lOGO
SIDXEY TV. BENSOS
The maximal absolute velocity corresponds to that calculated for a zeroorder reaction. The eutectic lead chloride-manganous chloride appears to be an exactly additive mixed catalyst, but in the mixed crystal barium chloride-lead chloride slight promotion is detected. I n the compound BahlnCld the activation energy is lowered to the value of the endothermic heat of reaction. Tn-o of these effects are tentatively explained as due to an increase of the dipole moment of the active doublet. The authors wish to t,hank Dr. G. Drikos for construction of the apparatus and preliminary experiments, and especially Prof. G. Rlat'thaeopoulos and Dr. Nakris, who for a year offered generous hospitality to our air-damaged laboratory and thus enabled us t o finish this investigation. REFERESCES (1) BALAXDIS, h.,A S U Li\-.mov.%,0.: Pci. Repts. State Univ. 11oscoil- 2, 237 (1934,; Zentr. 1936, 11, lj.28. (2) CRAXFORU, S.R . : T r a n s . F a r a d a y Poc. 42, 576, 580 (19461. &:.: ItBunion i n t e r n . phys. chini., P a r i s 1928, 214. (3) GRIlI1f. 13. G . , A S U SCHWAJIBERGER. (4) RVDKO~TSKI, D . 11..TRIFEI., A . G., .\SI) FROST, Ai.W.:t71xaYn.Khem. Zhur. 10, 277 (1935);Z e n t r . 1936, I, 3667. ( 5 ) SAXDOKSIXI, C . : . i t t i acrad. Liiicei 20, 11. 646 ( l o l l ) ; 21, I, 208 (1912). (6) SASDOXSISI, C.. ASU SLLRP.~, G . : A t t i accad. I h c e i 20, 11, 61 (1911).
Farads!- Soc. 42, 689 (1946). (S) SCHTAB, G.-AI.; ASI) C R I ; ~ : R E . . : %. pliysik. Chem. A144, 243 (1929); B6, 406 (1929). (9) SCHWAB, G.-11.. . ~ I DRIKOS, I G . : %. physik. Chem. A186, 405 (1940). 110) SCHW.AB, G.-?\I..< t s u SCHI-T.TES, 15.: %. physik. Chem. B9, 265 (1930). . : J . P h y s . Cliem. 60, 127 11946,. (7) SCHWAB. G . 3 1 . : Trans.
I. R E D U C E D DEXSITIES O F LIQUID5
Since van der TJ-aals first proposed the theory of corresponding btates in the last century, the concept has become a convenient tool in dealing with the thermodynamic properties of both gases a t high pressures and vapors. Because of the close relation between vapors and liquids and their identity above the critical temperature, most authors have tacitly assumed that both could be treated from the standpoint of corresponding states. The concept has thereby also gained wide usage in the field of liquids. The first theoretical justifications for the application of the law of corresponding states to gases and liquids were made recently by DeBoer and Xichels (2)
CRITICAL DENSITIES OF LIQUIDS
1061
and Pitzer (6), respectively. The arguments presented b y these authors indicate that the law of corresponding states may be used as a first approximation in dealing with gas imperfections and n-ith liquid properties. Pitzer has shown that for liquids the sufficient conditions for such an approximation are the assumption of a universal intermolecular potential of the form d U(R/Ro)and a time-average spherically symmetrical force field for molecules in a liquid. A and Ro are specific parameters for a given liquid and R is the intermolecular distance of a pair of molecules. Since R in a liquid is simply related t o the volume, we see that aside from the constant -4, the volume of a liquid is a convenient parameter in computing its physical properties. Pitzer's development is such that T and V become the simplest choices for independent variables of the liquid. This theoretical justification Jvas a long time overdue, since there has been a great deal of experimental evidence for applying the law of corresponding states to liquids. In addition there are good grounds for considering that the boiling point, T B ,is approximately a corresponding point. Empirical relations supporting this view of T Bare: 1. Trouton's rule: IS',‘^^^ = 21 cal./mole-"A. at T B , 2. Guye-Guldberg rule: T B / T , E O n 0.60 ( T , = critical temperature). (As we can see in tables 1-4, n-ith perhaps one or tn-o exceptions O B lies within a range of values from 0.57 to 0.70 Tvith a mean value a t about 0.63.) The additional fact that many approximate empirical expressions for liquids and vapors can be thron-n into a reduced form lends additional support to a corresponding law. Typical are the van der TTTaalsexpression for vapor pressure:
-
- 2.75 for most liquids); the Mathias and Cailletat law of ( rectilinear diameters Log T
=
a
k 1 - - (k
:
-
( a 1 for most liquids); and the various equations of state such as those of van der Waals, Dieterici, and Berthellot. The most recent I\-ork on reduced equations of state for liquids is the empirical representations of Bauer et a7. (1). By redefining reduced variables 0 and V to include the range from melting to critical, they get striking uniformity of representation for reduced volumes, compressibilities, and surface tensions for liquids through this range. Reduced densities at the boiling poi)zt I n the light of this information it was therefore not completely surprising when a study of reduced densities of liquids a t the boiling point showed fairly good
constancy. The data plotted in figure 1 and presented in tables 1-5 represent the best available critical values for ninety-six liquids. For normal liquids and even most liquids which are considered associated, it
10G2
230(
2 15C
2
ooc
I850
I700
I55C
I40C
I25C
I IOC
FIG.1. Plot of reduced liquid densities ut the boiling point against log I', (ntino.split,rrs I . (The dotted lincs Khich bound solid lines I and I1 reprebent t h c rurige of valurs. + 3 p r r cent .)
was found that the reduced density u t the boiling point, A B , is approsiinately constant with a mean value of 2.68. The vertical solid line labelled I in figure 1
CRITICAL DENSITIES O F LIQUIDS
represents this mean value. The two dashed lines parallel t o this line represent a range of values of =t3 per cent from 2.68.
11,
1lytlrog:r.h . . . . . . . , I le1 i i i n i . ....... , . . . . . . . Se o I i
Se
... I 11hen applied t o the liquid +tate. The foim of equation 1 ( a n he r l c r i ~etl 1)y combining van dri. Ii-aaL’ equcition for vapor prebmie?, log x = 2.’71(1 - 1, u i t h his equation of \tat? The latter 111 the J-icinity of the hoiling point ieduccc to equation 4. If \\e c\p;tnd rquation 1 11y the 1,inomial thcoiem a n d ncglect highrr-oitlw tc’i rn- in \ye find:
e))!
Ai, = 3.00 - o.s9(-)Jj
A B CALCULATED F R O Y \‘ASDER 1VA.U.S EQCATIUS (F.QU.\TION 4’
BH
0.57 0.60 0.63 0.65 0.70
2.35 2.30 2 25 2.22 2.11
(7)
Alj CILCCLATED IROM BERTHELLOT 1:Ql:ATIOS
(EQUAIIOS 6 )
2.68 2 64 2.60 2.56 2.47
S o w , substituting the value of O B from the vapor pressure equation this jecomes: -1,
=
3.00 -
0.89 ~~
1 - 0.37 log x
1,s 2.11 - 0.33 log x vhicli coniparea well with equation 1.
Deviations of liquid behatior f y o m equation I Pitzer (6) has defined a “normal” liquid a3 one which obeys the assumptions if time-average spherical symmetry and a universal potential function.
Such liquid could be represented by a reduced equation of state, and me see from urve I (figure 1) that to the first approximation A B = 2.68 may he taken :is a
t
1088
SIDNEY W. BENSON
point in such a representation. K e may interpret the nature of this approximation as assuming that normal liquids will have a common reduced boiling point of @ B = 0.63. From this point of view “associated” liquids will have lower reduced boiling temperatures and correspondingly higher reduced boiling densities.’ Equation 1 may then be looked upon as a better approximation which takes into account the fact that the boiling point is not quite a corresponding point. Thus it is not surprising to find an improved relation between A B and another of the reduced variables, in this case r B . As is evident from figure 1, equation 1 may be taken empirically as representing an expected behavior for liquids. It is then a matter of interest to understand the reasons for deviations from this behavior for the fen- liquids that do deviate. Direct visual observation of the critical yolume is almost never reliable because of the rapid change of volume in the vicinity of the critical temperature. For most liquids, the density undergoes an exponential change of from 20 per cent t o GO per cent in the 1°C‘. temperature interval preceding T,. This corresponds to a change of about 4-5 per cent in the 0.1”C. interval below T,. Because of this rapid change, reliable values of the critical volume are almost ah-ays made with the aid of the laxv of rectilinear diameters, extrapolating the mean density of liquid and vapor to the critical temperature. I n most cases it has been shown necessary to modify the law by adding higher-order temperature terms to obtain agreement better than 1 per cent (5, 8). For these reasonc it n-as decided to use 1 3 per cent as a point of departure in discussing the deviation of A B from equation 1. Four liquids have values of A B loIT-er than equation 1 predicts. They are helium (- 11 per cent), hydrogen (- 6.5 per cent), neon (-4 per cent), and phosphine (- 10 per cent). (Helium was not included in figure 1 for reasons of space conservation.) The lon- values for the first three are not surprising, since they have very low critical and boiling temperatures which lie in a region in n-hich they show quantum effects (6). The decreasing magnitude of the deviation Jrith increasing T , for these three liquids is in agreement n i t h such a viewpoint. The loir value for phosphine is surprising since its structure, relatively high critical pressure (64.5 atm.), and low reduced temperature ( 0 . 5 i )~vouldseem t o align it with associated compounds which tend t o have high reduced denqities. The orthobaric densities reported for i t are of poor accuracy ( & 5 per cent or worse) and it is quite probable that the deviation is due to experimental error rather than u real physical effect.’ There are nine liquids ivhich show positive deviations from equation 1. Three of these are acetone ( + 4 per cent), isobutyl formate (+4 per cent), andisobutyl acetate (+43 per cent), represented in figure 1 by the three circles just outside of the dashed line parallel to 11. For all of these it seems very likely that the 1 For Pitzei’s “peifert liquid,” a t c j B = 0 63 hc gives t h e values l / r = ~ 23 1, AB = 2.53, arid AS,,, = 15 6 cal /iiiole-cA * Such predictions were made in the couise of this s t u d ? io1 man) critical values taken from t h e Inte, ? i u t i o ~Cr ~ l~ t z c a lTables a n d Landolt-Bornstem T h e j were invariably confirmed 11 hen further research i n t h e literature uncovered more recent (and more reliable) measurements of these same critical constants.
LRITIC.IL DESSITIES O F LIQL-ID>
1069
experimental values are in error. I n the case of tn-o additional liquids, ethyl n-butyrate and ethyl isobutyrate, both with deviations of 4- 3 per cent (double circle figure I ) , it is almost certain that the experimental values are in error hy this amount. The four remaining liquids shon-ing positive deviations are carbon disulfide ($23 per cent), hydrogen cyanide (+23 per cent), acetonitrile ( t l l per cent), and propionitrile (+8 per cent). The high value for carbon disulfide may he understandable on the basis of a dense packing in the liquid structure which is compatible with its linear structure and the chain-forming tendency of the sulfur atoms due to the availability of 3d orbitals. This \\-auld be in agreement with similar observations which have been made on both this liquid and its analogue CJeS by Frank (3) and Hildebrand (4). This is apparently cooperative phenomenon for the liqiiid state, since the vapor densities at the boiling point seem to be almost “normal”. The low reduced boiling temperature ( O B = 0.58) and the high critical pressure ( i 5 atm.) are both characteristic of such association in the liquid. The extent of the deviation ( + 2 3 per cent), however, seems rather large and it would be worthwhile to repeat the experimental determinations of the orthoharic densities for thi; liquid. The relatively large deviations of the nitriles and their regular change \\.it11 decreasing molecular n-eight indicate that these deviations are not due to experimental error but are real effects. From the viewpoint of equation 1 they seem t o form a special class of liquids. Hydrogen cyanide is unquestionably unique among liquids. Quite aside from its chemical properties and instability (explosive), its physical properties are extreme in almost all respect?. Its dielectric constant (95) is the largest linou-n. T t has an abnormally high boiling point (25.SOC.) for its molecular weight of 27. I t s critical ratio (RTc/’PcT7c) is 5.36, again the highest linoirn ralue for a liquid, indicating considerable association even at the critical temperature. (The expected value for bbnormal”liquids is 3 . 7 . ) -Us0 significant is its vapor density at the boiling point (1.19 g., l.), which gives a “calculated” molecular weight for the 7-apor of 34. (This is a deviation of 26 per cent compared to expected deviations of 5 per cent for most liquids, and indicates a high degree of association in the vapor state.) Its entropy of vaporization at the boiling point is 22.2 cal.1 mole-”A.,a value IT liich is not high but in fact is almost normal. However, if this r-alue is corrected for vapor association, the corrected entropy of 27 cal./mole-OA1. :ompares with that of water. Further evidence for the abnormality of the nitriles compared to other liquids b to be found in their high dipole moments. Table 7 shows the dipole moments ~f some associated compounds. These values for the physical constants of hydrogen cyanide indicate a possible xplanation for the unique position n-hich its reduced boiling density occupies n figure d ( A B = 3.4). The value for A B is compatible with the simultaneous xistence of association for hydrogen cyanide both in the liquid and in the vapor.
If we assume a high tlcgrcc of :issociation in lwth licluid and ~ - a p oeven r up to the critical point, these will tend to compensate each other and tend to make almost normal such constants a\ 0 , ( 0 . 6 2 ) , the critical pressure (50 atm.), and the entropy of vaporization (22.2 c ~ ~/mole-"-L). l. The responsibility for the high reduced boiling density may then be placed i ~ p o nthe existence of long chains which pack closely in the liquid. The linear 5triicture of hydrogen cyanide and its high dipole moment both favor such an explanation, The same explanation would also account for the behavior of the methyl and ethyl c y m i d e ~ . 'I'hc latter of course show milch qmaller discrepancies than the hydrogen cyanide. D I'HTSIC .iL PROPERTIE5
Tables 1-5 show that I\-ith the exception of strongly associated liquid>, A B = 2.138. For the larger number of "regular" liquids which fit thib relation, the density at the boiling point m;ty be used to compute their critical volumes with an accuracy of 1 3 per cent, These computed values of the critical volume can then in turn be used to calculate, with similar a(wracy, other physical constlints 1)ipole vionients oj .some associated compourrtls ( 7 ) . . . . .
.....
........
.
CO>IIPOUSV
DIPOLE
.
I .Si
. . . . . . . . . . . . . . . . .
1.53
-
X'
Acetoiir
.
.
uomsr
~~~
1.08 2.8 3.4 3 .4 1.; 2 .oo 2.07 ~
.
p1ic1i ;is the van der T V d - ' h. F i u ~ h c rii, t h r caritical temperature is luion-n, then the constant 01 in thc 1:~wof leetilinear diameters and the coefficient of expansion of the liquid may br calculated. Conversely, if the critical temperature is not linon n but the coef€ic+nt of liquid expansion is known, then both the critical temper:iture and the constant a may be calculated.
T'an der 11-aals' c o t i s t m f 0 a n d oiticul columes
The critical volumei of regular liquids may be computed froin the density at the boiling point hy mean.; of the relation :
For regular liquids, the ratio of V , to b is about 2.20 (see table 8). Substitiiting this into equation 9 we find for h :
In table 8 valueb oi I', and b computed from equations (3 and 10, re+pectively, are compared with observed values for representative liquids taken from the literature. Law oJ' wctzlznear diameters and coeficieiat o j 1zqiud expanszou 'I'he equation for the lan- of rectilinear diameterq, 1)L
may be solved for
(Y
+ I), = 2D, + a ( T , - 2')
(11)
At the boiling point the
in terms of the other quantitie..
COMPUUSD
- .-
-
.
-
-.
~
,
cc./flLole
.lcctonc . . . . . . . . . . . . . . . . . . . . . . . . . . .
216
--
2i6
76.4 256 2iS
123,s 312 208
122 314 293
13.3
256
141 286 105 'S2 1:33,2 56. I :XI7
no. 0 74.4 122.2
cC.,'777O/C
99.4 32.2 115.4
20s
95 35 116 128 56 142 1x3
13s , 56.2 142.4 1 39 . 2
6i
63.8 141.2 86.5 1:34.-1 37.1
14i
2% 1nt 285 I32 S2 31G 92.\
130 S6
130 60 35 1-18 -12 3 ?J 54
10.x 141. i 39.1
--0
51 . s X.1
I
11s ~
~
~~. ~
2.18 2.33 2.22 1 .OD 2.20 2.19 2.14 2.21 2.02 2.25 2.10 2 .:33 2.11 2.17 2 ,:30 2 , :3:3 2.17
~
. . ~
gas density (Do) is negligible compared to the density of the liquid espresbion for (Y reduces to:
~
(Ill)
-
ant1 the
Table 9 sho\\-s the value* oi cy. calculated from equation 12 with values taken from the literature. The average absolute deviation is about 4 per cent. By differentiating equation 11 11-ith respect t o 1 ' Tve obtain a relation between Y and the coefficients of' expansion of the liquid and vapor:
Thermodynamically, it may be shown that :
10i2
SIDNEY W. BEXSON
A t the boiling point, n e may nbsiime the vapor to be nearly ideal and also neglect the'volume of the liquid compared to the vapor. Equation 14 then becomes :
=
2 (EJ I'
AEb
-
(15)
-Applying Trouton's rule, (L@/R?')~ 9.5, also Pt, = 1 atm., and finally multiplying by 1.05 to take into account the average deviation of vapors from ideality at the boiling point, n-c have :
T-I131,12!I P o i n i ~ ~ l e from rs the iuii
(I)
rectilinear diameters
JI is the molecular weight, T b is tlie boiling point in O X . , and (dD,/aT)s,t, i.; jn grams per cubic centimeter. S o t e also that it is positive in contrast t o the negative coefficient obtained for liquids. Substituting from cquation 16 into equation 13, we find:
(
Tlie partial aD1 z7)is always negative, so that cz is alivays greater than this quantity.
(aI =
sat.
-
1
aD (--')
D L dT
sat.
= coefficient of liquid expansion.)
CRITICAL D E S S I T I E S O F LIQUIDS
1073
Equation 17 thus provides a method of calculating CY from the coefficient of expansion of the liquid at the boiling point. I n table 9 values of calculated from this relation are compared with observed values. The agreement is better than that obtained n-ith equation 12. The average absolute deviation is about 2.3 per cent. Equations 12 and 17 may be combined and solred for T C :
18)
Equation 18 malies it possible to calculate T,filom quantities measured a t the boiling point, the density and the coefficient of expansion of the liquid. Yalues of T,thus computed are shon-n in table 9. For these regular liquids, the average absolute deviation of T , (calculated) is about 7”A. or about 2 per cent. SUMMARY
I . For most liquids, the reduced density at the boiling point is equal to 2.68
3 per cent. Only hydrogen, helium, neon, and phosphine fall belon- this range of \-alues. 2 . The empirical relation A B = -0.422 log 7rB 1.981represents A B t o within 1 3 per cent for both associated and regular liquids and ma>-be taken as a representative liquid behavior. 3 . From the standpoint of ( 2 ) above, there are four very “abnormal” liquids.caihon disulfide, hydrogen cyanide. methyl cyanide. and ethyl cyanide. It is suggested that their abnormality arises from their ability to form cluse-paclied chains in the liquid. 4. For regular liquids, the ielation in (1) abore may be used t o compute critical volumes from densities at the boiling point: T’, = 2.68/Db. 5. Yalues for van der \T7aa1s’ constant b may be computed from 1 1 0 by the equation: b = 1.22 Db. ‘6. Yalues of the constant CY in the lan- of rectilinear diameters may be computed from Dh,7‘h, and 1’, by means of: =t
+
c y =
0.254L)b ~2‘, - l ‘ b
i . Yalues of CY may also be computed from Tb, Db, and the coefficient of espan;ion of the liquid at the boiling point ( a i ) from the relation: CY
= fflIj*-
8. Values of 7‘, m:ty be computed from l’c= ?‘a
+ 0.264 __ ff1
0.122JI
2’6,
[
I&, and +
CY^,
0*122,] T i Db CY^
using the equation:
1074
-4LEX.kKDER SCHdlVBERG -4SD 3IOHAJIICD Z . I l d
B.IR4KAT
;\IOLECULBR-ITEIGHT 1)E'I'ERAIISXTIOSd O F SOS-T~OL.YL'IIX SUBST-WCES B-iSED O S T H E ET'XPOR.~TIOSJ~ELOC'ITIES OF THEIR SOLUTIOKS .4T,ESA4SDER SCHOSBI3RC;
. 4 \ ~
JLOH.1hII~D Z.iI
