T H E E F F E C T OF SURFACE FORCES ON MELTING1 W. A . PATRICK
AND
W. A. KEMPER
Department of Chemistry, The Johns Hopkins University, Baltimore, Maryland Receiaed December 84, 1937
The preceding paper ( 5 ) described calorimetric observations from which the lowered melting temperatures of certain substances adsorbed on silica gel were inferred. In this paper calculations are made of the melting temperatures to be expected if capillary condensation is the cause of adsorption, and the calculations are compared with the calorimetric observations of the preceding paper. MELTING TEMPERATURES O F ADSORBED SUBSTANCES
It is well known (1) that certain adsorbents are supposed to have capillaries in which vapors condense to liquids, and that the vapor pressure of the condensate increases as the capillaries fill and the stress arising from surface forces decreases. The temperature at which this condensate will freeze depends upon how the solid forms. There are several conceivable ways in which the solid might form, each affecting the stresses on the two phases and consequently the freezing temperature, differently. The solid may form on the walls, narrowing the capillaries. It may form entirely within the body of the liquid, causing but little change in the surface of the liquid. Or, it may form by a process of evaporation and sublimation entirely outside the liquid. I n this case the vapor pressure of the liquid remaining would be that of the same amount of liquid with no solid present. The stresses on the solid formed would also be different in these cases. I n the last case, the vapor pressure of the small solid particles might be increased by a surface tension of the solid, if such a force exists. It will here be assumed that: (1) the stress o n the unfrozen sorbate i s independent of the amount offrozen sorbate present,2 and (2) the frozen sorbate is not under a n y stress f r o m surface forces. The melting temperature will then be calculated as the temperature a t which adsorbed liquid and normal solid have the same vapor pressure. This article is based on a dissertation submitted by W. A . Kemper to the Board of University Studies of the Johns Hopkins University in partial fulfillment of the requirements for the degree of Doctor of Philosophy, June, 1934. In this paper the assumptions made are printed in italics. 38 1
382
W. A. PATRICK AKD W. A. KEMPER
The vapor pressure of the adsorbed liquid, p , is estimated by the Patrick isotherm (6) XlVl
=
();"
I n this, in accord with assumption I, Xi, the amount of melt on the gel, is used in place of the total amount of sorbate, customarily used in the equation a1 temperatures above the melting point of the sorbate. u l , U, and 25 are the specific volume, the surface tension, and the normal vapor pressure of liquid sorbate at the temperature concerned. The subscript 1 denotes a property of the liquid state and the overrule (-) refers to the value for the unadsorbcd or normal state. k and n are constants viliich characterize the adsorbent. This equation with the same values of k and n gives the vapor pressure isotherm at any temperature and, with the exception of water, for any sorbate The normal vapor pressure of liquid, p, at temperature T , in terms of i t s value Ijmat the nornial melting point, is given by the Clapeyron equation
where ti, and H Lare the heat contcnt of vapor and liquid and R is the gas constant. K i t h this value of p , equation a becomes
The vapor pressure of solid at the temperature T , in ternis of its value a t the riorninl melting point, T,, also pm,is
Equating p and p , from equations b and e, one obtains t h e melting ternperature, T , corresponding t o the amount of melt, X L :
or
EFFECT O F SURFACE FORCES ON MELTING
where C,
and C,
383
are the specific heats of liquid and solid sorbate and
H , is the heat of fusion a t t h e normal melting point. i l n approximate equation, which neglects the change in the heat of fusion with temperature, is
With these equations (d or d'), when k and n are knowii for the adsorbent, one can calculate Xl,the amount of sorbate that can exist as liquid a t temperature T . Any sorbate in excess of this is solid. This temperature is termed the melting temperature corresponding to the amount of melted sorbate, X I . Calculations of this function, Xl, were made for t h e four compounds studied in the preceding paper (5). The values of k and n were thone which best fitted a set of adsorption isotherms for benzene and water which were obtained in the laboratories of t h e Silica Gel Corporation on a gel of t h e same type as ours ( 7 ) . The similarity of the two gels was verified by comparing their water content when activated, a definite characteristic of a gel. Our activated gel contained 0.0325 g. of water per gram of silica as compared to 0.0324 g. of water per gram of silica reported for the gel from which the adsorption data were obtained. The following are the resulting values : log
Xivi
= 1.13
+ 0.46 log?P
for organic sorbates, and
for water where X i is grams of water (above 3.24 per cent) per 100 grams of activatcd gel. The specific heat of p-nitrotoluene, which has not been reported, was estimated as thc mean of the molar specific heats of p-xylene and p-dinitrobenzene. The values obtained were C,, = 12.6 + 0.09912' and C,,i = 21.17 0.085T cal. per "C. mole. The other data for the calculations were obtained from accredited sourceb. The actual values of A'& were not calculated from the observatiorts in thc previous report. However, the values of X i , calculated from equation d, are shown in figure 1. Since the amount melting increases with temperature until melting is complete, the obserrcd temperature of maximum melting can be compared with calculations of the temperature at which the sorbate would be eonipIetely melted. The temperature at which the
+
384
W. A . I'ATRICK A S D W. A. KEMPER
heat capacity n as a mxxiniiim \$as taken as the temperature of maximum melting T aluej of X i equal to the total amount of qorbate present in t h e v experiments a r iiidicatcd ~ on the curves hy the horizontal lines. A c~)inparisonof the+ t n o i. gil en in table 1.
-
w 0
075
k 050
0
\ m w A 0 025
150'
200'
300'K
250'
FIG.1 Calculated amount of unfiomn soibate
Sumbered horizontal line indicates total sorbate present in the experiment indicated
TL413LC 1 Comparison of obserwd temperature o j maximum melting with calculated temperature at which the sorbate would be completely melted SUUSTAHCE
Benzene . . . . . . . . . . . . . . . . . . . . . . . . . . ..' Water. . . . . . . . . . . . . . . . . . . . . . . . . . . . . ' Naphthalene . . . . . . . . . . . . . . . . . . . . . . . p-Kitrotoluene.. . . . . . . . . . . . . . . . . . . . . . . . .
,I
"X.
1
'K
17
1
219
44 50 57
I
307
'
229
231 278
1
260 313 283
The observed temperatures are not as low ab the calculated values. T h e discrepancy is believcd t o be due to a hysteresis. The differelice is largest in the case of water, and n ater is knovl-n t o show more hysteresis than other sorbatcs. CALCULATIONS OF T H E APPAREPJT H E k T CAPACITY OF THE ADSORBED PHASE
In the previous paper ( 5 ) values ncre obtained for the difference in apparent beat capacity of the adsorbrd phase and normal qo!id sorbate. If the tlieory of capillarity i\ correct, thic difference conyiqts OF the following terms : (1) ( C p ~ C,,)X,, the difference in heat capacity of adsorbed liquid and normal solid times the amount of adsorbed liquid.
385
EFFECT OF SURFACE FORCES ON MELTING
dX (2) ( H 1 - H,) 2, the heat of fusion times the amount of melting per dT degree increase in temperature.
-' (3:;
(3) X 1
2, the heat of compression for the change in pressure
accompanying one degree rise in temperature. dw (4) - H , minus the heat of wetting per unit area times the amount
n,
of surface newly wetted by melt per degree rise in temperature. (5) w dH, -,d T the heat capacity of the surface layer. The evaluation of each of these terms for the amount of sorbate contained on 100 g. of silica gel follows. I n the calculations the change in density of adsorbed liquid with pressure will be neglected. The heat capacity of a liquid is independent of pressure, if thermal expansion i s assumed to be linear. For
Hence the first term is The amount melting per degree is obtained from the derivative of equation d with respect to temperature, u being represented by A BT and
+
;3by E, the coefficient of
cubical expansion. The expression is
The heat of fusion is
where P1 is the hydrostatic pressure on the adsorbed liquid. The value of the last term, the heat of expansion, is obtained from the equation of state:
The second term of this is neglected so that
386
W . A. PATRICK AND W. A. KEMPER
I n order to change from the auxiliary function P : to T the following procedure can be used: P Lis related to the vapor prcssurc of the liquid by the equation for the vapor pressure of a liquid under hydrostatic pressurc,
RT In p / f i
=
vlPl
Then, using the value of p / p from equation a, one obtains
Finally by means of equation d the value of v l P l is found in terms of T
This i8 a fundarnrnral equation and also could have becn obtained from
d F i = dF, where F denotes free energy. Using equation f , and simplifying, one then obtains for the heat of fusion,
The second term of the heat capacity, (H1 - H , )
dX -'d,T can now be obtained
by multiplying equations e and g. d s the amount of liquid is increased by melting, the negative pressure decreases and the liquid is subject to compression. The increase of pressure on the liquid with temperature is found by taking the derivative of equation f with respect to temperature. When v l is represented by u 1 =- v:, (1 + EAT), the equation of state become?
(This coefiicicnt of Z)Z is small and was neglected above in calculating the correction to thc heat of fusion for the expansion of newly melted liquid.) The heat of compression is then
E1'FECT OF SVRFACE FORCES OK MELTING
387
The heat coiitent per unit area of surface,
+
nhere A \$as defined by u = A UT. Activated silica gel is aqsuined to be covered TTiith a monomolecular film of water. As melted sorbate fills the capillaries, this water-air interface is replaced by a nater-qorbatc interface. The heat evolved will be t h e lirat content of t h e water-air iiiterface less that of the water-sorbate interface By .lntonoiv's rule (2), n-hich is ITalid if the molecules of the second liquid are not oriented at the nater interface, this diflerence is equal t o the heat content of t h e sorbate-air interface, which ih A times the area. Thp change in arcn of surface, W , is calculated by considering a differential rleinent of the capillary ab a cylinder. The element of volume is d d h and the element of area is 2ardh. Hence
and since
dX1
dW
dT
= - -a -dT
Using the value of cipl from equation f, one obtains for the fourth term,
Since the heat content of the wrfacr,
and A is indrpcndent of thc teinperature,
The sum of the a h b e fir^ terim reprrvnt-, on the basis of the assumptions that h a w been made, the apparent heat capacity of the adsorbed subbtanee in exceqs of it> heat capacity in its nornial solid state, if capillarity is the c a u v of its nd5orptioii. C'alculation of all of thrse quantitieq for thr four wb.tances reported in
388
W . A. PATRICK AND W. A . KEMPER
FIG.2. Calculated coniponent parts of extra heat capacity of naphthalene associated with 100 g. of activated gel
Y 5
3
53 0 150'
250'
200'
300'K
FIG.4
FIG.3
FIG.3. -4dditional heat capacity of benzene adsorbed on 100 g. of gel FIG.4. Additional heat capacity of water adsorbed on 100 g. of gel
15
10
H \
5
"< 0
150'
200
FIG.
D
5
FIG.6
FIG.5. Additional heat capacity of naphthalene adsorbed on 100 g. of gel FIG.6. Additional heat capacity of p-nitrotoluene adsorbed on 100 g . of gel
EFFECT OF SURFACE FORCES ON MELTING
389
the previous paper has been made. Figure 2 shows t h e calculated increase in heat capacity of adsorbed naphthalene and t h e components from which it was evaluated. Figures 3 to 6 show the calculated and observed values for the increase in heat capacity of benzene, water, naphthalene, and p-nitrotoluene when adsorbed on the gel. The observed values lie on the high temperature side of the calculated values, and once more the poorest agreement found is in the case of water. The hysteresis would cause the values observed on heating to lag behind those calculated for equilibrium conditions. DISCUSSION
One might suppose that the small size of the solid particles would have some effect on the melting point. Meissner (4) observed a melting-point lowering of a fraction of a degrce with crystals 0.8 I.( thick. However, Meissner’s experiments and most of the theoretical treatments have been concerned with a situation in which the particles were within t h e melt. I n the present experiments the simplifying assumption has been made that the solid has normal vapor pressure. Kubelka ( 3 ) , who discussed various theoretical equations, proposed a similar simplification for sorbates in silica gel or charcoal. H e derived a n approximate equation for the nielting-point lowering as a function of the radius of the capillaries, in which he assumes that the heat of fusion is constant with the temperature. I n this paper we have calculated the temperature at which normal solid is in equilibrium with an adsorbed phase, which follows the Patrick isotherm modified for a supercooled liquid. I t is, of course, possible that the use of some other isotherm equation based upon some other theory of adsorption might give equally good results. However, the assumptions made contain intrinsically a good deal of t h e idea of capillarity. To this extent, accordingly, such agreement as exists between the calculationq of the present paper and the calorimetric observations lends some support to the theory of capillary condensation. A closer agreement could not be expected without eliminating Ihe hysteresis phenomenon. SUMMARE
1. The Patrick equation for adsorption was extended below the normal melting point of the sorbate by using as values for normal vapor pressure, surface tension, and specific volume, those of supercooled liquid sorbate. 2. A general equation was obtained for the melting temperature of substances condensed in po+ous adsorbents. 3 . Calculations were made of the melting temperatures of the four substances adsorbed on silica gel reported in the previous paper ( 5 ) . The calculations used only two constants characterizing the gel, k and n of t h e Patrick equation. The values employed for these were obtained from
390
W. A . PATRICK AND W. A. KEMPER
a study of adsorption isotherms of benzene and wat,er on this gel at higher teniperatures. The temperatures a t which the last portions of the sorbates were calculated to have melted averaged only 14°C. lower than the temperatures a t which the maxima in the heat capacities of the systems had been observed. It is believed t h a t t'he calculated temperatures are lower because of hysteresis. 4. Further equations were derived from the various t'erms which compose the difference between heat capacity of t'he sorbate in the adsorbed st,ate and in its normal solid state. 5 . The results of these calculations were compared with t h e observations reported in t,he previous paper (5). REFERENCES (1) FRIEDEL, G . : Bull. SOC. franc. minbral. 21, 5 (1898). (2) FREUNDLICH: Colloid and Capillary Chemistry, 3rd edition, translated by Hatfield, p. 99. E. P. Dutton and Co., Xew Pork. (3) KUBELKA: 2. Elektrochem. 38, 611 (1932). (4) MEISSNER:2 . anorg. allgem. Chem. 110, 169 (1920). ( 5 ) PATRICK AND KEMPER:J. Phys. Chem. 41, 369 (1938). (6) PATRICK .AND hIcG.~v.Ke: J. Am. Chem. sot. 42, 946 (1920). (7) Silica Gel Corporation: Unpublished reports Kos. 1001A, 1004, 1009, 1010 (1925, 1926).
