MAGNETOCHEMICAL INVESTIGATIONS. PART V.* THE DIAMAGNETISM OF BINARY LIQUID MIXTURES.

BY W. ROGIE ANGUS AND (THE LATE) DONALD V. TILSTON.

Received 10th April, 1946.

1 . Introduction.

The magnetic behaviour of binary mixtures of liquids was first in- vestigated by Smith and Smith’ in 1918. They showed that for the mixtures acetone-water, acetone-ethyl alcohol, acetic acid-water, and acetic acid-benzene the specific susceptibility of the mixture was an addi- tive function of the susceptibilities of the two components, and varied linearly with the concentration (wt.- yo) of one component. Trifonow, in 1924 2 and 1926,~ measured the susceptibility of several systems, but found that only in the systems benzene-m-xylene and ally1 mustard oil-dimethylaniline was strict additivity displayed. The other mixtures studied by him gave susceptibility-comp3sition curves which were either concave (benzene-nitrobenzene, benzene-SnC1,) or convex (benzene-CS,, acetone-CS,) to the compo8ition axis. For two systems, curves possessing maxima were obtained ; chloroform-acetone showed a maximum sus- ceptibility a t 5 1-5 mo1.- yo of chloroform, and stannic chloride-ethyl acetate gave a maximum at a composition corresponding with the com- pound SnCl, . 2EtOAc. Trifonow’s work, however, appears to have attracted little attention at the time, and it was not until 1931 that further data on such mixtures were published.

In that year Trew and Spencer 4 found wide deviation from additivity in several binary mixtures, and although these results were later shown to be fallacious by the original authors, their publication stimulated an immediate re-investigation of some of the systems by Buchner,5* 6

Ranganadhan,’. 8 and Rao and Sivaramakrishnan. 9l lo Buchner in- vestigated the completely miscible pairs C2H,0H-CS2 and acetone-chloro- form, and the incompletely miscible pairs phenol-water and methyl alcohol-CS,. Each mixture showed only small deviations from additivity, and measurements of the susceptibility a t three different field strengths proved that the results of Trew and Spencer could not be due to an extreme dependence of susceptibility on field strength. Ranganadhan investigated

Diamagnetism and the Hydrogen Bond,” by Angus and Hill, Trans. Faraday Soc., 1940, 36, 923, will henceforth be referred to as Part IV.

‘Smith and Smith, J. Amer. Chern. Soc., 1918, 40, 1218. 2 Trifonow, Mitt. wiss.-techn. Arbeiten Republik (Russ), 1924, 13, 10, II ;

3 Trifonow, Ann. Inst. Anal. Physico-Chem., Leningrad, 1926,3, 434 ; Chem.

* The article entitled

Chem. Zentr., 1925, 11, 386.

Zentr., 1927, I , 2634, Trew and Spencer, Proc. Roy. Soc. A , 1931, 131, 209. Buchner, Z . PhysiR, 1931, 72, 344. Buchner, Nature, 1931, 128, 301.

7 Ranganadhan, Indian J. Physics, 1931, 6, 4 2 1 . * Ranganadhan, Nature, 1931, 127! 975. 9 Rao and Sivaramakrishnan, Indzan J. Physics, 1931, 6, 509. lo Rao and Sivaramakrishnan, Nature, 1931, 128, 872.

9 221

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2 2 2 MAGNETOCHEMICAL INVESTIGATIONS the mixtures benzene-CC1 4, acetone-chloroform, acetone-water, and C,H,OH-water ; each showed small deviations from additivity, but to the least extent in the non-polar mixture. Molecular deformation by dipole-dipole interaction and compound formation were invoked to explain the deviations. It was held, in agreement with Buchner, that the effect of these on the magnetic behaviour could not be large. Rao and Sivara- makrishnan found that the mixture susceptibility was additive for eight pairs of simple non-polar and polar organic substances. van Aubel l1 summarised the position at this time, and pointed to the similarities in the behaviour of the density and the refractive index of mixtures.

Since 1931, numerous papers have appeared on this topic, the majority of which experimentally confirm the existence of small but real deviations from additivity. Agreement is complete, however, neither on the reality of the deviations, nor on their possible causes. A review of previous work, together with graphical data on organic acids, ethers, alcohols, and ketones, and their mixtures has been given recently by von Rautenfeld and Steurer.12

The present work was undertaken in the hope of making a quantitative comparison of available data and of providing a more certain basis for the discussion of the experimentally ascertained deviations. An in- vestigation by Angus and Hill l3 of the magnetic behaviour of substances capable of torming hydrogen bonds suggested a method for dealing with the data. Details of this method are given below, and its use in the investigation of the numerical results published by various workers * is illustrated by selected examples.

2. The Calculation of Apparent Susceptibility. If a mixture contains weight fractions wl and w 2 of two components

having specific susceptibilities of x1 and x2, then the susceptibility of the mixture, according to the additivity law is

Where real deviations occur, the mixture susceptibility, Xmix, is the sum of Xsdd and of the deviations, Xdev,

Xapp, the apparent or partial susceptibility of one component, will then be a function of w, and will have a value, which, substituted in (I) will make Xsdd = Xmlx. Hence

and

Xadd = W i X i + Wzx2. - (1)

Xmix = Xadd + X d e v - * (2)

Xmix = ~ 1 x 1 + W ~ X ~ ~ P P * * (3)

(4)

It will be noted from (2) that, when Xdev = 0, then Xmix = wlXl + w,xz and

a ( 5 ) %X1 + w2x2 - WIX1

X Z ~ P P = = x 2 * w2

11 van Aubel, Nature, 1931, 128, 455. 12 von Rautenfeld and Steurer, 2. Physik. Chem., B, 1942, 51, 39. 13 Angus and Hill, Trans. Faraday Soc., 1940, 36, 923. * The published data of Garssen (Compt. rend., 1933, 196, 541)~ Kid0 (Sci.

Rep. Tohoku Univ., 1932, 21, 385), Rao (Indian J . Physics, 1933, 8,483), Salceanu and Gheorghiu (Compt. rend., 1935, 200, I~o), Trew and Spencer (Trans. Faraday Soc., 1936, 32, ~ o I ) , and Ventkataramiah ( J . Univ. Mysore, 1942, 3, 19) have been investigated completely although they are not referred t o in detail in thls paper ; they conform to the general trends exhibited by the data presented.

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W. R. ANGUS AND D. V. TILSTON 223

Generally, however, deviations are encountered, and, hence, from ( 2 )

and (4) 7

(6) Xadd -k X d e v - % X i - w 2 X 2 + X d e v - X d e v

- x 2 + - W 2 W2 W,

. _ Xsapp =

The change of susceptibility, an index of chemical or physical reaction in the mixture, is then given by Xzapp - xz = Axzepp. Therefore,

AX2app = Xdev/wplz - - (7) Rao and Sriraman l4 and Rao and Aravamudachari l5 subjected their

own results on formic acid-water and CS,-phosphorus to an equivalent process, calculating the " change of susceptibility per 100 g. of solute." In their notation, this quantity is IOO (xz - xl)/c, where x, is the calculated value of Xmjx = Xadd = W1x1 f W 2 x 2 ; xi = observed value = Xmix, and c = weight percentage of constituent = w, x 100. Hence

C (8 ) - X d e v x I00 = ~

x 2 - x 1 Ioo = Xadd - Xadd - X d e v

IOOW, W 2

Rao and Sriraman found that this quantity was constant over a con- centration range of 29-79 wt.-yo of formic acid, but showed " a tendency to decrease at (the) higher concentrations." This claim of constancy is somewhat illfounded, since the lowest value, 0-020 , occurs at 51 yo, flanked by values fairly constant a t 0.033. Part of Rao and Sriraman's data is reproduced below in Table I, together with the results of the

TABLE I.-THE SYSTEM FORMIC ACID-WATER.

0'000 0.290 0'360 0'490 0.510 (0.575) (0' 5 75 1 0.695 0.710 0.790 1'000

Xmix.

0.7200 0.6292 0.6105 0.5701 0' 5 705 0'5434 0.5414 0.5055 0.5058 0'4834 0'4464

Rao's Method.

Xad d .

- 0.6408 0-62 I 7 0.5863 0.5808 0.5630 0.5630 0.5303 0'5261 0'5043 -

Present Method.

0.4069 0.4159 0.4141 0.4269 0.4128 0'4094 0.41 13 0.4 I 84 0.4206 (0'4464)

- 0.0395 0.0305 0.0323 0.0195 0.0336 0.0370 0.0351 0.0280 0.0258 -

present method of calculation applied to the same data. The similarity of the values of (Xadd - ~ m i x ) / W e / z and A X z a p p shows the essential equiv- alence of the two methods.

In the data on the phosphorus-CS, system, (Xadd - Xmix)/ZLJz increases from 0.06 at 22 wt.-yo phosphorus to about 0.15 at 12 yo, and then falls rapidly to 0.06 again at 3 yo, although the value at 6.5 Yo is still 0.14. The values are more erratic the lower the concentration. Rao and Aravamudachari l5 again remark on the tendency to decrease at the higher concentrations.

The behaviour of this second mixture is typical of that of the great majority of the 69 sets of available data on which we have carried out our calculations. As an illustration, the results of recalculating the data of Trew and Watkins on n-butyl alcohol-iso-propyl alcohol

14 Rao and Sriraman, J . Annamalai Univ., 1938, 7,. 187. 16 Rao and Aravamudachari, Proc. Indian Acad. Scz., A , 1940, 12, 36r. 16 Trew and Watkins, Trans. Faraday Soc., 1933, 29, 310.

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224 MAGNETOCHEM ICAL INVESTIGATIONS mixtures after conversion of the published mol.-% into &.-yo are given in Table 11.

TABLE II.-RECALCULATION OF TREW AND WATKINS' DATA ON N-BUTYL ALCOHOL-ISO-PROPYL ALCOHOL, TAKING T H E LATTER AS SOLUTE.

Mo1.-%. I Wt. -%.

0. 0000 0'0979 0.2036 0'2660 0.3998 0'4914 0'5955 0.6980 0.7988

1'0000 0'8974

0'0000 0.0809 0.1716 0.2271 0.3506 0'4392 0.5442 0.6519 0.7630 0.8762 1'0000

0.7916 0' 7944 0.7969 0'7975 0.7962 0'7974 0.7963 0'7947 0.795 1 0.7936 0'7939

- 0.8283 0.8222 0.8185 0.8046 0.8048 0.8004 0.7966 0.7964 0'7939 (0'7939)

- 0'034 0.028 0 ' 0 2 4 ~

0.006 0.003

0.011 0'01 I

0'002 0'000 -

The table shows clearly the rapid decrease of A X z a p p as the concentra- tion is increased, decreasing by two-thirds between 10 and 40 yo. TABLE III.-RECALCULATION OF DATA RECORDED BY VARIOUS AUTHORS FOR

ACETIC ACID-WATER MIXTURES. (CONCENTRATIONS I N 5% RANGES.)

AXzapp for acetic acid. wt.-yo AcOH.

Ref. I. Ref. 17. Ref. 18. Ref. 19.

0-5 5-10 10-15 15-20 20-25 25-30 30-35 35-40 40-45 45-50 50-55 55-60 60-65 65-70 70-75 75-80 80-85 85-90 90-95

- - 0.018 - 0.004 - - 0'010 - 0.008, - 0.003

0.001, - 0.005

- 0'002 -

- 0'002 - 0.003 0.003 -

- 0'002, 0'000

- 0'001

- - - - 0.000 - - -

- 0'010 - - - 0'001 - -

- 0'002 0'00 I 0.003 -

- -

- 0.020 - - - - - - - 0'000

- 0'012

- 0.099 - 0.046 - 0.01g

- - - -

0.538 0.580 0.520 0.525 XACOH.

0.720 0'720 0.714 0.720

When the data of different authors are compared, after calculating Axeapp, an almost complete inconsistency is immediately revealed as may be seen by reference to Table 111 in which the results of Smith and

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W. R. ANGUS AND D. V. TILSTON 2 2 5 Smith, Rao and Narayanaswamy,I7 Varadachari, l 8 and Sibaiya and Venkataramiah 19 on acetic acid-water mixtures are compared. I t should be noted that, in the calculations, the value we have used for the sus- ceptibility of water is that given by the individual author; but the absolute numerical value used is of little importance, provided that it is consistent with the measurements on the mixtures. The data of Smith and Smith,' and those of Rao and Narayanaswamy l7 show a decrease in the numerical value of Axzapp with increasing concentration of acetic acid, but, whereas the first set is large positive at low concentrations, the second is large negative. Varadachari's results l8 change, with the exception of the result for 21.4 yo, from large negative at low concentra- tions to small positive at high concentrations. Sibaiya and Venkatara- miah's results 19 are verv different, the value increasing numerically to a large negative maximum at 70 %, then falling to lower negative values at 80 and g o yo. Moreover, the experi- mental curve (Xmix-Wa) based on Sibaiya and Venkatara- miah's results differs con- siderably from those given by the other authors [Fig. I ) in that it shows a deep mini- mum at approximately 75 yo acetic acid, corresponding with a deviation of 12.5 yo from the additive value; and, further, it should be noted that they used the falling drop method of measurement,20 which would be more likely to give erroneous results owing to rapid changes in viscosity, density, and surface tension than the Quincke and Gouy methods used by the other investigators.

That these effects are in no way connected with the ionising power of acetic acid and water is borne out by the data on solutions of acetone in chloroform presented in

0.540 fleiyhf fradiibn of ucezic ecid -Wz.

FIG. I.-xmlx-wz curves for acetic acid-water.

The Xmix scale, correct for curve 4, must be displaced upwards by 0.027, 0.053 and 0.079 for curves 3, 2 and I. The continuous lines re- present " additivity " values. The broken line joins adjacent experimental points on curve Xmix.

QVO 0.20 0.50 040 03 0.60 070 om

Table IV (p. 226), in which concentrations have been grouped into TO % ranges. Here again, the large values of Axzspp at the lower concentrations are encountered ; in this case, all but the results of Rao and Varadachari 23 are mainly negative. The exceptional case cannot be lightly dismissed, since, although four mixtures only were examined, the examination would appear to be both careful and precise, as evidenced by the constancy of the sus- ceptibility values measured at different temperatures for each mixture.

The explanation of this irregular and uncertain behaviour was puzzling, but examination of the data for mixtures, in which additivity may be expected and for which " additivity " has been established by experiment, suggested a possible explanation. A convenient system is that containing

l7 Rao and Narayanaswamy, Proc. Indian Acad. Sci. A , 1939, 9, 35.

1Q Sibaiya and Venkataramiah, Indian J. Physics, 1932, 7, 393. zo Abbonnenc, Compt. rend., 1930, 190, 1395.

Varadachari, ibid., 1935, 2, 161.

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226 MAGNETOCHEM ICAL INVESTIGATIONS benzene and carbon tetrachloride, where both molecules, being non-polar, are hence not liable to large mutual polar distortions. The data of Seely 24

on this system are extensive and show little of the large irregularities TABLE IV.-RECALCULATION OF DATA ON ACETONE-CHLOROFORM

MIXTURES.

Axzapp for acetone. wt.-yo

Ace tone. Ref. 7. Ref. 9. Ref. 21. Ref. zz. Ref. 19. Ref. 23.

- - 0.006 0'004 - 0'000 -

- 0'001 - -

0-10 10-20 20-30 30-40 40-50 50-60 60-70 70-80 80-90 90-100

0.003 - 0.008 - 0.003

- 0.005, - 0'002

0'00 I - 0'001 - 0.006 -

- 0'102 -

- 0.049 -

- 0.007 - 0.005 0'001 - - -

- - 0'00g

0'000 -

- 0.003 - 0'002 - 0'00 I - 0'000

- - 0'001 - 0'020 - - 0'010 - 0.005

0.006 - 0'001

0'002 -

- -

- 0.057 - 0.041 - - -

- 0'010 - -

0.5897 0.5789 0.580 0.5971 0'595 0.592 Xcoiaez

XCHClS 0.4986 0.50 3 7 0.4966 0.505 0'495 0'494

carbon tetrachloride usually encountered. Seely finds, however, that I . I n . , c , .. ... , I nas a large negative temperature coemcient or susceptimiity ( - 0.0005

x I O - ~ per 10' c.), in contrast to Azim, Bhatnagar, and M a t h ~ r , ~ ~ who found the susceptibility independent of temperature. Moreover, the

TABLE V.-RECALCULATION OF SEELY'S RESULTS ON CC1,-C6H6 MIXTURES.

Ax2app for carbon tetrachloride. Wt. fraction

CC14, I I IOO. 30°. 400. 1 500 C. zoo.

0.0033 0' 002 8 0.0027 0*0023 0'0020 0'0012 0*0008 0.0005

I

0.21259 0.30803 0.38968 0'44515 0'54031 0.66264 0.77834 0.89028

0.0038

0.0037 0.003 I 0'0029

0.0016 0.0013

0.0039

0'0021

0.0023

0.001g 0-0014

0.0004

0'0020

0'0012

0'0001 -0'0001

0.0047 0.0048 0.0044 0*0040 0.0038 0.002g 0*002 3 0'0020

0.0057 0.0056 0.0052 0.0047 0.0046 0.0037 0.003 I 0.0027

0-4290 1 0.4285 0.4305 0.4300 0.4295 XCCl4

susceptibilities of all the mixtures varied with temperature. Seely, however, claims that the law of additivity holds good for the system; this, of course, is not incompatible with the unexplained temperature

21 Cabrera and Madinaveitia, Ann. Soc. Espunola Fis. Quim., 1932, 30, 528 2.2 Rao, Indian J. Physics, 1933, 8, 483. 23 Rao and Varadachari, Proc. Indian Acad. Sci. A , 1934, I, 77

Seely, Physic. Rev., 1936, 49, 812. 2 5 Azim, Bhatnagar and Mathur, Phil . Mag., 1933, 16, 580.

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W. R. ANGUS AND D. V. TILSTON 227

relations. Table V shows the values of A x z a p p for carbon tetrachloride, a t various temperatures and concentrations, calculated from Seely's results.

Here again the behaviour is similar in general to that of the other systems, although differing in some respects. The larger values occur at low concentrations, but the decrease is gradual, leaving, for the results at 50" c., a considerably high value for A x z a p p at 89 yo carbon tetra- chloride. The great difference between the five columns of the table may be traced to the fact that, whereas the temperature coefficient is negative for carbon tetrachloride, it is positive for the mixtures (o.oooz per 10" for 44-5-89 Yo CCl,, 0.0001 for 21.3-39-0 %I. If one considers, not the difference between xe and Xzapp, but the difference between xZapp for 89.0 yo carbon tetrachloride and for the other concentrations, then the values of the new quantity, A'xzaPp, are similar for each temperature, and closely resemble in behaviour the values of A x z a p p given in Tables I-IV (see Table VI) .

TABLE VI.-RECALCULATION OF THE RESULTS IN TABLE V.

Wt. fraction of CCI,.

0.2 1259 0.30803 0.38968

0'44515 0.54031 0.662 64 0' 7 783 4

A'Xzapp for carbon tetrachloride.

soo. 1 200.

0.0024 0'002 I 0'0020 0.0015 0'0013 0-0005 0'0002

0.0028 0.0023

0-0018 0.0015 0.0007 0.0003

0'0022

0'4598 I - 0'4597

30°.

0*002; 0.0026 0.0024 0.0018 0.0016 0.0008 0.0003

0.4600

40'.

0.0027 0.0028 0.0024

0.00 I 8 0.ooog 0.0003

0'0020

0.4602

50° C .

0.0030 0.0029 0.0025

0.001g

0.0004

0'0020

0'0010

0.4604

From the data in Table VI it will be seen that there still remains a small positive temperature coefficient of A'xzsPp and that the decrease of A'xzapp is more gradual a t the lower concentrations than the decrease of the corresponding A x z a p p of Tables I-IV. But it is of more importance to note that a mixture, reputedly additive, does, in fact, show deviations, that these deviations are of the same progressive type as those shown by mixtures containing at least one polar constituent, and are not small and irregularly positive and negative as might be expected. The possibility that this behaviour was fictitious and resulted from the method of cal- culation or from minor inaccuracies accentuated by the method of calculation was therefore investigated.

3. Errors in Apparent Susceptibility. The effect of small errors in the variables concerned in its calculation

on the value of A X z a p p can be found mathematically by the following process.

Then,

Consider a function u of x, y and z.

24 = u(x, y , z).

3U au au 3% 3y 32 du = -dx + -dr + -dz. . - (9)

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228 MAGNETOCHEMICAL INVESTIGATIONS If now x, y , and z are experimentally measured quantities, and any one set of x, y, and z has errors dx, dy, dz, then du represents the corresponding error in u. We may make dx, dy, and dz finite, and write

In the present instance the function u is Axzapp, a function Xmix, w2, x1 and x2. x1 and x2 must be retained as variables since, although they are theoretically " constant," they are as much subject to error as any other experimental measurement. w, is not retained since i t can be ex- pressed as a function of w2 alone. Although xmix and w2 are not inde- pendent variables in the general sense, they are independently subject to error and so must both be retained.

Hence,

Therefore,

A{AXza*pI

The effects of the errors in the four variables on the behaviour of

1. The Effect of Errors in x,.-If only xa is subject to error, then AxZapp can now be discussed.

The effect of a positive or negative error in x2 will be to reduce or increase Axzspp, respectively, by the same amount. When non-linear deviations are absent, Axzspp will be constant and equal, but opposite in sign to, the error in xB. This error may be the cause of the change in sign of Axzapp often observed at the higher concentrations (80-90 yo of component 2), as illustrated by Table I11 (ref. 18), Table IV (ref. 7, g and 22), and the data for 10' c. in Table V. In these examples the effect of Ax2 would be superposed on a variation induced by errors to be described below.

2. The Effect of Errors in xl.-If only xi is subject to error then

The error in Axtspp is thus opposite in sign to that in xl, for wl and w, can only be positive.

For the results on any one system Axl will be constant if the same material was used in making each mixture so that the curve of will, as equation (15) shows, be a rectangular hyperbola asymptotically ap- proaching the two reference axes, w2 = o and A(Ax2app) = 0. The be- haviour of the experimental values of Axzapp closely resembles this in the rise or fall at low concentrations ( w 2 + 0) and the approach at the higher concentrations (w, --f I) to a limiting value of zero. Combining this effect with that of errors in x2, a curve is obtained which reflects all the main characteristics of the experimental curves derived from, say, the data of Trew and Watkins on n-butyl alcohol-n-propyl alcohol and

Equation (14) can be re-written

W2[A{AX2.PP} - Ax11 = - AXl. (15)

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W. R. ANGUS AND D. V. TILSTON 229

n-butyl alcohol-iso-propyl alcohol mixtures. Fig. z shows these experi- mental curves together with the hyperbolze (assuming Ax, = 0) which best fit the data.

FIG. 2.-xzapp-W, curves from the data of Trew and Watkins.16

FIG. 2a.-n-C4H,0H-iso-C,H70H. FIG. zb.--n-C,H,OH-n-C,H,OH. 0-0-0 = exptl. curves. Hyper- O-O--O = exptl. curves. Hyper- bola based on average value of Ax, for bola based on average value of Axl for n-C,H,OH (- 0.0064) derived from n-C,H,OH (- 0.0044) derived from the

3. The Effect of Errors in w,.-If w 2 were the only variable in error,

X i - Xmix

the exptl. curves. exptl. curves.

then

. Aw, * (16) A{AX28PP) = ~

WZ2

The sign of the errors in A{Axzapp} will depend on the s i p of (xl-xmix), as Table VII shows. It will be noticed that if the real deviations from additivity are small or I x1 - xz I large, then x1 !mix 2 x,, over the range o < w, < I . Hence, for most real cases the sign of the error in Axaapp will depend in the same way on the sign of (rl - x,). As w, -+ 0, the value of the error A{AXzapp) corresponding to a given error Aw, will increase rapidly, and the curve of A ( A X , ~ ~ ~ ) against w, will theoretically approach the axis w, = o asymptotically. At the TABLE VII.-ERRORS IN ru,. high& cancentrations the curve should approach A{Ax,app} = 0. This be- Sign,of E ~ O I - S

holds for an error in w,, constant in both sign and

is not true, and the result

scatter the points on each

haviour, however, only In Wa.

magnitude. Generally this + of the imperfection is to -

Sign of Errors in Axnapp.

values of A { A X , ~ ~ ~ } as w, = o is approached. Superposing this effect on the previous two, the curve A,yzapp-w2 may now be expected to be a rectangular

9 *

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230 MAGNETOCHEMICAL INVESTIGATIONS hyperbola approaching wa = o and AXzapp = - Axz asymptotically, and with the experimental points becoming more and more widely scattered about the hyperbola as w, -+ 0. This scattering will not, in general, be symmetrical since evaporation of the more volatile component, which is most likely to produce large errors in w,, will be such that Aw, will have a tendency towards either positive or negative values and will thus both shift and, as the Awz variation is not hyperbolic, distort the hyperbola produced by Ax,.

4. The Effect of Errors in Xmix.-For errors in Xmix alone

Here again a hyperbola would be found if A x m i x were constant in sign and magnitude but, since neither of these conditions will generally be

TABLE VIII.-BEHAVIOUR OF HYPOTHETICAL Xdev CURVES.

Curve. % of I for Max. Dev.

50

70'7

25

Max. Dev. %.

2'00

2.09

2-29

Lt. wp+ 0

XdevlWz.

0.05

0.05

co

Behaviour of A ~ ~ ~ ~ p - w ~ curves.

Ax, = - 0.0063 ; hyperbola visibly approaches the straight line Axzspp=o-05 ( I - W ~ ) at ca. 40 yo.

Ax,= -0.0063 ; hyperbola visibly approaches the curve A X ~ ~ ~ ~ = O - O ~ ( I -wz4) at ca. 40 %.

Axl = -0~0063 ; hyperbola visibly approaches the curve Axzapp.= -0.05(1 -w,-*), itself very like a hyperbola, at ca. 40 %.

fulfilled, the effect of Axmix will be to reinforce the scattering due to Aw,. As the theoretical A ( A ~ g a p p } - A ~ m i x curve is hyperbolic, there will be no distortion of the Ax, hyperbola.

We may now consider the effect produced in systems in which real deviations do occur. Equation (7), Axaapp = X d e v / w z , will hold only when there are no experimental errors in Xmlx, w,, x1 and x,, but, generally, the effects described above will be superposed on the Xdev/W, curve. It will be immediately obvious that the lower the concentration the more will the hyperbolic effect predominate. The real AxzaPP-w2 curves will be hyperbolae, greatly distorted at the higher concentrations, and ap- proximating more and more closely to the ideal hyperbolae as w, -+ 0. Xdev/wz may itself approach an infinite limit as w, -+ 0, in which case the experimental curve will be almost indistinguishable from a hyperbola -and more particularly so when experimental errors and scattering are large.

Details of three curves, which are very similar to the experimental Xmix-Wa curves, for mixtures of two hypothetical substances of suscepti- bilities 0.500 (component I ) and 0.700 (component 2) are given in Table VII I and in Fig. 3-6. For the first curve of Table VIII both positive and negative deviations are shown, together with the theoretical curves of A x Z a p p - ~ , , calculated for an error of - 0.0063 in the susceptibility of component I. This represents an error of about I yo, not excessively large, nor indeed uncommon, in this type of measurement. Deviations of as much as z % are rarely encountered, so that these three Axeapp-wz

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W. R. ANGUS AND D. V. TILSTON 23 1

curves show the effect of real deviations even more strongly than can be expected, or, indeed, is found in the experimental curves, Even so,

FIG. 3a.-Error hyperbolz based on the "true " xdev/w, curve for Fig. 3b (b).-central . . . line.

(a) Ax1 = - 0'1000. (b ) Ax1 = - 0.0253. (c) Axl = - 0.0063.

Wt. fiactrbn of compned 1. 0-10 0-3 m u 0-70 om

FIG. 3b.-Hypothetical mixture curves for Xmix = Xadd + k(w, - W s 2 ) .

(a) k = 0'20. (b) k = 0'10. (c) k = 0.05.

it is clear that the concentration range up to 40 yo would have to be excluded from a discussion of the deviations.

A&+?P. l-00. 090. 0.M.

............ ...... -07v -0-20.

(c

( - ............. UL. fiachon of component 2. 0:/0 0.50 U?O 0.70 0.q

FIG. 4a.-Error hyperbolae as FIG. 4b.-Hypothetical mixture curves for Fig. 3a. for Xmix = Xadd $. k(w2 - W a 2 ) .

(u) K = - 0'20. (b ) K = - 0'10. (c) k = - 0 '05 .

Before discussing the determination of the errors, reference must be made to two other recent applications of this argument.

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232 MAGNETOCHEMICAL INVESTIGATIONS It has been observed in recent years that the molar polarisation of

one component of a binary mixture, calculated by the usual additivity

1.00. 0.90.

0.60.

FIG. 5a.-Error hyperbolz as for Fig. 3a.

0.90. 0.80.

0.60. 0.50.

Wt.fiac&ion of component 2. 0-10 0-30 0.50 0.70 0.90

FIG. 6u.-Error hyperbolse as Fig. 3a, but the true xdev/w2

Fig. 3b (u) is not shown.

FIG. 5b.-Hypothetical mixture curves for Xmix = Xadd + k(ruz - wz6).

(a) k = 0.10. (b) k = 0.05. (c) k = 0.02~.

for FIG. 6b.-Hypothetical mixture curves for for Xmix = Xadd f k(z02-u~d)~ .

(a) k = - 0.20. (b ) k = - 0.10. (c) k = - 0.05.

relation, P, = (P, - P,n,)/n, (where Pl, P, and P12 are the molar polar- isations of components I, 2 and solution, and n, and n, are the mol. fractions of the components), shows anomalous behaviour in very dilute

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W. R. ANGUS AND D. V. TILSTON 233

solutions, less than 0.0005 M . ~ ~ - ~ ~ Davis, Bridge, and Svirbely 30 and Lander and Svirbely 31 have shown that this anomalous behaviour, which takes the form of a hyperbolic deviation at low values of n,, the P,-n, curve approaching the n, = o axis asymptotically, was due to an error in PI, and that the sign of the error determined the sign of the deviation. The error could, in this case, be determined by an extrapolation to n, = o of the curves of dielectric constant against n,. This gave the true value of PI for the solvent used in the mixtures, and the value of AP,.

Another application is found in the investigation of the molar de- pression constant of camphor. Meldrum, Saxon and Jones,32 and Pirsch 33

noticed that the molar depression constant, K , of camphor and camphor dibromide increased rapidly at low concentrations of solute. They both observed that the curve of K against molar concentration of solute was independent of the nature of that solute. Pirsch also studied bornylamine as solvent, and i t can be seen from his data that, although for small values the calculated and observed molecular weights are very close, the cor- respondence is not as good with solutes of higher molecular weights, considering solutions of approximately equal weight-per cent. of solute. This behaviour is similar to that of camphor and camphor dibromide since the higher the molecular weight is the lower is the mol. fraction (at constant wt.-%), and hence the greater the deviation. Ricci 34 shcwed that these hyperbolic anomalies could be explained if i t is assumed that AT, the depression of the melting point, is subject to a small error, constant in sign and magnitude. Thus K = AT/m, where m is the molar concentra- tion of solute, and

A K = - - - A{AT) AT , A m s m m2 .

The first term represents a hyperbolic deviation, the second a scattering, becoming progressively wider towards the lower concentrations. Ricci suggested that the melting and grinding processes, used in preparing a sample for a melting point determination, decomposed the camphor to a small and fairly constant extent. This explains, of course, why the same K/m curve is obtained for all solutes, and the sign of such an error is compatible with the observed trend of K at the lower concentrations. Ricci’s diagram, where the first term has been applied as a correction, illustrates well the scattering as lower concentrations are attained. De- composition to the extent of only 0.005 M. is sufficient to account for Meldrum’s results.

4. Determination of the Errors. The success of Svirbely and Ricci suggested that the constant errors

occurring in the present work could be evaluated by some suitable pro- cedure. The value of such a determination lies in the fact that it would enable one to find the Axzspp-w2 curve for the real deviations, removing the present hyperbolic concealment. Two methods suggested themselves ; one an extrapolation of the experimental x m l x - ~ z curves to the w, = o axis, the other a determination from the calculated Axzapp-w, curves.

1 . Extrapolation of the xmrx-w, curves.-This method corresponds t o the extrapolation method of Svirbely et aZ.29 for determining AP,.

Halverstadt and Kumler, J . Amer. Chem. SOL, 1942, 64, 2988.

Svirbely, Abland and Warner, ibid., 1935, 57, 652.

2 7 Hoecker, J. Chem. Physics, 1936~4, 431. 28 Maryott, J. Amer. Chem. Soc., 1941, 63, 3079.

3 0 Davis, Bridge and Svirbely, ibid., 1943, 65, 857. 31 Lander and Svirbely, ibid., 1944, 66, 235. 32 Meldrum, Saxon and Jones, ibid., 1943, 65, 2023. 33 Pirsch, Ber., 1932, 69, 1229. 34 Ricci, J . Amev. Chem. SOL, 1944, 66, 658.

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234 MAGNETOCHEMICAL INVESTIGATIONS However, even a casual inspection of published x m i x - w z curves, or of curves constructed from published data, reveals in the most convincing manner the difficulty of successful extrapolation t o w, = o and w, = I. When experimental figures are so erratic that it is difficult to decide whether deviations exist or not, none but the simplest extrapolation can be applied. There is, however, little value in a linear extrapolation, since this assumes that the deviations do not exist, and will merely reduce the A x z s p p - ~ z curves to a symmetric, but scattered, distribution of points about the A x z a p p = o axis. Real deviations will in this way be reduced to errors in the extrapolated values of the susceptibilities of the two components.

2. Determination from the A x a a p p - ~ s curves.-If we neglect the effect of X d e v on the hyperbolae, we can attempt a determination of Axl. Thus, from (14)

I f then we plot not A { A ~ g a p p } , since A { A ~ z a p p } = A x z a p p - Xdev but

Axaapp against 1/w2, then both the slope and the intercept on the axis I/W, = o will have values equal to - Axl. This procedure can only be justified when Xdev-W, is small compared with A x s a p p , i.e. at low values of w,. However, the scattering effect of errors in X m i x and w, is so large in this region-most A x z a p p - ~ z curves are wildly zig-zag below w2 = 0.5 -that an extrapolation is impossible. Extrapolation for large values of w, is meaningless, since x d c v / w z is no longer small compared with Axzapp, and the value of Axl obtained will represent not the hyperbolic region of the curve but the real deviations.

w2

5. Conclusions. The behaviour of the apparent susceptibility of almost every one of

the 69 binary organic liquid mixtures studied points to the existence of considerable experimental errors in the published data on the dia- magnetism of such mixtures. Although the greater part of the quali- tative evidence is in favour of small but real deviations from additivity, no quantitative measure of any accuracy can be derived. The method of calculation, proposed by Angus and Hill,13 although having considerable possibilities when accurate and self-consistent data become available, cannot be applied with any profit to the data so far recorded, since it tends to accentuate the effect of small errors of measurement. In this connection i t is interesting to note that the data of Angus and Hill are strongly hyperbolic, as might be expected from the low concentrations involved.

The work presented above suggests certain important points in connection with an experimental investigation on the subject. The first, somewhat elementary, but to which, apparently, adherence has been by no means invariable, is that the pure components, used in the measure- ment of " solvent '' and " solute " susceptibilities, must be used in pre- paring the actual mixtures. The absolute values of their susceptibilities are of little importance, but x1 and x, must accurately represent the susceptibilities of the components. The second point is that the errors in Xmix and w, should be as small as possible. Accurate values of these variables will allow of an extrapolation of the mix-^, curves, and, by decreasing the scattering, of the determination of Ax, from the calculated A x Z a p p - ~ , curves. If this is done, then a correction for Ax, can be applied to the values of Axzspp and the effect of the real deviations freed from the objectionable hyperbolicity observed at present.

If these conditions are fulfilled, then the method will undoubtedly lead to quantitative conclusions regarding chemical and physical changes

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W. R. ANGUS AND D. V. TILSTON 23 5 in such mixtures and, by a correlation of the magnetic data with those on other physical properties (viscosity, surface tension, refractive index, etc.), will provide a possible test for theories of the structure of liquids and of molecular interaction.

6. Summary. The calculation of the apparent susceptibility ( ~ 2 a p p ) of one com-

ponent of a binary liquid mixture is discussed and a relationship between this value and the magnitude of deviations (Xdev) from strictly additive behaviour is derived. The change of susceptibility, an index of chemical or physical reaction in the mixture, is given by Ax2spp = Xdev/Wz. Over sixty sets of published data have been examined, but detailed discussion of Axzepp values is restricted to the systems formic acid-water, acetic acid-water, n-butyl alcohol-n-propyl alcohol, acetone-chloroform, and benzene-carbon tetrachloride. The progressive type of deviation shown by all systems, and amply demonstrated by those selected, suggested that the behaviour arose from the method of calculation or minor in- accuracies in certain variables on which Axzapp depends. The effect of errors in these variables has been computed and indicate that the real magnitude of observed deviations is not readily ascertainable quanti- tatively at present owing to the fact that the employed method of calculation can only profitably be applied when self-consistent and accurate data become available.

R6sum6. On calcule la susceptibilit4 apparente (xzapp) d’un composant d’un

mklange binaire liquide ; on discute ce calcul et on ktablit une relation entre cette valeur et l’importance de l’kcart (xaev) du comportement stricte- ment additif. L’4cart progressif, montr6 par 69 systhmes examinks, sugghre que cela est dQ & la m6thode de calcul ou A de 16gbres inexactitudes dans certaines variables dont dkpend AxzaPP; on a calculk l’effet d’erreurs sur ces variables.

Zusammenfassung. Der Artikel bespricht die Berechnung der scheinbaren Suszeptibilitat

(xZappLeiner Komponente in einem binaren Flussigkeitsgemisch. Es wird eine eziehung zwischen diesem Wert und der Grosse der Abweichung (Xdev) von streng additivem Verhalten abgeleitet. Der fortschreitende Charakter dieser Abweichung, der in 69 untersuchten Systemen gefunden wird, deutet darauf hin, dass diese auf die Berechnungsmethode zuriick- zufuhren ist oder auf kleine Ungenauigkeiten in bestimmten Variabeln, von denen AXzapp abhangt ; der Effekt solcher Ungenauigkeiten ist berech- net worden.

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