NOTES
798
a number of important physico-chemical relations represent a thermodynamic method which is superior in any way to what I suppose he would call the traditional or classical method.” Since then, there have been at least seven papersS which, on the whole, clarify and support Brqhsted’s formulation of thermodynamics. One of theseac strongly supports one of Brqinsted’s criticisms of traditional thermodynamics and anothergd shows that Brgnsted’s postulates are completely equivalent to the classical laws and calls his methods elegant and concise. None of these authors, however, including Brginsted himself, mentions the fact that Gibbs* discussion of the “Modifications of the Conditions of Equilibrium by Electromotive Force” is a very good example of what Brginsted, many years later, called the “work principle. ” The “work principle” is exceedingly simple. It is very nearly the same as the ‘(principle of work” used in the discussion of machines, even in highschool textbooks of physics6; or the ‘(energy principle” in a current textbook of “College Physics.’16 For example, if the turbine and generator of a hydroelectric plant are considered as a single machine in which losses due to friction and other causes may be neglected, the input of work is equal to the output, or
+
( P ~I Pg~)6Kg
(Pel
- Pep)GKe = 0
in which 6Kg is the weight of water which flowed from the height P,1 to the height P,z and 6 K e is the quantity of electricity which flowed from the electric potential P e l to the electric potential P e z . If, on the other hand, a thermodynamic system includes the reservoir, tail race, turbine, generator and power lines of a hydroelectric plant, the work principle says that the loss in gravitational work is equal to the gain in electric work. This may be expressed by the same equation. Without naming any specific principle, Gibbs4 obtained a similar result (Pel
- P&Ke
+ (Poi - P d K o + (PgI - Pg2)SKg = 0
in which “The first term represents the increment of the potential energy of electricity, the second the increment of the intrinsic energy of the ponderable matter, and the third the increment of the energy due to gravitation” when a quantity of electricity 6Ke passes from one electrode of a galvanic cell to another under equilibrium conditions. Gibbs, of course, defined his “intrinsic potentials” in terms of mass. So, 6K, was equal to 6Kg; but SK, can be expressed in moles and Pel - P,z can be the difference in what we now call chemical potentials. In that case, if we neglect the effect of gravity, the last equation above becomes
VOl. 59 (Pel
- Pe2)sKe + (Pcl - Poz)sKc = 0
the well known relation between the electromotive force of a cell and the chemical potentials of the ion involved. These examples show that the work principle, far from competing with the methods of traditional thermodynamics, is itself one of them; and that at least one derivation of a principle of physical chemistry with the aid of the work principle is simpler than that usually given in the textbooks. Since the work principle is not in conflict with the usual methods of physics or thermodynamics, such derivations can be inserted in any course in physical chemistry or thermodynamics without any modification whatsoever in the rest of the course except, perhaps, to admit that it is sometimes permissible to distinguish between t,he different kinds of energy in a thermodynamic system.
INTRINSIC VISCOSITY ,4ND KINEMATIC VISCOSITY BY CHARLESTANFORD Contribution from the Department of Chemistry, Sbals University o / I o w a , Iowa City, Iowa Received March BS, 1055
The quantity of most interest in the viscometry of macromolecular solutions is the intrinsic viscosity, [q], defined as the limiting value, a t zero concentration, of the relative viscosity increment, Le.
&1 c+o L
I‘[
7
c2 = L 9 0 c -L +~ 3 dc
(1)
where 7 is the viscosity of a macromolecular solution of concentration c (customarily expressed in grams per 100 ml. of solution), and qo is the viscosity of solvent. Most frequently used for viscosity measurement are capillary viscometers. The flow time of a liquid in such a viscometer is, however, a measure, not of viscosity, but of kinematic viscosity, v. For a given viscometer v = f(t) = At - B / t , where A and B are constants determined by calibration with liquids of known viscosity. The true viscosity is related to kinematic viscosity through the density, p , ie., q = pv, so that, in terms of capillary flow times 1
( w-) L d -=
1
d(Pf(t))
(2)
~ ~ f de t o) dc where the subscript zero refers to measurements with solvent. To obtain intrinsic viscosities by equation 2 thus requires accurate measurement of both flow time and density as a function of concentration. Since accurate density measurements are laborious, it would be simpler to measure the quantity [ v ] POVO C-N
dc
(3) (a) B. Leaf, J . Chem. Phys., 18, 89 (1944); (b) D. MaoRae’ J. Chem. Educ., 23, 366 (1946); (0) H. H. Steinour, ibid., 26, 15 (1948); (d) V. K. La Mer, 0. Fuss and H. Reiss, A n n . N . Y . Acad. Sci.. 51, 605 (1949); Acta Chem. Scand., S, 1238 (1949); (e)
T.
Rosenberg, ibid., 3, 1215 (1949); (f) P, Colmant, ibid., 3, 1220 (1949); ( 9 ) H.Holtan, Jr., J . Chem. Phya., 19,519 (1951). (4) J. W. Gibbs, “Collected Works I,” Longmans, Green, and Co., New York, N. Y.,1931,pp. 331-333. ( 5 ) N. H. Black and H. N. Davis, “New Practical Physics,” The AIacniillan Go., New Yotk, N. I-.,1933, p. 30. (0) Sears-Zemanskv. “Collene 2nd Ed., Addison- Wesley - Physics.” . l’tib. Go., Cambridge, Mass., 1953, pp. 460-1.
It is sometimes suggested’ that the difference between p and p o be ignored in the relation between q and v, so that q = pov. If this is done, then, by equation 2, [q] = [VI. The purpose of this paper is to show that this equality will, in general, be incor(1) E.u.,P. J. Flory, “Principles of Polymer Chemistry.” Cornell University Press, Itliaca, N. Y.,1953, p. 309.
NOTES
August, 1955
799
rect (though the error is likely to be inappreciable for most solutions of linear polymer), but that a simple relation between [TI and [VI can be derived which does permit the evaluation of intrinsic viscosity without accurate density measurements. Differentiating q = pv and expanding p and v in terms of Taylor series, we have so that, a t the limit of c [VI = [VI
=
0, by equations 1 and 3
+ -PO1
d dc
L 2
c+o
(4)
T o evaluate dpldc, let a solution contain gl grams 0.030 of solvent and g2 grams of macromolecular solute. 0 1 2 3 4 Let fll and f l z be the corresponding partial specific c (g./lOO i d . ) . volumes, so that the total volume is B = glfll Fig. 1.-Viscosity data for isoionic bovine seruni albumin @2. The relation between c and g2 is then, c = in 0.01 M KCl a t 25". (Straight lines determined by least 1OO(g2/V), the volume being measured in ml., so squares.) that curve shows extrapolation of the same data according to equation 3, using flow times only. The interwhere the partial specific volumes are the limiting cepts, determined by the method of least squares, are, respectively, [v] = 0.03704 and [ V I = 0.03436, values at zero concentration. difference between these being 0.00268. The The density may also be related to gl and 92, ie., the difference calculated by equation 8, using 82 = P = (91 gd/V, so that 0.7343and a measured po of 0.9975, is also 0.00268. Since the reproducibility of the data does not justify use of the last significant figure given for [?], it Since the relation between p and c must be inde- would clearly have sufficed to know flz within an accuracy of AO.01, in order to obtain an accurate pendent of gl, we obtain, from equations 5 and 6 value of [77]from flow times only. Acknowledgment.--This investigation was supported by research grant NSF-G326 from the the last equality arising from the fact that the lim- National Science Foundation, and by research iting value of fll a t c = 0 is merely the reciprocal of grant H-1619. from the National Heart Institute, solvent density. Combining with equation 4 then of the National Institutes of Health, Public Health Service. gives
+
+
(3) M. 0. Dayhoff, G. E. Perlinann and D. A. BIacInnes, ibzd., 1 4 , 2515 (1952).
The last term in equation 8 may be negligible for many polymer solutions, especially when the intrinsic viscosity is large. For polyisobutylene in benzene, for example, a t 20°, flz is 1.063 ml./g. and po is 0.87g2 so that this term becomes 0.000747. In other polymer solutions it may be as high as 0.003, which would not be negligible for viscosity measurements at relatively lo\v degrees of polymerization. The last term in equation 8 is especially important for aqueous solutions of proteins, where its value is of the order of 0.0025. Since the intrinsic viscosity of such solutions is often between 0.03 and 0.04, this term may thus amount to nearly 10% of the intrinsic viscosity. It is, however, easily evaluated, since the partial specific volume need not be known with high precision. As an experimental test of the validity of equation 8, we may examine data for serum albumin recently obtained in this laboratory by J. G. Buzzell. The upper line of Fig. 1 shows an extrapolation according to equation 2, both density and flow time being determined a t each concentration. The lower (2) W. R. JCrighmiiu a n d .'I J. I'loiy, J . A n s . Chem. Soc., 7 5 , 5254
(1953).
A SIMPLE ICINETIC METHOD FOR SOME SECOND-ORDER REACTIONS BY MARVINC. TOBIN General Research Organization, Olin Mathieson Chemical Corporation, New Haven, Connecticut Received March $9, 1966
The determination of rate constants and the general treatment of kinetic data are most simple if some function of a concentration variable is expressible as a linear function of time. This note indicates that the integrated kinetic expressions for four complex reactions involving second-order steps' may be recast to give such a function. The usefulness of this formulation goes beyond these four limited cases, since it has been found that the expressions derived are very useful as empirical equations.2 They may sometimes be used to reduce fragmentary data when the experimental curves are sigmoid in shape. The analysis is carried out in detail for the sim(1) S.Glasstone, "Textbook of Physical Chemistry," D. Van Nostrand Co., Inc., New York, N. y.,1945, yp. 1OG1, 1071. (2) J. P. Fowler and h l . C. Tobin, THIEJOURNAL, 58, 382 (1954).
