• No se han encontrado resultados

H. HILDEBRAND - THE THEORETICAL BASIS O F RAOULT’S LAW

N/A
N/A
Protected

Academic year: 2019

Share "H. HILDEBRAND - THE THEORETICAL BASIS O F RAOULT’S LAW"

Copied!
6
0
0

Texto completo

(1)

J. H. HILDEBRAND

THE

THEORETICAL BASIS

O F

RAOULT’S LAW.

BY E. A. GUGGENHEIM, M.A.

Received 27th July, 1936.

The foundations of the thermodynamics of ideal solutions were laid by van t’Hoff, but his treatment and formula ale applicable only to extremely dilute solutions.’ Although this restriction was realised by van t’Hoff himself, his laws have often been misapplied by later authors to concentrated solutions. In particulax his osmotic pressure formula which has the same form as the formula for the pressure of a perfect gas is even now regarded by some writers as an accurate law, although a t very high concentration of the solute i t becomes absurd, since i t gives

a finite value for the osmotic pressure in the limiting case of pure solute. The accurate expressions of the ideal laws applicable to solutions of any concentration were first given by van Laar,2 but his formulation of them was unnecessarily complicated. Their simple formulation is due to G. N. LewisS and to W a ~ h b u r n . ~ These ideal laws can, of course, be expressed in various forms thermodynamically equivalent t o one another. The simplest statement of these laws is that each species in a perfect or ideal solution obeys Raoult’s law in the form : partial vapour pressure1 mole fraction is constant a t constant temperature and external pressure. For a binary mixture containing n, moles of the species I , and n, moles of the species 2, the conditions that it should be perfect are then

Pl

= PlOnli(%

-!-

T22)

.

1 . 1

P2

= P2°n*l(n1

+

nz!

.

1’2

1 van’t Hoff, 2. physikal. Chem., 1887, I , 489. 2 van Laar, i b i d . , 1894, 15, 457.

Lewis, G. N., J . A m . Chem. SOL, 1908, 30, 665.

Washburn, 2. physikal. Cham., 1910, 74, 385.

Downloaded on 14 March 2013

Published on 01 January 1937 on http://pubs.rsc.org | doi:10.1039/TF9373300151

(2)

where

p,,

p,

denote the partial vapour pressures in the mixture and

plO,

p,O

denote the vapour pressures of the pure substances a t the same temperature and external pressure. From ( I . I ) , (1.2) i t is easy to deduce that the two species mix in all proportions a t constant temperature and pressure without any heat effect or volume change. But the converse is n o t true. The problem we want to discuss is the following: if two species mix in all proportions a t constant temperature and pressure without heat effect or volume change, what further conditions is necessary for mixtures to be perfect? We might hope to obtain an answer by means of statistical mechanics or thermodynamics or experiment. Actually the final answer is still unknown.

The application of statistical mechanics to the problem requires the use of some simple model for the molecules. The laws of perfect solutions have been deduced for a model of spherical molecules of approximately the same size; actually the sizes of the two species must be assumed sufficiently alike so that each molecule of either species is directly sur- rounded by the same number of other molecules.6 Up to the present no one has succeeded in applying statistical mechanics to a more general model.

The thermodynamic method of attack gives no definite answer, but i t is of value in showing the type of law that might be obeyed by mixtures ot molecules of different sizes. Let us denote by G the Gibbs function (or Gibbsian free energy) for a mixture of n, molecules of the species I and n, molecules of the species 2 a t the temperature

T

and the pressure P and by Gl0, G20 the Gibbs functions for one mole of each of the

two pure species a t the same temperature and pressure. Let us further define a function +(T, P, q, %) by the relation

G - n,G,O - n,G20 = RTJn,log n1

+

n2 log 2 2 -

+

+ ( T , P, n,, n,,}. 2.1

\

n , + n , n,+ n2

Then i t is easily shown that, if the two species mix in all proportions a t constant temperature and pressure without any heat effect, C$ is inde- pendent of

T ;

similarly, if they mix without volume change,

+

is in- dependent of

P.

For the type of mixtures that we are interested in, we may therefore write +(q, %) instead of C$(T,

P,

n,, H~). Moreover

+(%, This is all

that can be deduced from thermodynamics alone. From (2) we deduce for the partial vapour pressures the formulae :

must be homogeneous of the first degree in n,,

G.

But we know that kaoult's law (1.1) must be obeyed by species I in the limit that n2/n, + o and Raoult's law (1.2) must be obeyed by species z

in the limit that nl/n, + 0. To satisfy these conditions i t is necessary that

+

+

o

both when %/q --+

o

and when q/n, 3 0. If we assume that +(%,

3)

is a smooth function of the simplest possible kind, we can ex- press it as a series of the form

4 (al, n,) = n - 2 { A

,,

+

A %+"2

+

A

(=-)

%+ 2

+

.

.

.

+

A

( =')

ni

+

na

+ . . .

}

.

4.1

ti Guggenheim, Proc. Roy. SOC., A , 1932, 135, 181 : Proc. Roy. Sot., A , 1935,

148, 304.

Downloaded on 14 March 2013

(3)

E. A. GUGGENHEIM I53

This form of series is more symmetrical with respect to the two component species than the power series in n,/(n,

+

n2) used by van Laar. From (4)

we deduce

n,

-

(zr +I)%, n

f A r nl-957, (n:;;;)r

+

* *

.}.

5.2

The number of terms required in these series will be determined by the accuracy aimed a t and for most purposes two terms will be sufficient. We shall therefore set A,, A,, etc., equa.1 to zero. The formuke then reduce to

The question remains : what values might A, and A, reasonably have ? If the species I and 2 have identical sizes and shapes then

A,

and A, may safely be assumed equal to zero. reasons from this that the values of A, and A , will always be zero. (It must be remembered that we are considering only pairs of substances such as mix without heat effect or volume change.) But there is no obvious reason why A,, A, should not be determined by the ratio of the volumes of the two molecular species, becoming zero in the special case that this ratio is unity.

Hildebrand

Suppose, for instance, we tentatively set

.

8.1

where V1, V2 are the molar volumes of the two species at the same tem- perature and pressure. (If the two species have different coefficients of thermal expansion or different compressibilities, these values of A , and

A,

will not be absolutely independent of temperature and pressure, but the deviation from constancy may be neglected for ordinary varia- tions of temperature and presssure.) With these values for A,, A, we have

6 Hildebrand, Solubility, Reinhold Publishing Corporation, 1936, p. 20.

Downloaded on 14 March 2013

Published on 01 January 1937 on http://pubs.rsc.org | doi:10.1039/TF9373300151

(4)

Our formulae ( 7 . 1 ) ~ (7.2) for

p,,

p,

become

These rather unattractive formulae can be transformed into simple forms when the species in question is dilute. Thus when n,

<

n, formula

(10.1) becomes correct to terms in (nl/nl

+

n.J2 and neglecting terms in

t%/nl+

d3,

which can be written in the alternative form, correct to terms in

(%In1

+

%I2,

.

12.1

n,

p1

=

lZ,

q V 1 + %V,

where

Similarly when n,

<

rtl formula (10.2) is, correct to terms in (%/n1

+

n2)2,

equivalent to

.

12.2

n,

Pa=

hn

v

1 l+nzVz'

.

13.2

where

. .

We see then that our particular choice of values for A,, A, leads to Henry's law for the dilute component in the form: partial pressure of dilute component directly proportional to its volume concentration. A priuri, there i s no obvious theoretical reason why in dilute solution ( 1 2 . 1 ) ~ (12.2) should be more or less accurate than ( I . I), (1.2) and therefore the values (8.11, (8.2) chosen for A,, A, are not obviously less reasonable than the value zero usually assumed. That the values of A,, A, should be zero, is the simplest possible assumption, but it is only one amongst other possible assumptions.

The deviation between formulae (10.1)~ (10.2) on the one hand and formulae ( I . I ) , (1.2) on the other are measured by the exponentials which we may call activity coefficients and denote accordingly by fl, fi; thus

Downloaded on 14 March 2013

(5)

E. A. GUGGENHEIM 155

We give a table of values off,, f2 for various compositions taking the value of v2/u3 as 1.25. We see that the deviations of J1, f2 from unity are mostly less than I per cent. It follows

that even if the true laws were of TABLE O F fi A N D f 2 FOR 7'2/% = "2.5. the form (IO.I), ( I O . ~ ) , the usual

formulae ( I . I ) , (1.2) would be a very good approximation provided the molecular volumes do not differ by more than 25 per cent. For greater differences in size we cannot predict anything.

In the absence of any definite answer to our question either from statistical mechanics or from thermo-

-~ ~-

*l n F F Z

0'100 0.167 0 - 2 5 0 0.500 0.667 0.750 0.900 0-goo 0'833 0.750 0.500 0.333 0.250 0-100

f 1.

o-gg?,

1.007

1.008 1.006 I '000

1'012

1'0001 f 2.

0.9996 0.998 0.996 0.994 1.007 1.028

I '000

dynamics there remains the test of

experiment. It is well known that mixtures do exist which obey the laws of perfect solutions in the forms ( I - I ) , (1.2). It is probably less widely realised how scanty are the experimental data. The only examples for which these laws have been accurately established are ethylene bromide

+

propylene bromide and benzene

+

ethylene chloride, both mixtures having been studied by von Zawidski.7 The following mixtures were investigated by Young and Fortey : 8 chlorobenzene

+

bromobenzene, ethyl acetate

+

ethyl propionate, toluene

+

ethyl benzene, n-hexane

+

n-octane, benzene

+

toluene. They measured in each case the boiling-points of mixtures of molecular proportions I : 3, I : I and 3 : I a t various pressures. By interpolation they obtained the total vapour pressures a t given tem- peratures and compared these with the values computed according to the laws of perfect solutions. They found exact agreement for the mixture chlorobenzene

+

bromobenzene and small but definite deviations for all the other mixtures. It is to be noted that they obtained no data for the partial vapour pressures of the two constituents. of the mixture. Measurements were made by Linebarger

*

on the mixtures benzene

+

chlorobenzene, benzene

+

bromobenzene, toluene

+

chlorobenzene, toluene

+

bromoben- zene. Young and Fortey lo referring t o these measurements write : " The method employed by Linebarger does not appear to give good results with volatile liquids, and his observed vapour pressures differ somewhat widely from those of Lehfeldt and Zawidski." More recently Calingaert and Hitchcockll have studied the mixtures butane

+

pentane, butane

+

hep- tane, pentane

+

heptane. They did not measure the partial vapour pressures directly, but studied the pressure

-

volume - temperature relations and deduced thereform the partial vapour pressures by a method both ingenious and theoretically sound, but of such a nature that a small error in the data (such as due to a trace of impurity) might lead to large errors in the final results of the computations. Their data extend only up to 20 per cent. of the lighter constituent for the first two mixtures and

up t o 35 per cent. in the third. For the mixtures butane

+

heptane they find agreement with Raoult's law in the form ( I - I ) , (1*2), for the other two mixtures they find disagreement. Their partial pressure curves for the latter two mixtures cannot be correct as they contradict the Gibbs-Duhem- Margules relation. The only quantitative evidence referred to by Hildebrand6 in support of Raoult's law is the single sentence, " It is

worth noting that the system butane-heptane obeys Raoult's law very

Young, J . Chem. SOC., 1902, 81, 768 : Young and Fortey, ibid., 1903, 83, 45.

Linebarger, J . -4m. Chem. SOC., 189.5, 17, 615, 690. (Unfortunately I have Calingaert a n d Hitchccck, J . A m . Chem. Soc., 1927, 49, 750.

-

'I von Zawidski, 2. plysikal. Chem., 1900, 35, 128.

not been able t o obtain access t o these papers.) l o Young and Fortey, J . Chem. SOC., 1903, 83, 6 3 .

Downloaded on 14 March 2013

Published on 01 January 1937 on http://pubs.rsc.org | doi:10.1039/TF9373300151

(6)

accurately ” with a reference to Calingaert and Hitchcock. N o mention

is made of deviations found by the same authors for the systems butane

+

pentane and pentane

+

heptane, nor of the fact that the data extend only from o per cent. to 20 per cent. of butane.

In short, the only reliable, accurate and direct evidence for partial vapour pressures of a binary mixture obeying the laws (I-I), (IYZ), are von

Zawidski’s data for ethylene-bromide

+

propylene bromide and for benzene

+

ethylene chloride. The ratios of the molar volumes are for propylene bromide/ethylene bromide 104/86 = I ‘2 I and for ethylene chloride/benzene 94/89 = 1.06. Even for the former mixture the distinction between formulae (I*I), (I-z), on the one hand and f o m u l z (10-I), (IOQ), would be

hardly detectable. The same applies to the mixtures studied by Young and Fortey. I think i t would be of great interest to have data for mixtures such as benzene

+

diphenyl or benzene

+

triphenylbenzene and I am

planning to make measurements on mixtures of this type.

The S i r William Ranzsay Laboratories

of Inorganic and Physicnl Chemistry, University College,

Laandon *

Downloaded on 14 March 2013

Referencias

Documento similar

The private field itself, so important and not included in the PSB Law, could have only been approached from a more general law that also regulated

Later, in this section, after having defined, for a set of sorts S and an S-sorted signature Σ, the concept of Σ-algebra, for a Σ-algebra A = (A, F ), the uniform algebraic

No diffraction peaks corresponding to the isolated components have been observed for both DA:HSP and DA:SP system, which suggests the formation of mixed homogeneous

Although these refinements of the parameters (which are expected to reinforce the difference in radius of HD 209458b and TrES- 1, thus providing a theoretical challenge for

For the non-substituted thiourea and all the studied derivatives, besides the strong S-Au bond, it has been observed an Au· · ·H-N interaction similar to a hydrogen bonding, which

On the one hand, theoretical simulations carried out for H 2 /Pd(111) and H 2 /NiA/(110) have shown that a large proportion of the incoming molecules are rotationally excited

So, in our design all these factors were the same in both groups, except for the family environment (the treatment variable) in which the subject has been brought up

The marginalized posterior distribution of f for models built upon the T1 simulation, which have different prescriptions for how central and satellite velocities are assigned