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Some Statistical Mechanical Models of Elastic Polyelectrolytes and Proteins
1.T. L. Hill and M. F. Morales, Arch. Biochem. and Biophys. (to be published).
2.For example, M. F. Morales and D. J. Botts, Arch. Biochem. and Biophys. (to be published).
2.See also J. Riseman and J. G. Kirkwood, J. Am. Chem. Soc. 70, 2820 (1948).
3.T. L. Hill, Trans. Faraday Soc. (to be published).
4.L. Varga, Hung. Acta Physiol. 1, 1 (1946);
4.A. Szent‐Gyorgyi, Biol. Bull. 96, 140 (1949);
4.J. Gergely, Enzymologia 14, 220 (1950);
4.J. Gergely and K. Laki, Enzymologia 15, 272 (1950). The model is due originally to Laki.
5.T. L. Hill, J. Chem. Phys. 18, 791 (1950).
6.It should be pointed out that (1) the “rest length” referred to by Gergely and Laki is the maximum length and not the elastic rest length used here; and (2) the application made by Gergely and Laki is to an (enzymatic) steady‐state condition (see reference 2).
7.F. T. Wall, J. Chem. Phys. 10, 485 (1942);
7.F. T. Wall, 11, 527 (1943); , J. Chem. Phys.
7.P. J. Flory and J. Rehner, J. Chem. Phys. 11, 512, 521 (1943);
7.P. J. Flory and J. Rehner, 12, 412 (1944); , J. Chem. Phys.
7.P. J. Flory, Chem. Revs. 35, 51 (1944);
7.P. J. Flory, J. Chem. Phys. 18, 108, 112 (1950).
7.See also Kuhn, Pasternak, and Kuhn, Helv. Chim. Acta 30, 1705 (1947);
7.J. J. Hermans, Trans. Faraday Soc. 43, 591 (1947);
7.H. J. White and H. Eyring, Textile Research J. 17, 523 (1947);
7.L. R. G. Treloar, Proc. Roy. Soc. (London) 200A, 176 (1950).
8.For example, A. Katchalsky, Experientia 5, 319 (1949);
8.Kuhn, Hargitay, Katchalsky, and Eisenberg, Nature 165, 514 (1950).
9.T. L. Hill, J. Chem. Phys. 17, 762 (1949).
9.a A. D. McLaren and J. W. Rowen, J. Polymer Sci. 7, 289 (1951).
9.b T. L. Hill and J. W. Rowen, J. Polymer Sci. (to be published).
9.c F. W. Boggs (personal communication)
9.and J. Chem. Phys. 20, 632 (1952).
10.See, however, Fuoss, Katchalsky, and Lifson, Proc. Nat. Acad. Sci. 37, 579 (1951).
11.T. L. Hill, J. Chem. Phys. 17, 1125 (1949).
12.Compare J. J. Hermans and J. Th. G. Overbeeck, Rec. Trav. Chim. 67, 761 (1948), Appendix I.
12.The electrostatic theory in the present paper is similar to that given by Hermans and Overbeeck, except for geometry (R small) and nonlinearizing the Boltzmann factor (R large). Incidentally, we are investigating two refinements of the Hermans and Overbeeck theory, namely, use of an ellipsoidal model instead of a sphere (to allow following the transition from a sphere to a rod on adding charges) and inclusion of the appropriate Flory‐Rehner free energy of mixing terms for a single polymer molecule (P. J. Flory, J. Chem. Phys. 17, 303 (1949)).
12.See T. L. Hill, J. Chem. Phys. 20, 1173 (1952).
13.The ion being adsorbed is of course part of the electrolyte hence this simplification (single symmetrical electrolyte) may not be possible in some cases. In ATP adsorption, the concentration of ATP in solution is in general low enough to neglect its contribution as part of the electrolyte (this neglect is of course not necessary when the linear approximation is used).
14.See E. J. W. Verwey and J. Th. G. Overbeeck, Theory of the Stability of Lyophobic Colloids (Elsevier Publishing Company, Amsterdam, Holland, 1948), Eq. (17).
14.a J. W. Rowen and R. Simha, J. Phys. and Colloid Chem. 53, 921 (1949).
15.A. Katchalsky, American Chemical Society meeting, September, 1951. This paper included both experimental and theoretical developments for this case (with ).
16.W. T. Astbury, Proc. Roy. Soc. (London) 134B, 303 (1947).
17.M. L. Huggins, Chem. Revs. 32, 195 (1943).
18.L. Pauling and R. B. Corey, Proc. Nat. Acad. Aci. 37, 261, 729 (1951).
19.J. Frenkel, Kinetic Theory of Liquids (Oxford University Press, London, England, 1946).
20.Actually, and (below) can be free energies (including entropy terms) without changing the equations below. See E. A. Guggenheim, Trans. Faraday Soc. 44, 1007 (1948).
21.This restriction is made for simplicity but is not necessary. See Appendix I.
22.R. H. Fowler and E. A. Guggenheim, Statistical Thermodynamics (The Cambridge University Press, Cambridge, England, 1939).
23.E. A. Guggenheim, Proc. Roy. Soc. (London) 183A, 213 (1944);
23.E. A. Guggenheim and M. L. McGlashan, Proc. Roy. Soc. (London) 206A, 335 (1951);
23.E. A. Guggenheim and M. L. McGlashan, Trans. Faraday Soc. 47, 929 (1951). We believe the criticism of our paper (see reference 26) by Guggenheim and McGlashan to be incorrect. This point will be discussed elsewhere.
24.C. N. Yang, J. Chem. Phys. 13, 66 (1945).
25.Y. Y. Li, Phys. Rev. 76, 972 (1949).
26.T. L. Hill, J. Chem. Phys. 18, 988 (1950).
27.T. S. Chang, Proc. Cambridge Phil. Soc. 35, 265 (1939).
28.Reference 22, p. 432.
29.T. L. Hill, J. Chem. Phys. 15, 767 (1947), discusses the relation between “loops” and hysteresis.
30.For example, H. B. Bull and M. Gutmann, J. Am. Chem. Soc. 66, 1253 (1944);
30.H. B. Bull, J. Am. Chem. Soc. 67, 533 (1945);
30.T. Alfrey, Mechanical Behavior of High Polymers (Interscience Publishers, Inc., New York, 1948);
30.H. R. Kruyt, Colloid Science, Vol. II (Elsevier Publishing Company, Inc., Amsterdam, Holland, 1949);
30.J. B. Speakman, J. Text. Inst. 24, 273 (1933);
30.J. B. Speakman and M. C. Hirst, Trans. Faraday Soc. 29, 148 (1933);
30.H. Burte and G. Halsey, Textile Research J. 17, 465 (1947);
30.W. T. Astbury and S. Dickinson, Proc. Roy. Soc. (London) 129B, 307 (1940);
30.J. B. Speakman and L. Peters, J. Text. Inst. 39, 253 (1948).
30.a As an alternative to considering only the case as in the text, one can follow the same procedure as in reference 26. That is, although the quasi‐chemical relations obtained will not involve in a completely satisfactory way, we can use the quasi‐chemical configurational expression of Eq. (88) and still take In this case an additional factor is inserted in Eq. (88), where (Fig. 10). The only effect on Eq. (90) is to replace in γ by
31.Reference 14, p. 25.
32.L. Pauling and R. B. Corey, Nature 168, 550 (1951).
33.M. G. M. Pryor, Progress in Biophysics, edited by J. A. V. Butler and J. T. Randall (Butterworth‐Springer, Ltd., 1950), Vol. I.
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