Jan., 1962
NOTES
tion of shear stress or shear rate. This requirement is fundamental so that any valid experimental evidence to the contrary must be of considerable concern and any claim of such data deserves critical examination. Recently Philippoff and Gaskins2 have presented dat,a which thev claim shows the initial deviation from Newtonian behavior of an essentially monodisperse (with respect to molecular weight), random coil polymer to be an odd function of shear rate. Their data are presented in a graph where the ordinate appears8to be (?I- vg)/(qo- ss) us. log ?; where 7 is the viscosity of the solution at the shear rate f, vo is the so-called zero shear viscosity, and qs is the viscosity of the solvent. In the same figure they plot functions labeled qsp/(l z) and qsp/(l 2". The functions appear to be 1/(1 b y ) and 1/(1 ai.2)vs. log i., respectively, where the parameters a and b have been selected to make the agreement between the function and the experimental data as good as possible. On the basis of their figure it is claimed, and indeed it appears true, that the function (1 b?;)-' gives the best initial fit. Unfortunately, this test lacks real significance. T o show this one need only note, as Philippoff and Gaskins have, that the data corresponding to (7 vS)/(qo- vs) > 0.90 ,scatter so badly as to preclude analysis and then to show that "data" calculated from a suitable even function also appear to be better fit by (1 bi.)-' than by (1 over a range comparable to that of the experimental data. Such a comparison is made in Fig. 1. Here, as with the experimental data of Philippoff and Gaskins, the best fit appears t o be given by (1 hi.)-', but in this example it is known that the correct function is positive and even. The difficulty apparently lies with a tendency to confuse the two distinctly different criteria: (1) goodness of fit over an appreciable range a t low rates of shear, and (2) goodness of initial fit. The confusion is perhaps unavoidable because of the relatively large experimental errors a t low rates of shear and the necessity for estimating initial curvature. I n any case, it is apparent that such a test is unreliable. As B matter of incidental interest, the positive even function plottcd in Fig. 1 is the familiar4.6
+
+
+
+
+
'I='lN
xfl sinh-1 83
4-a Br
recast for the present purpose as -L = ' I N ? + 'I-..?
- 'Ia
\ "I
\
-
'I@tla
4'lo
-VJ
106
104
103
Y
Fig. 1.-Comparison of n positive even function of viscosity us. shear rate with (1 ay2) -1 and (1 b y ) -I.
+
+
combination gives a reasonable fit to the data of Philippoff and Gaskins as the reader easily may verify for himself. One may note that these constants are also in fair agreement with the constants chosen by Kim, et d16 to represent the data of Yang' on the same polymer under similar conditions. This correspondence is trivial to the present argument except as it shows that the positive even function in Fig. 1 is not some strange thing concocted for the present purpose. The essential point of interest here has not been what even function to choose. It has been to point out that results such as those of Philippoff and Gaskins do not provide a valid test of the form of the initial deviation from Kewtonian flow, as has been alleged. (0) W. K. Kim, N. Hirai, T. Roe and 1%.Eyring, rhd., 31, 358 (1900). (7) J. T.Yang. J . Am. Chem. Soc., 80, 1783 (1958).
T H E APPARENT MOLAR VOLUME OF SODIUM HYDROXIDE AT IXFINITE DILUTION AND THE VOLUME CHANGE ACCOMI'ANYlNG THE IONIZATION OF WATER' BY AGNESBODANSZKY AND WALTERN A U Z M A ~ U N Frick Chemical Laboratorv, Princeton Unrvereit?/,Prrncelon, N . J . Heceised July 80,1061
sin-10-i
BY 'I'he constants have been chosen so that 0.45 ('IN - 'I~/('IO x O l 4 v o - 13 = 0.55 B = 10-4 The reason for this choice was simply that this (1) F. R. Eirich, "Rheology-Theory and Applicationr," Vol. I, 'lo
c
+
+
+
0.50
177
'18)
Academia Press, Inc., New York, N. Y., 1956, Chaptat 16, J. G. Oldroyd. (2) W. Philippoff and F. II. Gaskins, J . Phya. Chem.. 88, 086
(1959).
13) The ordinate axia is denoted by "Spec~fio Visaosity,
duced)." (4) T. Ree and H. Eyring, J . A p p l . Phys., 28,793 (1055). (6) T. Ree and H. Eyring, ibid., 26, 800 (1066).
')sp
(Re.
According to O w n and Brinkley2 the volume change accompanying the reaction II+( m aq.)
+ OH-(
aq.)
+€120
(I)
is 23.4 ml. at 25". This result was derived from the best available data on apparrnt molar volumes of the common miiieral acids, alkali hydroxides and their salts, extrapolated t o infinite dilution. Direct measurements of the volume change of this rcmtion at finite concentrations using dilatometers give, however, considerably smaller values. Weber and Nachmannsohn3 observed values of 21.0 ml. when (1) This work was supported by a grant from the National Science Foundation. (2) B. B. Owen and 5. R Brinkley. Chem. Rem., 29, 461 (1941). (3) H. H. Weber and D. Nachmanneohn Bzochem. 2,.204, 215 fl029).
NOTES
178
0.1 N nitric acid is added to 0.1 N sodium hydroxide and 20.7 ml. when 0.1 N HCl is added to 0.24 N sodium hydroxide. Measurement,s by one of us4 give a volume change of 20.8 ml. when equal volumes of 0.1 N hydrochloric acid and 0.1 N sodium hydroxide are mixed to form 0.05 N sodium chloride. It is difficult to believe that the changes in apparent. molar volumes that would accompany the infinite dilution of these electrolytes would amount to several ml. We therefore have rcexamined the volume changes on dilution of the substances that are involved in this react,ion. Redlich and Bigeleisen5found that at 25" in t,he concentration range 0.0035 to 0.19 N 6t,hc apparent molar volume of hydrochloric acid is giveii by rp = 17.830 + 1 . 8 6 ~ % - 1.1511.1
(2)
For sodium chloride a t 25" Wirth' found betivceii 0.04 and 4 M q5 =
16.435
+ 2.01OdlG + 0.052M
(3)
The molar volume of water a t 25" is 18.069 ml. Studies of the apparent molar volume of sodium hydroxide have been made only a t much higher concentrations, and the extrapolations to infinite dilution are therefore less reliable. Guckers concluded from a survey of data in t,he literature that cp = -6.7
+ 4.18.\/3?
(4)
but the measurements on which this result is based appear to have been made almost entirely in solutions more concent,rated than 1 M . Owen and Brinkley2 concluded that a value of -6.8 for the apparent molar volume of sodium hydroxide a t infinite dilution is consist,ent with values obtained from the ionic volumes deduced from measurements of ot,herhydroxides and salts. None of the volumes of these othcr hydroxides was obtained from data with solutions more dilute than 0.5 to 1&I. Akerlof and KcgelcsQmeasured the apparent molar volumes of sodium hydroxide between 1 and 25 m at temperatures between 0 and 70". At 25" and between 1 and 3.7 m their equations give cp = -5.944
+ 3.98301/n2
(5)
1,aninan and M a P give apparent
