng. Chem., 23, 763 (1931).
E. Morawetz, J. Chem. Thermodynamics, 4, 455 (1972).
R. J. L. Andon, J. F. Counsell, D. A. Lee and J. F. Martin, J. Chem. Soc. Farad. Trans. I, 70, 1914 (1974).
R. J. L. Andon, J. F. Counsell, D. A. Lee and J. F. Martin, J. Chem. Thermodynamics, 7, 587 (1975).
L. D. Christensen and J. M. Smith, Ind. Eng. Chem., 42, 2128 (1950).
M. D. Vine and C. J. Wormald, J. Chem. Thermodynamics, 21, 1151 (1989).
V. Svoboda, V. Charvátová, V. Majer and J. Pick, Coll. Czech. Chem. Comm., 46, 2983 (1981).
J. E. Connett, J. F. Counsell and D.A. Lee, J. Chem. Thermodynamics, 8, 1199 (1976).
V. Svoboda, F. Veselý, R. Holub and J. Pick, Coll. Czech. Chem. Comm., 42, 943 (1977).
M. Månsson, J. Chem. Thermodynamics, 4, 865 (1972).
J. F. Counsell, J.H. S. Green, J. L. Hales and J. F. Martin, Trans. Farad. Soc., 61, 212 (1965).
E. L. Pace and J. G. Aston, J. Am. Chem. Soc., 70, 566 (1948).
J. Huisman and B.H. Sage, J. Chem. Eng. Data, 9, 223 (1964).
C. J. Wormald and T.K. Yerlett, J. Chem. Thermodynamics, 17, 1171 (1985).
T.B. Douglas, G. T. Furukawa, R. E. McCoskey and A. F. Ball, J. Res. NBS, 53, 139 (1954).
R.A. McKay and B.H. Sage, J. Chem. Eng. Data, 5, 21 (1960).
G. Waddington, S. S. Todd and H.M. Huffman, J. Am. Chem. Soc., 69, 22 (1947).
P. Hestermans and D. White, J. Phys. Chem., 65, 362 (1961).
J. F. Counsell and D.A. Lee, J. Chem. Thermodynamics, 5, 583 (1973).
W. Jr. Weltner and K. S. Pitzer, J. Am. Chem. Soc., 73, 2606 (1951).
T.K. Yerlett and C. J. Wormald, J. Chem. Thermodynamics, 18, 719 (1986).
R. A. Macriss, J. Chem. Eng. Data, 12, 28 (1967).
P. T. Eubank, L. E. Cedlel, J. C. Holste and K. R. Hall, J. Chem. Eng. Data, 29, 389 (1984).
Р. М. Варученко, Г. Л. Лальченко and В. А. Медведев, ж. Фuз. Хuм., 51, 992 (1977).
J.G. Aston, G. J. Janz and K. E. Russell, J. Am. Chem. Soc., 73, 1943 (1951).
S. Kapteina, K. Slowik, S. P. Verevkin and A. Heintz, J. Chem. Eng. Data, 50, 398 (2005).
W. M. Jones and W. F. Giauque, J. Am. Chem. Soc., 69, 983 (1947).
J. P. McCullough, D.W. Scott, R. E. Pennington, I.A. Hossenlopp and G. Waddington, J. Am. Chem. Soc., 76, 4791 (1954).
J. C. Philip and S. C. Waterton, J. Chem. Soc., 2783 (1930).
K. S. Pitzer and W. D. Gwinn, J. Am. Chem. Soc., 63, 3313 (1941).
V. Majer, V. Svoboda, J. Pechá ek and S. Hála, J. Chem. Thermodynamics, 16, 567 (1984).
W. F. Giauque and J. Gordon, J. Am. Chem. Soc., 71, 2176 (1949).
J. E. Mock and J. M. Smith, Ind. Eng. Chem., 42, 2125 (1950).
J. S. Chickos and J.A. Wilson, J. Chem. Eng. Data, 42, 190 (1997).
E. Morawetz, J. Chem. Thermodynamics, 4, 139 (1972).
S. Sunner and C. Svensson, J. Chem. Soc. Farad. Trans. I, 75, 2359 (1979).
W. Kozicki and B.H. Sage, J. Chem. Eng. Data, 5, 331 (1960).
G.H. Messerly and R. M. Kennedy, J. Am. Chem. Soc., 62, 2988 (1940).
K. S. Pitzer, J. Am. Chem. Soc., 63, 2413 (1941).
B.H. Sage, H. D. Evans and W. N. Lacey, Ind. Eng. Chem., 31, 763 (1939).
Н. Л. Ярым-Агаев, Е. А. Коган, В. Л. Рудин, and В. А. Титова, ж. Фuз. Хuм., 37, 1445 (1963).
Н. Л. Ярым-Агаев, В. Л. Рудин, В. А. Титова, and Е. А. Коган, ж. Фuз. Хuм., 35, 2285 (1961).
J. P. McCullough, D.R. Douslin, J. F. Messerly, I.A. Hossenlopp, T. C. Kincheloe and G. Waddington, J. Am. Chem. Soc., 79, 4289 (1957).
C. H. Rochester and J. A. Waters, J. Chem. Soc. Farad. Trans. I, 78, 631 (1982).
N. L. Helgeson and B. H. Sage, J. Chem. Eng. Data, 12, 47 (1967).
J.D. Kemp and C. J. Egan, J. Am. Chem. Soc., 60, 1521 (1938).
В. П. Лебедев, Е. А. Мирошниченко, Ю. Н. Матюшин, Б. П. Лебедев, В. С. Романов, ю. Е. Буколов, Г. М. Денисов, А. А. Балепин and ю. А. Лебедев ж. Фuз. Хuм., 49, 1928 (1975).
D. L. Hildenbrand and R.A. McDonald, J. Phys. Chem., 64, 1521 (1959).
J. P. McCullough, H. L. Finke, W.N. Hubbard, W.D. Good, R. E. Pennington, J. F. Messerly and G. Waddington, J. Am. Chem. Soc., 76, 2661 (1954).
D.W. Scott, G. B. Guthrie, J. F. Messerly, S. S. Todd, W. T. Berg, I.A. Hossenlopp and J. P. McCullough, J. Phys. Chem., 66, 911 (1962).
E. Donath, Z. Elektrochem., 32, 316 (1926).
P.A. Giguère and J. L. Carmichael, J. Chem. Eng. Data, 7, 526 (1962).
N. S. Osborne, H. F. Stimson and D. C. Ginnings, J. Res. NBS, 23, 197 (1939).
T.W. Richards and J. H. Mathews, J. Am. Chem. Soc., 33, 863 (1911).
V. Svoboda, V. Hynek, F. Veselý and J. Pick, Coll. Czech. Chem. Comm., 37, 3165 (1972).
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Jovanović, J.D., Grozdanić, D.K. A correlation for heat of vaporization of pure compounds. Korean J. Chem. Eng. 25, 1499–1508 (2008). https://doi.org/10.1007/s11814-008-0247-3
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DOI: https://doi.org/10.1007/s11814-008-0247-3