521s
NORMAN C. LI, JAMES M. WHITEAND ROBERTL. YOEST TABLE VI1
P A R T O F THE SYSTEM n-CsFId22F6COOH Mole fraction Dew point pressure, Bubble point pressure, WC6Fla mm. mm.
0 ,000 0,496“
30.0 142.6 183.0 0.755 174.7 201.5 1.000 219.2 Solutions of mole fraction less than 0.496 could not be used in the apparatus because no suitable stopcock grease was found for the gas buret.
Discussion The close resemblance of the system argonperfluoro-n-pentane to ideal behavior was to be expected because of the low polarizability of molecules of both substances and because of the general physical resemblance of the rare gases to the fluorocarbons. The system perfluoro-n-hexane-pentafluoropropionic acid deviates greatly from ideality, the pressures being greater than ideal. This might be expected in the case of substances differing as these do. The deviation, however, is probably not as great as it appears to be, because polymerization of the acid causes its true mole fraction to be less than indicated in Table VII. I t is of interest to compare the observed behavior of systems described by Tables I to VI with that predicted by the theory of Hildebrand4 and Scatchard.j This theory results in the equation K T In y1
=
Vi+2(61 -
&)2
in which the symbols have t h e following meanings : R is the gas constant, T the absolute temperature, y1 the activity coefficient of component I , VI the molar volume of component, 1, +2 the volume fraction of component two in the liquid solution and 61 and 62 are the “solubility parameters” of the two components. X similar equation may be used to calculate the activity coefficient of the second component. From the two activity coefficients one may calculate the pressures of the components over a solution. To do this, 6 for each component has [ C O N T R I B U T I O X FROM THE
Vol. 78
first been calculated by the usual method: 6 = (AE,,/V)*. The quantity AEv is the energy of vaporization per mole of the component a t 25’ and I/ is its molar volume when a liquid. Table VI11 gives the calculated values of 6. TABLE VI11 SOLGBILITY PARAMETERS AT 25’ Substance
CsFlo
6
6.09
W C S F X iz-C6Fu
c5,77
5.96
SFG
(CaF3)dIi
C6Fiz0
3.57
5.52
6.03
Since all of these values except that for sulfur hexafluoride are close to each other, the term (SI - fi2) is small for mixtures of these substances, unless sulfur hexafluoride is one component. The systems as predicted by the theory are very close to ideal, as shown in Table IX. TABLE IX CALCULATED TOTAL PRESSURE (IN MM.) AT 25” TEXS AT LIQL-ID MOLEFRACTION 0.500 System
CsFlnCsFiz
CjFioCbFia
Ideal Fromeq. 1
74.0 745.3
526.3 527.1
SF6CSF12
9289 12700
FOR
SYS-
C6FI4C6F14(CaFs13N C6Fiz0
110.1 114.2
230.0 230.1
The system SF6--n-CgF12is without question more nearly ideal than is predicted from the &values. In a case such as this, i t is common practice to use the system to calculate an arbitrary &value for the component which seems to misbehave. To make the system in question ideal, 6 for SFg would be 5 . i i . A value as low as 5.3 would still give reasonably good agreement with experiment. For other fluorocarbon systems involving sulfur hexafluoride a &value in this neighborhood should lead to better predictions than the value 3.57, given in Table VII. Acknowledgment.-This work was performed under contract with the Office of Naval Research. The authors are grateful to the Minnesota Mining and Manufacturing Co. for some of the chemicals. SEATTLE 5, WASHINGTON
L)EPARTMENT O F CHEMISTRY,
DUQUESNE USIVERSITY]
Some Metal Complexes of Glycine and Valine1 BY
NORMAN
c. LI, JAMES
Af, WHITE AND
ROBERT
L.
YOEST
RECEIVED APRIL 19, 1956 The formation constants of the nickel(I1) and copper(I1) complexes of glycine and valine a t different temperatures have been measured. From these data, the enthalpy and entropy changes for the reaction M z c -t 2A- = MA2 have been calculated for M2+ = nickel(I1) and copper(I1) ions, and d- = glycinate and valinate ions. Only data for the nickel chelates are in agreement with the “iceberg-forming tendency” of Frank and Evans and of Robinson. In 45% by weight dioxane, the entropy of formation of Ni(glycinate)z complex is much more positive than in aqueous medium, presumably in part because of “sorting-desorting” effect. A linear relationship exists between log k”.of nickel-glycine complexes and mole fraction of dioxane in the range N2 = 0 t o AT2 = 0.323, whereas the variation of pK2” of glycine with mole fraction of dioxane is not linear. The highest order cadmium complex of glycine is CdA3-; log kl = 1.8,compared t o log klkz = 8.1. These values suggest that in the highest order complex, two glycinates are chelated, but the third is not chelated.
Introduction For the purpose of obtaining a better understanding of the factors which cause complex forrnation to take place, it is desirable to consider the ( I ) YrIii, inve.;tijiatiun was iupported h J (:rant K ~ , .KSF-GIW i r c m the X a t i o n d Science f ~ < n r n d a t i o n
free energy change to be a consequence of the enthalpy and entropy changes accompanying complex formation and to measure the relative contribution of these factors2 Such a study is described herein for the Ilickel(I1) and copper(I1) (21 C . G S p i k e : ~ n dK .
\V Parr)-, T H I SJ U I J K X A I . , 76, 2720 (1953).
MBTALCOMPLEXES OF GLYCINE AND VALINE
Oct. 20, 1956
5219
complexes of glycine and valine. These two amino perimentally determined values, were added to the acids were chosen because the results are also of in- observed pH readings. The over-all formation constants, kf, of complexes terest in connection with the “iceberg-forming tendency” proposal of Frank and EvansI3inasmuch as were determined by means of the polarographic Robinson4 and Mason, Kampmeyer and Robinson5 method. Where only one divalent metal cation have used the iceberg picture t o interpret the was present in solution, the data obtained were thermodynamic and flow properties of aqueous used to calculate the number of groups, p , coordinated to each metal ion and the over-all formation solutions of amino acids. The effect of the solvent itself upon the “meas- constant by means of the equation6d ured” formation constants of the chelate com- ( E l / t ) c - ( E L / J= ~ pounds formed between the divalent metal ions - ( R T / n F ) In kf - p ( R T / n F ) In C4. (1) and glycinate can be demonstrated by varying the mole fraction of water in solution. As in previous where CAis the concentration of free ligand in solustudies,6the ionic strength of the aqueous solutions tion. I n order for this equation to be valid, the was 0.15. I n a medium of low dielectric constant, dropping mercury electrode reaction must be reversthere will be greater electrostatic interactions and ible. Since the reduction of the nickel ion and some of consequent ionic association. I n order that the are irreversible, we have deactivity coefficients of electrolytes remain com- the nickel complexes6c’10 parable in mixed organic-water media as in water,I vised the following polarographic method for the the ionic strengths of the solutions in 45 and 70% determination of the formation constants of nickel by weight dioxane were kept a t 0.004 and 0.001, complexes. I n essence, the half-wave potentials of solutions (i) and (ii) are determined: solution (i) respectively. consisting of 5.00 X M Cu(N03)2, 0.15 M Experimental Materials.--All chemicals were C.P. reagent grade prod- KN03,O. 200Mglycine, 0.100 M KOH; solution (ii) ucts. Stock solutions of copper(I1) nitrate were analyzed has the same composition as solution (i), except by addition of excess K I and titration of the liberated iodine; that 0.0495 M Ni(N03)2solution replaces 0.15 M stock solutions of nickel(I1) nitrate by precipitation with KN03. Since the first polarographic waves in dimethylglyoxime. Standard solutions of the amino acids were prepared directly from the vacuum-dried compounds, both solutions are due to the reduction of cupric and only freshly prepared solutions were used. Dioxane ion, the dropping mercury electrode reactions in mas refluxed with sodium and fractionally distilled, b.p. both are reversible. These half-wave potentials 101-101.5”. for solutions (i) and (ii) may therefore be given by Apparatus.-The pH measurements were made with a Beckman pH meter, Model G, equipped with external elec- the expressions trodes. Aqueous buffer solutions, p H values of 4 and 7 , were used to standardize the instrument. Polarographic current-voltage curves were made with a Sargent Recording Polarograph, Model XXI, and manually with a Fisher Elecdropode. A411potentials were measured against saturated calomel electrode (S.C.E.) and the half-wave potentials were corrected for the I R drop, as previously described.6 The characteristics of the capillary were: m = 2.33 mg. sec.-l, t = 3.74 sec. (open circuit) in water, a t a height of the mercury column of 50 cm.
Calculation of Constants The stepwise formation constants for equilibria of the type Malo-1 A = MA* are designated by the expression ko = (MA,)/(MAp-l)(A), and were determined by means of the Bjerrum method.* Since one does not get a true reading of pH in the 49 and 70y0by weight dioxane solutions, the observed readings were corrected. Correction factors, obtained by comparing the calculated values of pH9 for the alkali titration of glycine with ex-
+
(3) H. S . F r a n k a n d M . W. Evans, J . Chem. Phys., 13, 531 (1945). (4) A . L. Robinson, ibid., 1 4 , 588 (19+6). ( 5 ) L. S . N a s o n , P . hl. Kampmeyer a n d A. L . Robinson, THIS J O U R N A L , 74, 1287 (1952). (0) (a) iV C. Li a n d E . Doody, ibid., 74, 4184 (1952); (b) 76, 221 ( 1 9 % ; (c) N . C. Li, T . L. C h u , C . T. Fujii a n d J . M . White, ibid., 77, 859 (1956); (d) h-.C. Li a n d R . A. Manning, ibid., 77, 5225 (1955). (7) H. S . Harned a n d B. B. Owen, “ T h e Physical Chemistry of Electrolytic Solutions,” 2nd ed., Rcinhold P u b l . Corp., New York, N. Y.,1950. T h e activity coefficient of hydrochloric acid in water, i.r = 0.15, is a b o u t t h e same a s in 45% dioxane, fi = 0.004, a n d in 70% dioxane, f i = 0,001. ( 8 ) J. Blerrum, “Metal Ammine Formation in Aqueous Solutions,” P . Haase & Sons, Copenhagen, 1941. (9) T h e calculated values of p H were based on t h e p K t of glycine in t h e corresponding dioxane solution a n d ionic strength. T h e l a t t e r values were in t u r n calculated f r o m t h e values of K B of glycine (H. S. Harned a n d C . M . Birdsall, THISJOURAAL, 66, 1117 (1943)) a n d K w (H. S. Harned a n d I,. D, Fallon, ibid., 61, 2374 (1939)), which were rlrLrrminr(1 frcjln cell, without liquid junction
(El/Z)o,(i)
- (E~/z)s= - ( R T / n F ) In kf - p ( R T / n F ) In
(Ej/do,(ii)
-
(E1/z)s
C.k,(i)
(2)
=
- ( R T / f i F ) In ki - p ( R T / n F ) In
C.k,(ii)
(3)
Subtracting equation 3 from equation 2, one obtains (El/z)o,(i)
- (El/z)c,(ii)
=
-P(RT/nF) In
(cA,(i)/cA.Cii))
(4)
Since Li and DoodyGbhad previously shown that the values of log k f and p for the cupric complex of glycinate are 15.1 and 2 , respectively, the value of C~,ci)may be taken to be equal to 0.099 AI. The value of C ~ , ( i i )is then easily calculated from equation 4, and is equal to 0.099M - qx, where x is the molar concentration of the NiA, complex. The equilibrium constant of the reaction Ni++ f q A - = Ni.4,(2-g)+
can therefore be calculated and this is the over-all formation constant of the NiA, complex. The value of p is determined from an independent measurement such as potentiometric or conductometric titration.6b Results and Discussion The values of pK2‘ for the amino acids and the formation constants of the metal chelates are listed in Tables I and 11, respectively. The enthalpy and entropy changes for the reaction M + + 2A- = MA2 are calculated from the values of log klkz a t different temperatures in the usual manner, and are listed in Table 111. A consideration of t h e
+
(10) I. M. Kolthoff a n d J. J. 1-ingane, ”Polarography,” 2nd ed., Interscience Publishers, New York, L-. Y , 1952, p 486.
NORNAN C . LI, JAMES M. \VIIITE
5220
AND
ROBERT L. YOEST
Vol. 75
As seen from Tables I1 and 111, the enthalpy values for the formation of the Ki(g1ycinate)z and Ni(valinate)* complexes are essentially the same, whereas their stabilities differ by approximately 1.5 log units. This must mean, then, that the difference in stabilities is due predominantly to an TABLE I entropy effect. According to the "iceberg-forming tendency" ACIDDISSOCIATION CONSTANTS #lit' proposal of Frank and Evans,3 we might expect PK2' PKz' (70% by wt. the entropy of formation of the bl(valinate)z coin(aq.) (45% dioxane) dioxane) P = 0.1; 0.004 0,001 plex to be more negative than the corresponding quantity for the M(glycinate)z complex, both in Glycine, 25' 9.68 10. 13 11.18 aqueous medium. This could be because the iso30 9.52 10.02 11.08 propyl group in the valine structure imposes :L 40 9.21 9.79 greater iceberg-forming tendency on the utiValine, 25' 9.61 charged hi jva1inate)r complex than on the valinnte 30 9.46 anion. This would lower the partial molal entropy 40 9.15 of the product more than that of the reactant in the equation &I2+ 2%-= MAz and produce a TABLE I1 negative contribution t o AS. Since no alkyl CONCENTRATION FORMATION CONSTANTS IN group is present in the glycine structure no such log kr log k, log kikz influence on A S would be expected. Our data 011 ( A ) Aqueous medium, p = 0 . 1 5 the entropy of formation of the aqueous nickel(I1) S i ++-Glycinate coniplexes of glycinate and valinate therefore are in agreement with this hypothesis. 25" 5.97 4.95 10.92 The values of A S for the copper(I1) complexes 30 5.88 4.86 10.74 of glycinate and valinate, however, are about 40 5.72 4.70 10.42 equal, as are also the A S values found by Basolo Ni + +-Valinate and hlurmannl' for the copper(I1) coinplexes of 25 5.37 4.16 9.53 ethylenediamine, en, and N-methylethylenedia30 3.27 4.07 9.34 mine, meen, so that the iceberg hypothesis finds no 3.99 9.16 35 5.17 support from the data on copper complexes. 40 5.11 3.91 9.02 \Ye have found that in general the magnitudes of the formation constants of copper(I1) chelates are Cu ++-Glycinatea relatively insensitive to the nature of the side 25O 15,lO chains, a t least when compared to the correspond30 14.83 ing chelates of other metal ions. Thus, the values 35 14.60 of log k~ for the copper complexes of oxidized glutaCu ++-Valinate" thione and glycine differ by only 0.46 unit whereas 25O 14.76 the difference in the corresponding values for the zinc complexes12amounts to 2.74 units. Kumerous 30 14.51 35 14.28 other examples of this kind can be cited. As seen from Table 111, the values of A H for copper com(B) 45% by wt. dioxane, p = 0.004 plexes are relatively constant and, since formation Ni++-Glycinate constants of the copper complexes are also rela25O 7.16 6.06 13.22 tively constant, the values of A S must be also. 30 7.09 5.97 13.06 In 4S70 by weight dioxane, the entropy of for40 6.96 5.77 12.73 ination of Ki(glycinate)2 complex, 14.0 e.u., is much more positive than in aqueous medium. (C) 70% by wt. dioxane, p = 0,001 This could be in part because in the mixed dioxaneA3 ++-Glycinate water solvent, the solvent molecules are sorted 25O 8.51 7.24 15.75 with the more polar water molecules congregating 30 8.45 7.22 15.67 around the nickel(I1) and glycinate ions, while some desorting takes place around the uncharged a Polarographic data. Ki(g1ycinate)z complex. The question of how TABLE I11 homogenous a mixed solvent is in the presence of ENTHALPY ASD ENTROPY CHANGESFOR THE REACTION an ionic solute is important, and the sorting by M + + + 2.4hf& AT 25" ' ~ Frank I ' ions has been discussed by S ~ a t c h a r dand from a theoretical point oi view. In the inixecl Aqueous medium, p = 0.15, unless otherwise noted. ethanol-water solvents, for instance, the calculaAH, AS,
possible errors in log k,kz makes i t appear that A H may be uncertain by about 2 kcal., and A S by about 4 entropy units. The values of AH and A S in Table 111 have been rounded to the nearest half unit.
+
O
kcal.
Ni+ +-Clycinate Si++-GIyciiiate (457, dioxane, p = 0.004) Xi++-Valinate Cu +-Glycinate C u +'-\~alinate
-14.0 -14.0 --14..5 -21.0 - 211. :i
e.u.
2.5 14.0 -5.5 -1.0 0
(11) 17. Nasolo a n d I ! ) ; 1 4 , 3243 (1'332). ?'tie a u t h o r s friunrl t h a t when A H is d r t e r m i n e i l ralorimetrically, A S (Cuenn) = 7 , A S (Cumeenz) = 8 and t h a t w h e n A H is determined f r o m the temperature ?fleet on formation cim\t.ints,
~ S ( C u e n 9 )= 35, &S(Cumeenn) = 31,. (12) N. C . I,i, 0 Gaw-ron and (.., Basc,ias, ihid.. 7 6 , 224 ( l Y 5 4 j (1.9) G , S c a t c h a r d , J i (I) €I. S , Frank, z b d . , 23, 2023 (1933,.
Oct. 20, 1956
METAL COMPLEXES OF GLYCINE AND
5221
VALINE
tionI6 predicts that the molecules in contact with an ion are the more polar water molecules even though the mixed solvent is 99% alcohol. It is interesting to note that in by weight dioxane, the entropies of formationl6 of formic acid, acetic acid and propionic acid from proton and the corresponding anion of the acid are about 10, 9, 8 e.u., respectively, more positive than in aqueous medium, whereas the corresponding difference in A S values for the Ni(glycinate)2 complex is about 11 e.u. Figure 1 gives plots of log k“ of nickel glycinate complexes and pKN of glycine us. mole fraction of dioxane. Included as reference are data on pKY of acetic acid, calculated from ref. 16. These constants are in mole fraction concentration units, and are calculated from the concentration formation constants (molar concentration scale) listed in Tables I and 11, by means of the equation log k’ = log k
+ log (1000do/Mo)
12.5 12.0 11.5
(5 )
In this equation do and Mo are the density and mean molecular weight of the solvent, respectively. King15 and Adamson17have shown that it is a more sound procedure from a theoretical point of view to use mole fraction concentration units and that correlation with theory is much better when the constants are calculated on the basis of this scale. A linear relationship exists between log k N and the mole fraction of dioxane. Table I V gives mathematical expressions for the curves which fit experimental data within 0.05 log unit.
I 0
0.1
0.2
0.3
Mole fraction of dioxane. Fig. 1.-Log k N of Xi ++-glycine complexes and p k ” vs mole fraction of dioxane.
is in agreement with the accepted concept that in predominantly aqueous media, glycine is dipolar, +H~NCHZCOO-,and that in predominantly nonTABLE IV aqueous media glycine is less polar, HZNCHzLOG kN EQUATIONS FOR THE RANGE Nz = 0 TO N2 = 0.323” COOH. The pKz in predominantly aqueous while , that in predomi(70% BY WEIGHT DIOXANE)FOR NICKEL-GLYCTNATEmedia is then P K N H ~ + nantly non-aqueous mediamight come to be ~ K C O O H CHELATES Acetic acid, on the other hand, may well have the 25O log kiN = 7.71 6.9o-V~ same structure in water-dioxane mixtures within log kZN = 6.69 S.lON, the range studied ; hence the linear relationship 30’ 7.OOivz log klN = 7.62 between pKN of acetic acid and mole fraction of log kZN = 6.60 6.30N~ dioxane is as expected. a These equations fit the experimental points to 2~0.05 A survey of the literature reveals that the forlog unit. mation constants of nickel complexes of amino As can be seen from Fig. 1 and Table IV, the slope acids have not heretofore been determined by the of the log k l N vs. Nz line is greater than that of the polarographic method. This is probably because log kZN V S . Nz line. This is probably because the the polarographic wave for the reduction of effect of the dielectric constant of the medium is nickel(I1) is irreversible. As mentioned in the greater when divalent nickel ion is a reactant in section “Calculation of Constants,” we have forming the Ni(g1ycinate) + complex, than when determined the half-wave potentials for the reducunivalent Ni(g1ycinate) + is a reactant in forming tion of copper(II) complexes in two solutions (i) the uncharged Ni(g1ycinate)z complex. Of course, and (ii), both solutions have an ionic strength of changes in the effective solvation of the various 0.25. The values of EL/^)^,(^) and ( E I / J C , ( i i ) in species involved in going from water to 7Oy0me- aqueous medium a t 25’ are -0.369 and -0.186 dium would affect the formation constants also. v., respectively. With the use of equation 4 and Figure 1 shows that the variation of pKzNof gly- putting p = 2, we have obtained for Ni(g1ycinate)z: cine with mole fraction of dioxane is not linear, log kf = 11.2, in fair agreement with the more acwhereas the corresponding variation for acetic acid is curate value, log klkz = 10.92, obtained by the linear. The direction of the deviation for glycine Bjerrum pH method and listed in Table 11. Other values recorded for log klk2 are: 11.14,18 11.0,’’ (15) E. L. King, i n P. H. E m m e t t , “Catalysis,” Vol. 2, Reinhold 10.64. Publ. Corp., New York, X. Y., 1955, p . 377. (16) T h e entropies were calculated from t h e pK’s of t h e organic Table V lists the polarographic results for acids in water and in 45% b y weight dioxane listed in R. A. Robinson cadmium-glycine complex. A plot of -Ellz V S . a n d R . H. Stokes, “Electrolyte Solutions,” Academic Press, Inc., New
+ + + +
York, K. Y., 1955, p . 504. T h e p K values were determined from cells without liquid junction. (17) A. W. Adamson, THISJ O U R N A L , 76, 1578 (1954).
(18) A E. Martell and M Calvin, “Chemistry of Metal Chelate Compounds,” Prentice-Hall, Inc., New York, N. Y., 1952. p 528. (19) F. Basolo a n d T. Y. Chen, THIS J O U K S A L , 76, 953 (1954).
5222
Yol. i S
PAULJ. FLORY
-log CA,according to equation I , yields a straight line. The value of p is calculated from the slope of the line to be 3.04, so that the highest order complex is Cd(glycinate)3. The electrode reaction is reversible, and the average value of log Kf is calculated by means of equation 1 to be 9.94.
in which two glycinate ions are chelated but the third is not chelated. Douglas, Laitinen a r d Railar'O note that monodentate groups usually give a coordination number of four for the cadmium ion, but the coordination number of six seems to be more common for polydentate groups. In the structure postulated above, the coordination TABLE V POLAROGRAPHIC RESULTSFOR CADMIUM-GLYCINE COMPLEX number is six, with one of the positions occupied by the solvent. Each solution contains 5 X lo-* -If Cd(NOa)z, glycine We have also investigated polarographically half neutralized with KOH, KSOJ to keep 1.1 = 0.15, 25". the cadmium complexes of leucine, isoleucine and loe ki norleucine and found that the highest order com(kr = C&.lI,oinaie - Eii z - 1 O g CA kikzka) plex in each case is CdA3-, where A is the amino 0.000 0.583 acid ion. These complexes are less stable than the ,020 ,726 1.699 9.93 glycinate complex, presumably because of steric ,030 ,743 1.523 9.96 hindrance. However, since the polarographic ,040 .753 1.398 9.93 waves for these complexes are not strictly rever,050 ,762 1.301 9.97 sible, no quantitative values for the formation con,060 ,769 1 ,222 9.93 stants were calculated. *4v. 9.94 Acknowledgment.-The authors are indebted to The value of log klkz for the cadmium-glycine Professor Henry S. Frank, University of Pittscomplex is reported as 8.1,18obtained by the pH burgh, for suggesting the entropy studies and for method, so that log ka becomes 1.8. It is obvious helpful discussions, Mr. R. A. Manning for from these values that the tendency to form the carrying out some ofto the experiments. This rehighest order complex is extremely weak, and a possearch was sponsored by the National Science sible structure for the complex is therefore Foundation. (20) B. E. Douglas, H.A . Laitinen and J . C . Bailnr, J r . , JOURNAL,
THIS
72, 2184 (1950)
PITTSBURGH, PA.
[CONTRIBUTION FROM
THE
DEPARTMENT O F CHEMISTRY, CORNELL LTSIVERSITY]
Theory of Elastic Mechanisms in Fibrous Proteins BY PAUL J. FLORY RECEIVED APRIL30, 1956 This paper is concerned with problems relating to dimensional changes in systems comprising long chain molecules so constituted as to occur, under suitable conditions, in the state of high order characteristic of native fibrous proteins. Particular attention is given to the process of disordering of the molecular chains, which is treated as a reversible phase change between crystalline and amorphous states. Thermodynamic relations between the force f,the temperature T and the length L are developed for fibers of uniform constitution, for fibers whose properties vary axially, and for systems containing a second component (diluent) . The hitherto unexplained thermoelastic characteristics of typical fibrous proteins are readily accounted for by the hypothesis that decrease in length signifies melting of crystalline regions. Network structures formed by cross-linking polymeric chains in the oriented (crystalline) state are considered from the point of view of the statistical mechanical theory of elasticity of polymeric systems. Significant differences as compared to networks formed by cross-linking disordered chains in the usual manner are noted: the length to which the cross-linked fiber will shrink on melting in the absence of a force is predicted to increase approximately as the square root of the degree of cross-linking; a t extensions substantially greater than this relaxed length, the force of retraction should be independent of the degree of cross-linking. By combining these results of the statistical theory of elasticity with the thermodynamic relationship a(f/T) / b ( 1/T) = A H / A L , where AH and AL are the latent changes in heat and length accompanying melting, the force, length and temperature may be related over ranges which include the phase change. The elevation of the melting point which should result from cross-linking in the ordered state is treated theoretically.
studies.
('2) E. Guth, A n n . 'V,
Y.A c a d . S c i . , 47, 715 (1947).
