A MODIFIED EQUATION FOR PORE VOLUMEAND AREADISTRIBUTIONS
404 1
A Modified Equation for Pore Volume and Area Distributions in Finely Divided and Porous Materials
by P. T. John and J. N. Bohra National Physical Laboratory, New Delhi, India
(Received January
4, 1967)
The Pierce equation has been modified, and using this equation, the pore volume and surface area have been computed from the desorption isotherm using the film thickness and Kelvin’s radius. The equation is first solved for the pore volume using the pore radius, film thickness, and the observed desorbed volume. Substituting this value of pore volume in the equation, which relates the volume, surface, and radius of an open-ended cylinder, the surface area is determined. The computed values show good agreement with the experimental values.
Introduction The determination of pore volume and surface area distributions in a porous system gives an insight into its physical structure. Shull’ was one of the first to determine pore size distribution from gas desorption data. Barrett, Joyner, and Halenda2 have described a theory and method for calculating the distributions. Pierce3 has described a simplified method based on Kelvin’s equation radii for computation of the pore size and surface area distributions from nitrogen desorption isotherms. The method involves fewer parameters when compared to those of Barrett, et aL2 Orr and Dallavalle4 have followed the method of Pierce3 using modified values of film thickness. Their values seem to have been computed from an empirical equation, whereas t,hose of Pierce are from experimental observations. .4ccording to Shull’ it is probably better to use the flat-surface experimental data of film thickness. Method of Computation The BJH data2 published by Pierce3 are used in the present computations. The data are taken from the nitrogen desorption isotherm for bone-char. The derivation of the equation of Pierce, modified by the authors, is given below. Initially, a t saturation pressure Po, all the pores of adsorbent are supposed to be completely full. It implies that there is no surface available for adsorption. Let the vapor pressure be reduced in small decrements from Po t o PI. For ni-
trogen adsorption the capillary radius corresponding to any pressure P is given by Kelvin’s equation.
r = 4.14/ln (Po/P)
(1)
When pressure is reduced from Po to PI,capillaries of radii larger than r1 will be emptied. Let the pore size of capillaries of radius rl be RI and the corresponding film thickness is then tl. Hence
Ri
= ~i
+ ti
(2)
Let the mean values of pore radius, capillary radius, and film thickness in the vapor pressure range Po-PI be R 1 , ,?i and ti, respectively. Note that
RI = (RI + Ro)/2, 71 =
(ri
+
?0)/2,
and
ti
=
(ti
+ to)/2
The volume of vapor desorbed, when pressure changes from Po to P I , from the pores of mean pore radius, R1, may be looked upon as the difference between the pore volume of pores of mean radius, Rl, and the volume of vapor adsorbed on the walls of those pores. Let the volume of these pores be ul. The area contributed by these pores can be written as (1) C. G. Shull, J . A m . Chem. Soc., 70, 1405 (1948). (2) E. P. Barrett, L. G. Joyner, and P. P. Halenda, ibid., 73, 373 (1951). (3) C. Pierce, J . Phys. Chem., 57, 149 (1953). (4) C. Orr and J. hl. Dallavalle, “Fine Particle Measurements,” Macmillan and Co., London, 1959, p 272.
Volume 7 1 , Number 19 November 1967
4042
P.T. JOHNAND J. N. BOHRA
AAI = ( 2 s ) (O.O01558)/1O-*Z?1cmz AAl = (2s) (0.001558)/10-4fi1 m2
VI
(3)
Area AAl is determined after finding the value of vl (see eq 6). Pore volume, 01, multiplied by 0.001558 is the volume of the pore when v1 is expressed in cc of gas volume. Radius and thickness are measured in angstroms. The volume FI (in cc of gas) that remains adsorbed on the walls of the pores is given by F1 = AA!(&/3.6)0.23
= (2~~)(0.00l558)(0.23~i)/3.6 X 10-4Z?1
(5)
The amount desorbed (which is the observed value) when the pressure changes from POto P1 is given by
V , - VI
= AV1
VI
- F1
(10) Referring to eq 6 it may be stated that fl is zero since there existed no area from where evaporation could take place. Similarly, when vapor pressure is changed from Pz to P3
V Z- Va
(7) The volume of vapor (in cc of gas at STP) that remains on the walls of the pores of mean pore radius 8, is given by
- F3
~3
+
(11)
f3
Fs = AA,(ls/3.6)(0.23) where 13
=
(t3
+ t2)/2
and = (AAi
f3
+ AAz)(At3/3.6)(0.23)
where At3 =
12
- t3
Hence, the general equation can be written as
(6)
Substituting the values of F1 in eq 6, VI, the only unknown quantity in eq 6, can be determined. Consider the next decrement of vapor pressure from PI to Pz. A t Pz, let the pore radius, capillary radius, and film thickness be Rz, rz, and t2, respectively, and the corresponding mean values for the range PI-PZ be R2, P ~ ,and &, respectively. When pressure is changed from PI to Pz, there will be desorption from pores whose mean pore radius is I?,. Let the volume of these pores be v2. The contribution of area by these pores is given by
= AV3 =
where
(4)
The film thickness divided by 3.6 gives the number of layers of molecules, since one molecule is 3.6 A in size. A unimolecular layer of nitrogen on 1 m2 is equal to 0.23 cc of gas at STP. Substituting eq 3 in eq 4 we get F1
- Vz = AVz = V Z - Fz + f 2
Vm-l
- V m=
AVm =
- Fm
V,
+
fm
(12)
where Vm-l and V , are, respectively, the volume of all the pores whose radii lie between Rm-z and R,+ and Rm-1 and Rm. Fm
= (2~m) (0.001558tm/3 -6)(0.23)/
10-4Z?, = AAm(lm/3.6)(0.23) where
-
R m = (Rm-1
+ Rm)/2
and
A 9 2 = (2~~)(0.001558)/10-~Z?~ mz
and
AA ,= (2vm ) (0.001558)/10-4Rm fm
F2 = AAz(&/3.6)0.23
(8)
=
(AA1
+ . . + AAm-l)[(tm-I 3
(13)
- 1,)/3.6](0.23)
i=m--l
where
+ tl)/2
?z = ( t z
fm
i=l
In addition to the above-mentioned desorption, when the vapor pressure is changed from PI to Pz there will be thinning of the adsorbed layer from the walls of pores whose radii are larger than Z?Z. The area of such pores has already been found to be AAl. The volume fz evaporated from these walls is given by the expression f2
= AAi[(ti
- tz)/3.6]0.23
fz = AA1(At2/3.6)(0.23)
Hence The Journal of Physical Chemistry
AAt
= (At,/3.6)(0.23)
This type of calculation is carried out to a pore size where capillary condensation is almost negligible. Sampte Calculations. (1) Referring to Table I, consider the desorption when the relative pressure is changed from 1.00 to 0.967. Substituting the value of AA1 from eq 3 in eq 4,when m = 1 we get FI = 2~1(0.001558)(t1/3.6)(0.23)/fi1X lo-*
(9)
Fl = 2v1(0.001558)(15.25 X 0.23)/500 X lov4 where &/3.6 =
rtl
(a1 is the mean number of molecular
A MODIFIEDEQUATION FOR PORE VOLUME AND AREADISTRIBUTIONS
4043
Table I: Calculation of Pore Volume and Surface Area Distribution for Bone-char P/PQ
V
1.000 0.967" 0.961 0.953 0.950 0.947 0.941 0.937 0.932 0.927 0.921 0.915 0.906 0.897 0.886 0.870 0,850 0.823 0.808 0.788 0.763 0.733 0.695 0.642 0.580 0.488 0.352
152.0 145.2 142.5 138.0 135.2 130.5 127.3 123.5 119.2 115.3 110.3 105.9 99.8 93.0 86.3 77.5 68.5 58.7 54.6 50.3 46.5 ,43.0 39.2 ,3.5.7 ,31,8 '27.7 '23.2
0
R
v*
n
... 8.7 3.43 5.83 3.66 6.29 3.91 4.93 5.63 5.20 6.78 5.82 8.76 8.06 8.98 11.66 11.88 12.97 3.52 3.77 2.65 2.68 3.02 2.29 3.98 3.59 2.61 150.56 152.00
152
- 145.2 = V I
- 2~1(0.001558)X
+ fi
As stated earlier, fi = 0. Solving for vir we get vi = 8.700 cc. Substituting the value of vi in eq 3 we get =
(2) Now consider desorption from the relative pressures 0.967 to 0.961. In this case there is a wall-film contribution, fi, toward desorption, since there exists an area of 0.540 In2 a t relative pressure PIIPo. Using eq 12 when n =: 2 ~2
-
=
(Ro4- R1)/2,
F2
, . .
...
0.40 0.72 0.51 0.53 1.09 0.89 1.06 1.04 1.44 1.47 2.24 2.38 2.82 4.26 4.88 6.09 2.78 3.16 2.99 2.68 3.31 3.02 3.71 3.91 5.03 62.33
RZ =
+
fi
or
...
( R I f &)/2
.
. etc.
Solving for v2, we get 02 = 3.432 cc. Substituting this value of v2 in eq 13 when m = 2 , we get = 0.3448 m2
Hence, the cumulative area would be AA1
+ AAz =
i=2
At = 2=1
= 0.540 m2
2 X 8.7 X 0.001558/500 X
Vi - V2 = AV2 =
R,
A A ~
0.54 0.34 0.70 0.48 0.87 0.59 0.83 1.03 1.04 1.46 1.34 2.18 2.18 2.66 3.87 4.57 .5.77 1.77 2.13 1.68 1.90 2.48 2.23 4.59 5.08 4.64 56.91 64.57
AA2 = 2 X 3.432 X 0.001558/310 X
(15.25 X 0.23)/500 X
AA1
0.54 0.34 0.69 0.45 0.85 0.56 0.82 1.06 1.00 1.41 1.31 2.13 2.13 2.62 3.77 4.40 5.48 1.65 1.97 1.58 1.83 2.36 2.13 4 14 4.38 5.17 54.77
1.9 4.9 1.5 3.5 6.5 3.2 4.7 5.3 4.8 6.2 5.5 8.5 8.1 8.6 11.6 11.8 12.7 5.1 5.3 4.4 3.7 4.0 3.1 3.3 2.8 2.88 143.8
The mean pore radius corresponding to relative pressure 0.967 is taken as 500 A, 3. See eq 12. d Values determined by BJH.Z
AAC
...
. I .
* See ref
layers at relative pressure P1/Po= 0.967). Hence, substituting the values from Table I in eq 12 we get
A A ~
vd
Ue
19.0 ... 340 11.5 8.8 280 10.4 3.4 240 9.2 5.8 230 8.8 3.4 6.2 220 8.5 3.7 190 7.9 4.9 180 7.5 5.4 160 7.1 5.0 150 6.8 6.6 140 6.5 5.7 130 6.2 8.6 120 5.8 110 5.4 7.9 8.9 100 5.1 11.4 88 4.7 11.5 75 4.3 12.4 65 3.9 59 8.6 3.3 52 3.3 3.5 2.5 46 3.0 41 2.75 2.6 35 2.69 2.9 2.2 30 2.23 25 2.01 3.6 20 1.76 3.1 15 1.46 2.5 145.8 Total calculated Total experimental
...
0.5420
+ 0.3448 = 0.8868 m2
and the cumulative pore volume would be VI
+
i=2 212
=
C V=~8.700 + 3.432 = 12.132 cc
i=l
All the succeeding distributions of area and pore volume have been computed in the similar manner.
Results and Discussion
145.2 - 142.5 -- 212 - 2~2(0.001558)X (10.95 X 0.23)/310 X 0.540 X (11.5 - 10.4) X 0.23
+
The computed values of pore volume and surface area distributions are given in Table I, and the corresponding values by Pierce3 and BJH2 are also given for Volume 7 1 . Number 12 November 1967
4044
comparison. The pore volume computed by the authors is in very good agreement with the experimental values whereas those by Pierce and BJH are lower. The surface area obt,ained by the authors shows better agreement wibh the experimental value when compared to that by Pierce but is lower than that by BJH. I n the earlier m e t h o d ~ l -the ~ determination of pore volume involved the squares of capillary and pore radius. It must be noted t*hat errors are liable to occur in the values of capillary radius and film thickness due to the limitations of Kelvin's equation and the method of measurement of film thickness. Since they occur as squares, the error in the computation of pore volume will be greater and consequently it will be reflected in the value for the area. Since cumulative area is used in the succeeding calculations, the effect of earlier errors will give rise to more errors. The modified equation for computation of pore volume contains two terms in the numerator. The second term involves the product of R and t. I n the first step of computation (in the pressure range 1.000.967) the value of the second term is zero. I n the
The Journal of Physical Chemistry
P. T. JOHN AND J. N. BOHRA
succeeding computations the second term increases. Even for the last computation the value of the second term becomes only comparable to the first term. This implies that the error involved in the computations of pore volume by the modified equation is less when compared to that of Pierce. The computations of surface area by Pierce involve squares of the capillary radius whereas the modified equation avoids them. The computed value of the pore volume a t each step has been used by Pierce as well as by the authors for the computation of area in the same step. Since the pore volunie computed by the authors is more precise, as discussed earlier, their computed surface area will obviously be better.
Acknowledgment. The authors are thankful to Mr. G. D. Joglekar, scientist in charge of the division, for his continued int,erest in the work. We are also thankful t,o Mr. R. s. Sekhon for useful discussions. The authors express their gratitude t.0 the Director of the National Physical Laboratory of India for the permission to publish the paper.
