Critical temperatures and volumes of some binary systems

Chemical Engineering

Science, 1963, Vol. 18, pp. 715-724. Pergamon Press Ltd., Oxford.

Printed in Great Britain.

Critical temperatures and volumes of some binary systems D. C-H. CHENG~ Department of Chemical Engineering, Imperial College, London, SW.7 (Received 16 March 1963 ; in revised form 15 May 1963) Abstract-The effect of gravity on material distribution near the critical point is discussed and procedures for extending the sealed tube method to binary mixtures are suggested. One of the procedures is used to obtain critical temperatures and volumes of some binary mixtures. The behaviour of mixtures in sealed tubes is described and explained in terms of composition gradients. The critical temperatures are discussed in terms of the theory of random mixtures.

A NUMBER of critical temperatures of binary mixtures of organic compounds have been measured [l, 21. Many of these measurements relate to mixtures of aliphatic and aromatic hydrocarbons of low molecular weights, but most of them refer to two components of similar critical temperatures and volumes. It is important to extend the investigation of binary critical data to molecules of markedly different critical properties. The present study was undertaken as a means to obtain information about intermolecular forces between unlike aromatic molecules. Interaction between unlike molecules can be discussed in terms of the theory of random mixtures [I, 31. For molecules obeying the Lennard-Jones potential u(r) =

(-$.-) (f$“‘- [(;)”- (;J]

T; =

fJ”oo

(4) and for molecules of equal energy, it is (5)

(1) It has been shown [l] that although f, depends on fij and gii, to a good approximation it can be considered to be independent of gij. Thus

the critical temperature TC of a multicomponent system is related to that of a reference substance T&, via that of an equivalent pure substance T,‘. where

T’ is not simply determined, because while T,“, being the property of a pure substance, is given by conditions of mechanical instability, T’ is determined by conditions of material instability [I, 41. The difference between them has been calculated in terms of the equation of state of the system for a binary mixture; it has further been evaluated for systems obeying the van der Waals equation of state [l]. For molecules of equal size, it is

T,E= x,T;,

(2)

where

el2

+ =

x2Ti2 + xlx2e12TZjo

2f12

-fil

(6)

(7)

-f22

Interactions between unlike molecules are described by the combining rules [l, 51 9:19232 w311~22)1'2 fi2

d2

(3) and

3 4':.=~--&

Ufj i,j =

1, 2) . . .

tPresent address: DSIR Warren Spring Laboratory,

112

=-

Al

Q12 =%-Ill

+ B22 +g22)

(8)

(f11f22) (9

A value of f12 or T,‘, given by these rules would therefore indicate normal interaction. Larger values would suggest stronger interaction than usual, and vice versa. Stevenage, Herts.

715

D. C-H. CHENG PRINCIPLEOF THE SEALEDTUBE METHOD

Critical pressures and temperatures of binary systems are usually determined as functions of composition either from the pressure-composition loop at constant temperature or the pressure-temperature loop at constant composition. Critical volumes are seldom measured. However, these methods are laborious. In order to facilitate the direct measurement of binary critical temperatures the sealed tube method has recently been used. In the usual description of the sealed tube method, as applied to pure substances, the tube contains the liquid in equilibrium with its vapour, each phase being homogeneous and the temperature uniform. Considering a tube of uniform section, the position of the meniscus would be given by 1

P - P"

z _- pl Near the critical

- p”

point

pr - p= = p= - p” = A(T” - T)‘P [l, p.911. On substituting I

-=$l+

L

[

(11)

(11) into (10)

F - PC - z-p3-1

A(T’

(12)

If the tube is filled with the critical amount of material (p = p’), when it is heated the meniscus disappears at the middle of the tube (l/L = +) when the temperature becomes critical. Otherwise, the meniscus rises or falls out of the tube (I tending to f co) depending on whether p is greater or less than pc. In using the sealed tube method for binary mixtures [2, 61, this description of the method was assumed. The critical volume being unknown, estimated values were used in filling the tube with the critical amount of material. It is well known, however, that the meniscus of pure substances will disappear at all positions in the tube [I, 7, 81. The cause of this is the existence of large density gradients [9, p.631; lo] that are set up in the earth’s gravitational field due to the high compressibility of matter near the critical point. In particular, the work of SCHNEIDERet al. [8] shows clearly that the conditions at the disappear716

ing meniscus are critical and that the meniscus can disappear at all positions in the tube depending on the mean density. This makes the sealed tube method the simplest way of measuring the critical temperatures of pure substances as no great care need be taken in filling the tube. But for mixtures the usual description cannot be expected to hold. Accompanying the density gradient there will also be a composition gradient, leading to uncertainty in the local composition at the meniscus. The sealed tube method cannot therefore be applied to such systems without modification. The effect of gravity on the distribution of matter in sealed tubes is discussed in an Appendix. The distribution is specifically calculated for the van der Waals mixture. The results show qualitatively the parameters involved and allow procedures to be evolved for extending the sealed tube method to binary mixtures. Near the critical point, the distribution of material under gravity can be expressed as odd functions of the distance from the disappearing meniscus. Equation (A.lO) in the Appendix then shows that when the meniscus disappears at the middle of the tube, x’J and pm will equal the mean values Xz and p. The critical temperature as well as the critical volume can therefore be obtained from experimental quantities. It is, however, experimentally difficult to obtain the exact correct tilling, and so the value of L and T” corresponding to l/L = + have to be interpolated from data over a range of l/L. From equation (A. 15), the temperature at which the meniscus disappears depends not only on the mean density in the tube but also on the tube length. Thus, given X,, there are two parameters, namely wr or w2 and L, that are experimentally variable. In order to interpolate, either the tube length has to be kept constant and the amount of mixture in the tube varied, or the amount of mixture has to be kept constant and the tube length varied. In this work the latter method was used. EXPERIMENTAL Preparation of the Sample Tube A known weight of solid was degassed in a thick-walled glass capillary tube by alternately meit& sphere of dried, oxygen-free hydrogen

under an atmoand evacuating.

Critical temperatures

and volumes of some binary systems (ii) This sample of benzene was supplied by the Chemical Research Laboratorv, and had a stated purity of 99.990 + OGO2moles per centfrom freezing point measurement. Both samples of benzene gave the same critical temperature of 288.5”C, which differed significantly from other recent literature values of 288-7 [2], 288.8 [13] and 288.94”C [14]. They were used indiscriminately in making the mixtures. Naphthalene. (i) Molecular weight quality naphthalene, supplied by British Drug House Ltd., had been previously purified in this laboratory for solubility work [15]. Its melting point was 80.1”C. (ii) A purified sample was kindly given to us by North Thames Gas Board. Its purity was 99.978 i 0010 moles ner cent [16]. Thecritical temperature of various samples of-naphthalene (i) gave widely varying temperatures within the range 474.9-476.7”C according to treatment, showing mixture behaviour and signs of decomposition. Samples of naphthalene (ii) did not show mixture behaviour. but became slightly discoloured ; the disappearing temperature rose slightly with time at the rate of about O.l”C hr-l and extrapolated to zero time to give a critical temperature of 473~4°C. This differed from the recent literature value of 4752°C [14]. Sample (i) was used in cyclohexane + naphthalene mixtures, while both samples were used in benzene + naphthalene mixtures. Phenanthrene. Laboratory reagent grade phenanthrene, supplied by British Drug House Ltd., was recrystallized from methylated spirit four times, chromatographed in benzene through alumina columns and vacuum sublimed. Its melting point was 99.O”C. The fluorescence spectrum showed that there were not more than @l moles per cent of anthracene present. Anthracene. General purpose reagent grade anthracene, supplied by Hopkin and Williams Ltd., was purified as phenantbrene. Its melting point was 216.5”C.

Degassed liquid was then distilled into the tube and the samnle sealed off at a preformed constriction. The tubes were from 5 to 10 cm long; internal diameters were about 2 mm and walls 3 mm. It was estimated that the error in -?‘awas not more than 0.02. Measurement of critical temperatures The disappearing meniscus was observed by immersing the sample tube, placed in a stainless steel holder, in a molten nitrite + nitrate bath. The bath was well insulated and stirred vigorously; heating was by a controlled immersion heater. An observation window was provided at the front of the salt bath; another was placed towards the back for illumination bv sodium light 1111. After the tube had been incubated for some time ai a temperature at which there was no meniscus, the temperature was alternately lowered and raised slowly (at about O.Ol”C mit-l) and the temperature at which the meniscus appeared and disappeared measured on a calibrated thermocouple previously described [ll]. The position of the disappearing meniscus was also obtained. In the early stages of this work, the fractional position of the disappearing meniscus (of cyclohexane + naphthalene mixtures) was obtained by eye, but for the rest of the work it was calculated from the positions of the meniscus and the top and bottom of the capillary space measured against graduation marks cut on the stainless steel holder. These observations were repeated several times. the (tm, Z/L)being separated by periods of incubation of an hour or two to test that equilibrium had been reached. The length of the tube was then reduced by cooling its lower part in liquid air and heating the tip with an oxygen torch until the glass was drawn in to close the capillary space. (t*, Z/L)was again measured. The process was repeated for different I.. The critical temperature corresponding to the mean composition was determined from the plot of (tm, Z/L) by interpolating at, or sometimes extrapolating to Z/L = 4. It was estimated that the uncertainty in tc was about 0.5”C. (Critical temperatures of pure substances were accurate to + O.l”C.)

&.SULTS

Behaviour of sealed tubes-transient

Measurement of critical volumes The critical volume corresponding to ZZ was calculated from the internal diameter of the capillary tube, the known weights of materials in the tube and LC obtained from the plot of (L, Z/L) at Z/L = 4. The expected error in vc was about 10 per cent. Materials Cyclohexane. Spectrosol cyclohexane, supplied by Hopkm and Williams Ltd., was fractionally crystallized until the refractive indices of the crystals and liquor were the same [12]. It was dried by storage over sodium wire. The final samnle had a normal boiling point of 807-80.8”C and n% of 1:4261. Its critical temper&e of 280.2”C compared wei with the recent literature results of 280.2 [2] and 280.3”C t131. Benzene. (i) Analar grade benzene, supplied by British Drug House Ltd., was fractionally crystallized as above. The crvstal fraction was distilled, dried and stored in the presence of sodium wire. It had a normal boiling point of 801°C and a% of l-50(17.

behaviour

When a sealed tube was first put into the salt bath and alternately heated and cooled, the meniscus was observed to disappear and appear, the position of the disappearing meniscus and the temperature at which the mensicus disappeared altering with time. The first temperatures could differ from t” by up to 15°C. If the tube was underfilled, both the position and the temperature fell with time; if over-filled, they rose. The disappearance and appearance were slow processes. The meniscus on disappearing became progressively flatter, thinner and indistinct. Total internal reflexion from the under side of the meniscus eventually ceased to take place. But the refractivities above the below the meniscus were quite obviously different. The zone of high refractivity gradient persisted at the position of the meniscus as the temperature was raised. If the temperature

717

D. C-H. &JJNO

was held at the value at which the meniscus had just disappeared, a sharp meniscus reformed after a time. If the tube filling was greatly different from the critical, the meniscus would move out of the tube while in the transient stage. If, however, the filling was not very different from the critical, the meniscus would move, within a few hours, to a position where it would disappear and appear with a temperature that was substantially constant when the bath temperature was slowly varied about this value. The disappearance of the meniscus was still indistinct and less well defined than the appearance, but the zone of high refractivity gradient was soon gone after the disappearance. Attainment

of equilibrium

When the sample tube was left undisturbed at a temperature above the disappearance temperature for some time, and then the temperature was lowered slowly, the appearance of the meniscus, exactly like that of a tube of pure substance, was preceded by a white opalescence, the extent of which depended on the length of incubation. It might be about 1 mm thick, or, if the tube had been incubated for a long time (3648 hr), it might extend over the whole of the tube. This region became opaque at about the middle of its expanse, and finally resolved into two parts, with the thin, flat and rippling meniscus in between them. Raining in the vapour phase and bubbling in the liquid phase then followed and the meniscus might rise or fall to a steady position at the end of the raining and bubbling period. The appearance temperature and the final meniscus position might be slightly different from the first “steady” values. The temperature might be about 0.5”C higher, but the meniscus would take up a slightly lower position if the tube was under-tilled and vice versa. The rates of equilibration depended not only on the components in the mixture but also on the length of the tube and the mean composition in it. Thus while tubes of cyclohexane or benzene + naphthalene came to equilibrium relatively rapidly, within about 12 to 24 .hr, tubes of benzene + phenanthrene or anthracene took about twice as long. In general, the longer the tube the longer it took to equilibrate. Also, it was more difficult

to establish equilibrium with samples of about 50 : 50 in composition compared with tubes of low concentration in one of the components. The final attainment of equilibrium was at a rate much slower than the initial rates. Normally over the “final” 18 hr, the disappearance temperature did not alter by more than 0*5”C, while the fractional meniscus position by about 4 0*02O-03. When l/L was less than 0.5, t”’ and I/L vary together but did not seem to depart significantly from the equilibrium (P, l/L) curve. But when I/L was over 0.5, the equilibrium curve was much flatter than that given by points away from equilibrium. And further, when a tube was sufficiently over-filled, although initially the meniscus might disappear at, say, about I/L = 0.7, after a prolonged period of incubation, when the temperature was lowered, the meniscus would reappear with extensive opalescence at a value of I/L approaching 0.8 or 0.9, but with a temperature not greatly different from that when Z/L = O-7. Equilibrium behaviour

As the tube length was reduced, the meniscus disappeared higher and higher in the tube, while the disappearance temperature fell. The temperature was very nearly a linear function of tube length (Fig. 1). The fractional meniscus position, however, varied slowly at fust with the tube length as it was reduced, but later, small changes in the tube length gave rise to large changes in the fractional position (Fig. 2). The corresponding variation of temperature with fractional position is shown in Fig. 3. Critical temperatures

and volumes

The critical temperatures and volumes obtained are given in Table 1. The temperatures showed a positive deviation from the linear (x,t;, + x&), and can be fitted to a quadratic equation in mole fractions : tC = xltfl

+ x&z

+ x,x$3

(13)

The volumes showed a definite negative deviation from (xlvfi + x&), and can also be fitted to a quadratic equation

718

Critical

temperaturesand volumesof some binary systems

5

w 0

2.5

- 2.5

-5 FIG.

0 cyclohexane + naphthalene 0 benzene + naphthalene (i) l benzene + naphthalene (ii) + benzene + phenanthrene x benzene + anthracene

temperature as a function of tube length.

FIG. 1. Disappearing meniscus

Xx = 2s = f’a = 18 = 2s =

cyclohexane + naphthalene benzene + naphthalene (i) benzene + naphthalene (ii) benzene + phenanthrene benzene + anthracene

-6

0414 0.174 0.104 O-107 0.103

-

.4 -

*2 -

0

i 2

0

6

4 L,

e

IO

cm

FIG. 2. Fractional meniscus position as a function of tube length.

q cyclohexane + naphthalene 0 benzene l benzene + benzene x benzene

+ + + +

naphthalene (i) naphthalene (ii) phenanthrene anthracene

28 = X8 = 28 = %a= 22 =

0.414 o-174 0.104 o-107 0.103

3. Disappearing meniscus temperature as a function of fractional meniscus position. %2z = 28 = 28 = za = Rs =

0414 0.174 O-104 o-107 o-103

The mean values of 8 and 4 calculated from the data are also included in the table. The critical volumes of cyclohexane, benzene and naphthalene are taken from the literature [17-191. Most of the compositions were relatively stable and were only slightly tinted in the salt bath. But tubes of cyclohexane + naphthalene of E, = O-748 and 0.800 and of benzene + anthracene of XZ = 0.224 were badly discoloured. Tubes of benzene + phenanthrene of XZ greater than 0.15 gave disappearing temperatures that increased with time. The rates of increase were measured and the results at these compositions have been corrected by extrapolation to zero time. Two sets of data were obtained for benzene + naphthalene because two different samples of naphthalene were used. The two sets of critical volumes did not differ significantly, but the critical temperatures did; naphthalene (i) giving a mean 8 about 10°C higher than naphthalene (ii). This was very probably due to impurity in the former. It is likely that 8 .of cyclohexane + naphthalene is also too high by a figure of the same order of magnitude. 719

D. C-H. CHENG Table 1. Critical temperatures and volumes UC

(cma[g mole]-1)

0

0.215 0.245 0308 0.414 0.540 1

280.2 331-o 3371 351.5 313.1 391.9 413.4

cyclohexane + naphthalene 308 543 303 51.8 323 548 316 349 55.9 53-5 345 408

4

(cm3[g mole]-l)

157 53 108 0 70 78

541 benzene + naphthalene (i) 0 0.143 0.174 O-242 0458 0.413 O-682

288.5 324.7 332.9 348.7 394.0 391.4 431.9

260 211 282 212 302 307 346

19.6 849 841 83.5 85.9 795

81 29 128 106 94 68

829

benzene + naphthalene (ii) 0.099

313.2

0.104 0320 0.341 0.679 1

314.4 364.1 368.0 430.0 473.4

257 257 291 298 345 408

120 71.1 75.4 73.4 73.4

100

13.1

0 0.090 0.106 0.101 0.151 0.266 0.291 1

L48 0.064 0.087 0.103 1

288.5 340.5 348 350.5 3125 428 438.5 596.1

288.5 3165 324.5 341 348.5 600

196 195 76 54 70

benzene + phenanthrene 260 265 247 269 256 284 289 (554)

267 464 240 379 277 274 -

293.6

317

benzene + anthracene 260 2927 245 260 268.4 317.4 256 303.4 254 (554)

634 322 369 393

295.5

720

430

Critical temperaturesand volumes of some binary systems tzZ and tI of benzene + naphthalene were calculated statistically using the principle of least

LI

squares [20]. But because of the scatter and scarcity of the data, when the same method was applied * to the critical temperatures of benzene + anthracene and to the critical volumes of both these systems, unrealistic values oft&, 8, viz and 4 were obtained. A reasonable fit of the critical tempera- 0 tures of benzene + anthracene was obtained from t;* = 600°C (calculated from the ratio 7”/Tb = FIG. 5. Changes in equilibrium position of the disappearing meniscus as l-41 8 of phenanthrene and a boiling point of 342°C for anthracene [21]). The critical volumes were correlated using z& = 554 cm3 (g mole)- l, esti- mean composition X,. The position of the meniscus mated from LYDERSEN’Smethod [18]. The two would be high in an overfilled tube and vice versa. In the transient stage the exact behaviour is values of 4 obtained were not significantly differdetermined by the rates of diffusion and of temperaent. ture rise. As the tube equilibrates the compositions of the phases approach each other and the mean DISCUSSION value, and so the vapour composition increases From the knowledge of the existence of density while the liquid composition decreases. In an gradients in pure substances near the critical point, over-filled tube the locus of the disappearing the existence of composition gradients in mixtures meniscus would tend towards E, from the low and hence varying t’” in tubes of binary mixtures composition side, and t’” would increase with were predicted. The latter conclusion was demontime. Similarly, in an under-filled tube, t” decreases strated experimentally. Similar observations were with time. If the rate of heating is in excess of the made by CENTNERSZWER [9, p.6571. PARTINGTON rate of diffusion, local diffusion across the meniscus et al. [2] did not observe this phenomenon pro- can result in its disappearance. But if the temperabably because their mixtures were made up of ture is held steady, and diffusion in the bulk phases components with similar molecular weights and allowed to catch up, the meniscus will reform. critical temperatures. The observed behaviour can In the equilibrium state, if the equation of state be qualitatively explained in some detail in terms of the system is known, the problem is capable of of composition distributions. solution in principle as is shown in the Appendix. The distribution has to satisfy equation (A.6). Qualitatively, as the tube length is reduced, the The composition profile of the heavier component equilibrium distribution changes such that the 2(M, > M1 and tqz > t$& at t” is expected to be meniscus position rises (Fig. 5), hence X; as well as something like those in Fig. 4. It is divided by the t” decreases. The four systems studied have critical temperatures that show large positive deviation from the linear (x1 t; 1 + xZti2). This illustrates the generalization that binary mixtures whose components differ greatly in critical temperatures or volumes show large deviations from the linear [l, p.2161. Further, the critical temperatures are quadratic functions of mole fraction as required by the theory of random mixtures. The critical temperatures of the polynuclear 92 aromatic molecules can be compared with those FIG. 4. Changes in composition profile with time in the predicted by LYDERSENmethod [18] (Table 2). transient stage. 721

D. C-H. CHBNG Table 2. Experimental and calculated tc Cal. tc

expt tc

(“Cl

(“C)

288 465 606 609

288 473 596 600

80 218 340 342

benzene naphthalene phenanthrene anthracene

Table 3. Experimental and theoretical 8

expt e (“c)

cyclohexane + naphthalene benzene + naphthalene benzene + phenanthrene benzene + anthracene

54.1 73.1 293.4 295.5

The properties of anthracene, both ti2 and 0, seem to be just larger than those of phenanthrene. The same seems true of the boiling point, although literature values of the boiling point of anthracene vary from just under 340 to 360°C depending on authors [19, 21, 221. Because of the difference in shapes of the molecules it is possible that there may be a large difference in the boiling points. This effect might still be appreciable at the critical point, but the uncertainty in the critical data due to the scatter does not allow the differences in t& and 9 to be very significant. It will be assumed that for these systems of components of different energies and sizes the difference between T’ and T,’ is given by the sum of the contributions (4) and (5). Choosing either of the components as the reference substance does not alter the derived values significantly, and, T”,,= j(Tfl + T,C,) and z& = +(vfl + v&) will be used. The theoretical 8 can then be calculated from the combining rules. (The polarizabilities of cyclohexane, benzene and naphthalene were taken from SIMONS and HICKMAN [23]. For anthracene and phenanthrene, a value of 25 x 1O-24 cm3, estimated from those of benzene and naphthalene, was used.) The differences between the experimental and theoretical 8’s (Table 3) mean that the fi2’s are larger than the values given by the combining rules. This can be interpreted as indicating strong interactions between the unlike molecules. I-low-

cal. elsT&

cal. e

(“Cl

(“Cl

-22.3 -33.9 -964 -97.6

13.2 3.6 1.3 1.9

ever, the experimental 0’s do seem unusually large. The calculation of the theoretical 0 is based on the theory of random mixtures which is applicable strictly only to spherically symmetrical molecules. It is probable that the large deviations represent special properties of the critical state of mixtures of components that differ largely in sizes and shapes. More refined theory that takes these factors into account will no doubt lead to better understanding of these large deviations. Acknowledgement-The author is indebted to Prof. A. R. this work was carried out; and to Dr. J. C. MCCOUBREYfor helpful discussions. Dr. D. EVANS determined the fluorescene spectrum of phenanthrene. UBBELOHDEunder whose supervision

APPENDIX Efect of gravity in sealed tubes The condition Ref. [24, p.851 or

of equilibrium

under gravity

& + Mtgr’ = p; + Migr’

is given by (A.11

z+Mtg=O The chemical potential is a function of temperature and pressure 124, p.891 dpt = -&dT + p!dp (-4.3) and so at constant given by dP

temperature Mag

d,=-r=-where

722

v = CS xrPI

the pressure %xtMrg V

gradient

is

(A.41 (A.51

Critical temperatures

and volumes of some binary systems

There are as many equations in each of the above expressions as there are number of components in the system. When the equation of state of the system is known these will give the distribution of any property as a function of height. The prevalent distribution in a sealed tube of given length and containing a given mass of material is further determined by the principle of conservation of matter

L

o xipd, = ZipL

The properties will be close to those at the meniscus and the 1.h.s. of (A.13) can be expanded in terms of -- X(P (Up)”

Then on solving for these and substituting into (A.6)

- _ -(xyp;m

(A.6)

1

1 = M&m

- MZKIZ gL

1 = MaKn

- MIKZI gL

(A.15) where both xi and p are functions of z. There are again as many equations in the expression as there are number of components. Under conditions at which themeniscus stillexists, SUPPOSing that the distributions of the components are (xtp)’

xtp =

+

F:(x---I),

z <

=(xtp)” + q(z-_I),

1 )

z> 1

_.

+ JfF”dz (A.@

at which the meniscus disappears

(xap)” = (xip)” = (~cp)~,

T = Tm

(A.9)

and (AS) becomes [xtp - (x&%

= s;y(z-l)dz

(A.10)

1

P+;

where the parameters a =

I

cc f,j=l,Z

(u-b)=RT

xixjbij,

bsj = bjt

i,j=1,2

The chemical potential can be obtained and on substituting into (A.l) ,n

I

--

X~P

1 -(bp)m

(xrp)” I-

zjz where

2

YUP +l--bp---

bp

(b3~

-

(YiCP)”

1 -(bp)m

,,-I~)

ycc = btr + (2bla-bn-ha)

= - $$(z-O (l-x$

x

2bij

1 _ cbp)m

+

-2s; I

NOTATION

(A.12) b = cc

Cap)”

The conditions at the meniscus are critical and are functions of Tm only. Rr and p are experimental quantities and can be expressed in terms of the dimensions of the tube and the weights of the components in it. Thus (A.15) relates wi or ws, L, l/L and Tm. Actual calculations [25] do not, however, agree quantitatively with experimental results. The important discrepancy between theory and experiment is that the calculated variation of Tm with I/L is negligibly small over the range 0 < I/L < 1. This is probably due to the variation of xs with p in a van der Waals mixture at the critical point being smaller than in real mixtures.

a and b are given by ai = a5r

h3 +

(YiiY33PF

(All)

xix3ai3,

=

11 -@pW

The i equations in (A.10) determine the position and the composition of the disappearing meniscus, and hence the temperature Tm. The relation between these are not given explicitly, but in general, when XT = it and p* = p, the meniscus does not disappear at the middle of the tube, that is, I does not equal +L, expect when F;“(z- I) are odd functions of (z--I). The above equations can readily be applied to a binary mixture obeying the van der Waals equation of state

(

i

where

&3

_

where (x~p)’ and (xcp)” are conditions just either side of the meniscus. Integration of (A.6) using (A.7) gives

At the temperature

bP)”

(A../)

1

[%,i - (xrp)“]L = [(~tp)~ - (xcp)“ll+/;F;dz

nzp

64.13) (A.14)

723

A Constant a, b Van der Waals parameters ela = 2fia-fn-far F Molecular distribution, equation (A.7) f = e/e00 = T=/Tgo B = o/u00 = (U”/U&p3 B Acceleration due to gravity Kg3 Coefficients defined by equation (A.6) L Length of sealed tube I Distance of the meniscus from the tube bottom zz M Molecular weight m,n Exponents of the Lennard-Jones potential P Pressure R Gas constant Molecular separation ; Partial molar entropy T Absolute temperature Tm Temperature at which the meniscus disappears t Temperature “C tm Temperature at which the meniscus disappears “C Molecular interaction potential Partial molar volume Molar volume cm3 (g mole)-l W Weight of material in sealed tube Mole fraction ; Defined by equation (A.14) Distance from the tube bottom; height cm is the polarixability ; = eoe/ors, where OL

D. C-H. CHENG Kronecker delta Characteristic energy of the Lennard-Jones potential Defined in equation (13) “C Chemical potential Molecular density Collision diameter of the Leonard-Jones potential Defined in equation (14) cm3 (g mole)-i Superscripts b Normal boiling point c Critical point

I Saturated liauid nhase Meniscus when f = Tm v Saturated vapour phase - Mean value in sealed tube ’ ” Coexisting phases Subscripts i,j Running suffixes for different species x Mixture of composition x 0 Reference substance 1,2 Components in a binary mixture, 2 being the heavier m

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