3464
Langmuir 1997, 13, 3464-3473
New Approach to Determination of the Surface Phase Capacity in Liquid Adsorption on the Homogeneous Solid Surfaces A. Da¸ browski* and P. Podkos´cielny Department of Theoretical Chemistry, Faculty of Chemistry, Maria Curie-Słodowska University, Lublin 20-031, Poland Received January 21, 1997. In Final Form: April 14, 1997X
Theoretical foundations are presented to analyze the errors resulting from the determination of the surface phase capacity from the excess isotherms at the solid-liquid interface. It is shown that this analysis provides the answer to the question whether the reliable results are obtained within the assumed adsorption models. Several experimental systems are analyzed to illustrate the usefulness of the presented approach.
Introduction An essential difficulty concerning the solid-liquid adsorption systems for which the excess adsorption isotherms have been measured over the whole concentration region results from the fact that the amount of the solution undergoing adsorption cannot be experimentally determined. The Gibbs formalism typical for this interface, making use of the excess quantities, does not provide information about the structure and mechanism of the process.1 Therefore, the most important assumption refers to accepting the adsorption model which consists of the equilibrium system solid-surface phase-bulk phase. For such a system, the so-called surface phase capacity is defined as a number of molecules constituting a surface phase (e.g., as amount of substance per adsorbent mass). If it is assumed that the model of adsorption system and the corresponding isotherm equation are true, then the question arises to what extent the results obtained for the accepted adsorption system model are reliable. Formal satisfaction of the linear form of a given equation by the experimental data does not always give the results having a physical meaning. If the excess adsorption isotherms with the assumed value of surface phase capacity are simulated, then the application of linear forms of isotherm equations for their analysis does not frequently allow to reproduce the assumed values of adsorption parameters.2 On the other hand, the surface phase capacity is an important quantity in the study of adsorption from solutions. It characterizes sorption properties of the solid adsorbent and is necessary for the calculation of the surface phase composition. This quantity is also useful in the calculation of those thermodynamic functions which characterize competitive adsorption at the solid-liquid interface. A few years ago, Da¸ browski and Jaroniec3 published the comprehensive review on the methods used for analyzing the excess adsorption isotherms to determine the surface phase capacity. This review comprises about 400 experimental systems with binary mixtures of non* Author to whom correspondence should be addressed. X Abstract published in Advance ACS Abstracts, June 1, 1997. (1) Kipling, J. J. Adsorption from Solutions of Non-Electrolytes; Academic Press: London, New York, 1965. (2) Chmutov, K. V.; Larionov, O. G. Progr. Surf. Membr. Sci. 1981, 14, 237. (3) Da¸ browski, A.; Jaroniec, M. Adv. Colloid Interface Sci. 1990, 31, 155.
S0743-7463(97)00063-2 CCC: $14.00
electrolytes consisting of molecules of similar molecular dimensions. It means that the ratio of the molecular sizes is not significantly different from unity. For such liquid systems, the concept of surface phase may be safely used. However, in this review no discussion is presented on the reliability of the adsorption parameters obtained in terms of various methods. The aim of this paper is to present the theory enabling a numerical, statistical analysis of the errors made applying different methods for calculating capacities of surface phases formed on the homogeneous solid surfaces. This statistical analysis provides the answer to the question whether the reliable results are obtained within the assumed adsorption models. It allows to minimize the possibility of random results and to limit the error only to a systematic one which is a function of assumptions underlying the given isotherm equation. To this end, the definite models of adsorption systems and isotherm equations corresponding to them are employed. For adsorption on the homogeneous solid surfaces linear forms of the Everett equation3 are introduced. Their adoption is justified by their wide application in the literature on adsorption at the interface under discussion. Several experimental systems are analyzed to illustrate wide usefulness of our approach. Theoretical Let us consider the following equation to describe all the types of excess adsorption isotherms:4
nσ(n) ) 1
ns1,0xl1xl2(R12 - 1) R12xl1 + rxl2
(1)
where n1σ(n) is the reduced surface excess of the first component, s ni,0 (i ) 1, 2) denotes the surface phase capacity (per mass of s s /n1,0 , where woi (i ) solid) of the ith component, r ) wo1/wo2 ) n2,0 1, 2) is the molar cross-sectional area, R12 is the so-called separation factor,4 and xli (i ) 1, 2) denotes the equilibrium bulk mole fraction of the ith component. For the IBP model in which the bulk and surface phases are assumed to be ideal and r ) 1, eq 1 can be transformed into three linear forms proposed by (4) Da¸ browski A.; Jaroniec M.; Os´cik, J. Colloid Surf. Sci. 1987, 14, 83.
© 1997 American Chemical Society
Surface Phase Capacity Determination
Langmuir, Vol. 13, No. 13, 1997 3465
Table 1. Linear Forms of Eqs 2-5 and Definitions of the Parameters Yj, xj, Aj, and Bj code
linear form xl1xl2
E
nσ(n) 1
)
xl1xl2
( )(
Yj
1 1 xl1 + R12 - 1 ns
( )[ ( )(
)
nσ(n) 1
] ) )
R12 - r l 1 r ) s + x R12 - 1 R12 - 1 1 n nσ(n) 1,0 1
GE
1 - xl1
SR
)
nσ(n) 1
1-
xl1
)( (
1 1 1 1 + ns (R12 - 1) xl1 ns
nσ(n) 1
SEr
)-
σ(n) R12 - 1 1 n1 + ns R12 xl R12 1
xl1xl2 nσ(n) 1 1 - xl1 nσ(n) 1 nσ(n) 1 1-
Everett,5 Schiessler and Ro¨we6 as well as Sˇ isˇkova and Erdo¨s.7 The Everett equation (E):
xl1xl2 nσ(n) 1
)
( )(
)
1 1 xl1 + s R n 12 - 1
nσ(n) 1
)
( )(
)(
1 - xl1
)-
)
1 1 1 1 + ns R12 - 1 xl1 ns
(3)
)
σ(n) R12 - 1 1 n1 + ns l R12 x1 R12
(4)
Equation 2 was generalized by Schay and Nagy8 for the case when r * 1. The corresponding linear equation is called the generalized Everett equation (GE):
xl1xl2 nσ(n) 1
)
( )[
R12 - r l 1 r + x1 s R 1 R n1,0 12 12 - 1
]
(5)
s Our aim is calculation of the parameters ns (or n1,0 ) and R12. These calculations are made assuming the linear regression model of random variable Y toward variable x (in short, Y/x) and within it making a statistical analysis of the errors resulting from using linear eqs 2-5. We assume that liquid mole fractions x1l are the quantities which are measureable with negligible error, and we treat them as ordinary variables. In relation to the reduced surface excess, we state that it is the quantity which is subject to random error (so we treat it as the random variable). A hypothetical model of linear regression with one real variable x can be presented as follows:9
Y ) Ax +B
(
)
xl1
xl1
( )
r 1 s R12 - 1 n1,0
1 xl1
()
1 ns
nσ(n) 1
-
[
1 R12 - r s R12 - 1 n1,0 1 1 ns (R12 - 1) 1 R12
xl1
ns
(
]
)
R12 - 1 R12
expected that for any arbitrarily chosen point (xi, yi), i ) 1, ..., n the model equality is satisfied because it is to be satisfied only indifferently but not accurately. The linear model has an equivalent point representation:
(6)
where A and B are the model parameters which would be estimated from the data (xi, yi), i ) 1, ..., n. Equality 6 can be viewed as the definition of the conditional expected value E(Y|x) ) E(Y) ) Ax +B. That means that even knowing the hypothetical values of parameters A and B of general population, it is hardly (5) Everett, D. H. Trans. Faraday Soc. 1964, 60, 1803. (6) Schiessler, R.; Ro¨we, C. N. J. Am. Chem. Soc. 1954, 76, 1202. (7) Sˇ isˇkova, M.; Erdo¨s, E. Collect. Czech. Chem. Commun. 1960, 25, 2599. (8) Schay, G.; Nagy, L. G. J. Colloid Interface Sci. 1972, 38, 302. (9) Taylor, J. R. An Introduction to Error Analysis; University Science Books. Oxford University Press: Oxford, 1982.
i ) 1, ..., n
(7)
The random component (error) i is the difference between the observed value yi of random variable Y and its expected value E(Yi) ) Axi +B. The basic assumption for the linear regression model is that errors i have a normal distribution N(0, σ) for i ) 1, ..., n. The following linear form corresponds to eqs 2-5:
Yj ) Ajxj + Bj,
(
Bj
1 1 ns R12 - 1
yi ) Axi + B + i,
The Sˇ isˇkova-Erdo¨s equation (SEr):
nσ(n) 1
xl1
Aj 1 ns
(2)
The Schiessler-Ro¨we equation (SR):
1 - xl1
xj
xl1xl2
where j ) E, GE, SR, SEr
(8)
The symbols E, GE, SR, and SEr denote a linear form of the Everett equation, generalized Everett equation, Schiessler-Ro¨we equation, and Sˇ isˇkova-Erdo¨s equation, respectively. The linear forms of these equations and the definitions Yj, xj, Aj, and Bj are included in Table 1. “Real” values A and B are not actually known and therefore they are approximated with the estimators a and b obtained from the data (xi, yi), i ) 1, ..., n. The estimator is a function used for determining an unknown hypothetical parameter of general population. The hypothetical model is opposed to the approximate model:
Y ) ax + b
(9)
The point representation of the problem corresponding to the approximate model has the following form:
yi ) axi + b + ei,
i ) 1, ..., n
(10)
The corrections ei are called remainders. The remainders ei should be treated as estimation of the errors i of the hypothetical model. As a measure of deviation of the theoretical function given by eq 9 from the experimental function yi ) f(xi), i ) 1, ..., n, there is assumed to be n
Q ˆ )
∑e i)1
n
2 i
∑(y - ax - b)
2
)
i
i
(11)
i)1
called the sum of the squares of the remainders. To find function 9, the coefficients a and b should be chosen in such a way that the value Q ˆ is as small as possible. The procedure which is used to determine the unknown parameters a and b satisfying eq 11 is called the least-square method. At first, the partial derivatives of expression 11 in relation to the parameters a and b are determined, then equating them to zero, a set of equations is obtained which is solved with respect to these parameters.
3466 Langmuir, Vol. 13, No. 13, 1997
Da¸ browski and Podkos´ cielny
Table 2. Parameters ns and r12 Determined for the Equations E, GE, SR, and SEr by means of the parameters a and b ns
code
R12
E
1 aE
aE + b E bE
GE
r r(aGE + bGE) - bGE
r(aGE + bGE) bGE
SR
1 bSR
(aSR + bSR) aSR
bSer 1 + aSEr
-
SEr
Figure 1. Probability density function of Student-t. The sum of dashed areas is equal to probability R.
1 aSEr
coefficient F can be calculated by means of the data (xi, yi), i ) 1, ..., n from the equation
The coefficient a is calculated according to the following equation:10 n
n
a)
i i
i
i)1
n
i
i)1
n
∑x
i
i)1
i)1
(12)
n
2
R)
∑x )
-(
i
i)1
However, the coefficient b is determined from the following relation:
b ) yj - axj
(13)
where
xj )
1
n
1
n
∑x and yj ) n∑y n i
i)1
Q ˆ n-2
i i
x( ∑ )( ∑ ) 1
x2i - xj2
1
(15)
n
n i)1
y2i - yj2
where R ∈ 〈-1, +1〉. The value |R| is closer to unity, the better approximation of the data by the regression straight line is observed. In the case of 0 < R < 1, the straight line of regression forms the acute angle with positive sense of axis OX, whereas for -1 < R < 0 the straight line forms obtuse angle. Then a statistical test for the significance of correlation coefficient is made. Its result indicates the existence of correlation between the variables X and Y. With the lack of correlation, the obtained values of the parameters ns and R12 do not have a physical sense. Let us make a zero hypothesis H: F ) 0, equivalent to nonexistence of the linear correlation between X and Y against the alternative hypothesis K: F * 0.
i
i)1
From the determined parameters a and b for the equations E, GE, SR, and SEr, the values of parameters ns and R12 are calculated according to the expressions included in Table 2. Reliability of the parameters obtained in terms of a given linear equation depends on the results of the statistical analysis. This analysis includes (1) calculation of the sum of the squares of the remainders and remainder variance, (2) determination of the correlation coefficient and realization of the significance test for this coefficient, (3) determination of the significance test for the coefficients A and B of the regression straight line, (4) calculation of the mean standard deviations for the parameters ns and R12, (5) determination of the confidence intervals for the coefficients A and B of the regression straight line and of the confidence intervals for the parameters ns and R12, (6) determination of the confidence interval for the regression straight line, and (7) determination of the tolerance interval for the experimental values departing from the regression line. The sum of deviation squares, Q ˆ , obtained from eq 11 and the remainder variance:
s2yˆ )
∑x y - xjyj
n
n i)1
2
n
n i)1
n
∑x y - (∑x )(∑y )
n
1
(14)
are the measures of the errors made while describing the experimental system by using the linear regression model. The values Q ˆ and the remainder variance are close to zero, the fitting of a theoretical straight line to the experimental data is better. As a result, the obtained values ns and R12 are more reliable. The measure of the regression model linearity is a value F of the correlation coefficient.10 The value R of the correlation (10) Czermin´ski, J. B.; Iwasiewicz, A.; Paszek, Z.; Sikorski, A. Statistical methods for chemists; PWN: Warsaw, 1986.
The characteristics defined by
t)
R
xn - 2
x1 - R2
(16)
has a distribution of Student t-test with ν ) n - 2 degrees of freedom. The number of degrees of freedom (ν) is determined as a number of experimental points (xi, yi) diminished by a number of equations used for calculation of the regression straight line. The critical set (set of rejections of null hypothesis H) on the level of significance R is a set (W) defined by
W ) (-∞, -tR;ν > ∪ < tR;ν, +∞)
(17)
The significance level of test (R) above mentioned is a probability of rejection of true zero hypothesis. It is frequently assumed that R ) 0.05.10 Let us assume the significance level R ) 0.05. The values tR;ν (occuring in eq 17) are given in tables,11 whereby tR;ν is chosen in such a way that probability P that the random variable of distribution of Student-t of ν degrees of freedom is not smaller than tR;ν and equals R (see Figure 1). The above statement can be written
∫
P(|t| g tR;ν) ) 2
∞
tR;ν
g(t)dt ) R
(18)
(11) Oktaba, W.; Niedokos, E. Mathematics and basis of statistical mathematics; PWN: Warsaw, 1980.
Surface Phase Capacity Determination
Langmuir, Vol. 13, No. 13, 1997 3467
where g(t) is a probability density function and is expressed by the equation:
1 g(t) ) 1 1 1/2 (ν) β , ν 2 2
( )(
sn2 s(GE) )
1
)
t2 1+ ν
,
-∞ < t < +∞ (19)
(ν+1)/2
∫x
1 p-1
(1 - x)q-1 dx;
0
p > 0,
q>0
(20)
From the experimental data (xi, yi), the value tR of the characteristics t is determined from eq 16. If tR ∈ W, then the zero hypothesis is rejected which means that there is a correlation between the variables X and Y. Otherwise, the correlation between X and Y does not occur. The test of significance for the coefficients Aj and Bj (j ) E, GE, SR, SEr) provides the answer to the question if these coefficients are really different from zero. Insignificance of the coefficient Aj or Bj causes lack of a physical sense of corresponding adsorption parameters ns or R12 existing in the definition of these coefficients (Table 1). If the assumptions about the regression model are satisfied, then the random variables
ta )
a-A , sa
tb )
b-B sb
(21)
have the distribution of Student t with ν ) n - 2 degrees of freedom. The above sa is the mean standard deviation for the regression coefficient; sb is the mean standard deviation for the free term
sa ) syˆ
x
1
,
n
∑
sb ) syˆ
2
(xi - xj)
i)1
x
a , sa
(
∑ n
n
∑ i)1
∑
-(
tb )
)
)
( )
( )
( )( )
(27) Similarly, for the Schiessler-Ro¨we equation, it can be written:
( ) ( ) ( ) ( ) ( )( ) 2 ∂(ns)SR 2 2 -1 sbSR ) 2 sb2SR ∂bSR bSR
sn2 s(SR) )
sR2 12(SR) )
-bSR 2 aSR
2
(28)
-bSR 1 2 1 2 2 s +2 2 sabSR aSR bSR aSR aSR
sa2SR +
(29)
To the Sˇ isˇkova-Erdo¨s equation, there can be attributed the following relations:
sn2 s(SEr) )
(
(1 + aSEr)
)
2
-bSEr
sa2SEr +
2
2
(
(
(
)
1 2 sab (30) SEr (1 + aSEr)2 1 + aSEr -bSEr
)
∂R12(SEr) ∂aSEr
) )(
2 1 s2 + 1 + aSEr bSEr
2
sa2SEr )
( )
2
1
2 aSEr
sa2SEr
(31)
(22)
n
x2i
)(
2
-raGE 2 2 r 2 2 r -raGE 2 saGE + sbGE + 2 sabGE 2 bGE bGE bGE b2GE
sR2 12(GE) )
sR2 12(SEr) )
i)1
(
-r -r + r 2 sab (26) GE (raGE + rbGE - bGE)2 (raGE + rbGE - bGE)2
x2i
2 The variance sab can be calculated in a general case from
2
xi )
i)1
n
b sb
(23)
The set given by eq 17 is taken as a critical set on the significance level R. If the calculated value ta ∈ W (alternatively, tb ∈ W), it means that the coefficient A (B) is different from zero. Otherwise, the coefficient A (B) is equal to zero. Other characteristics used for estimation of reliability of the obtained values of the parameters ns and R12 are the mean standard deviations of these parameters. These quantities should be as small as possible, possibly close to zero. At first, the estimation of variances sn2 s and sR2 12 of the parameters ns and R12 is made for the linear equations under discussion. Having the variances sn2 s and sR2 12 it is easy to calculate the mean standard deviations sns and sR12 of the parameters ns and R12 finding corresponding square roots of the variances. For the Everett equation (E), the following relations are true:
( ) ( ) ( ) ( ) ( )( ) 2 ∂(ns)E 2 2 -1 saE ) 2 sa2E sn2 s(E) ) ∂aE aE
sR2 12(E) )
)
2
n
The symbol syˆ means the root of the remainder variance defined by eq 14. The null hypothesis H1 is made, according to which the coefficient A is equal to zero, i.e., H1, A ) 0, with the alternative hypothesis K1, A * 0. Analogous hypotheses H2 and K2 are made for B. These hypotheses are verified by means of Student t-test calculating the random variables
ta )
(
2 2 -r2 -r2 + r 2 s + sb2GE + a GE (raGE + rbGE - bGE)2 (raGE + rbGE - bGE)2
2
In eq 19, β(p, q) is a beta function defined by the integral
β(p, q) )
For the generalized Everett equation:
-aE 2 2 1 2 2 1 -aE 2 s + sbE + 2 sabE 2 b E aE b bE b2E E
∑x s2ab
)
-s2yˆ n
i
i)1
n
∑ i)1
(32)
n
x2i
∑x )
2
-(
i
i)1
where ab ) abE, abGE, abSR, abSEr. On determining the confidence interval for the coefficients Aj and Bj (j ) E, GE, SR, SEr), it is possible to determine the confidence intervals for the parameters ns and R12. It can be done when the definition of the coefficient A or B is expressed by only one adsorption parameter, i.e., by ns or R12 (see Table 1). The confidence intervals for the coefficients A and B on the confidence level 1 - R can be determined from the following inequalities:
a - satR;ν < A < a + satR;ν,
b - sbtR;ν < B < b + sbtR;ν
(33)
For the Everett equation AE ) 1/ns, then
1 1 < ns < E aE + sa tR;ν aE - sEa tR;ν
(34)
For the Schiessler-Ro¨we equation BSR ) 1/ns, then
1 1 < ns < bSR + sbSRtR;ν bSR - sbSRtR;ν
(24)
(35)
For the Sˇ isˇkova-Erdo¨s ASEr ) -1/R12 equation:
(25)
-
1 1 < R12 < aSEr - saSErtR;ν aSEr + saSErtR;ν
(36)
3468 Langmuir, Vol. 13, No. 13, 1997
Da¸ browski and Podkos´ cielny
Table 3. Information Concerning the Experimental Systems Investigated by Means of Eqs 2-5 liquid mixture
no. of adsorption system
component 1
component 2
adsorbent
T/K
ref to adsorption system
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
benzene benzene trichloromethane 1,1,1-trichloroethane pyridine dichloroethane ethanol methyl acetate benzene benzene methyl propyl ketone n-butanol tert-butanol n-hexane n-hexane
cyclohexane cyclohexane tetrachloromethane tert butyl chloride ethanol benzene benzene benzene ethanol ethanol benzene n-heptane benzene benzene benzene
silica gel spheron charcoal charcoal charcoal boehmite titania gel silica gel graphon graphon silica gel silica gel silica gel tin oxide gel tin oxide ppt
333 293 293 293 293 293 333 293 293 298 293 298 293 308 308
12 13 13 13 13 14 15 15 16 16 17 18 19 20 20
The next step is to determine the confidence interval for the regression straight line (or interchangeably, the confidence interval for the expected value E(Y)).10 The value yˆ i calculated for a given xi, i ) 1, ..., n from the approximate model equation
yˆ i ) axi + b
(37)
is a measure of estimation of the “true” expected value Ei(Y) ) Axi + B. We are going to define the following relation:
sy2ˆ i
)
s2yˆ
[
(xi - xj)2
1 +
n
n
∑(x
- xj)
]
k)1
(38)
yˆ i - Ei(Y) syˆ i
(xi - xj)2
1 + n
n
∑(x k)1
k
]
(43)
- xj)2
After satisfying the assumptions about the regression model, the characteristic
yi - Ei(Y) s ∆i
(44)
has the distribution of Student-t, with ν ) n - 2 degrees of freedom. The tolerance area under research can be written as
The square root of the above equation, i.e., syˆ i, determines the standard deviation of the mean value yˆ i from the “real” expected value Ei(Y). If the assumptions about the regression model are satisfied, then the random variable
t)
s∆2 i ) s2yˆ 1 +
t)
2
k
[
- Ei(Y) can be determined by
(39)
yˆ i - s∆itR;ν < E(∆i) < yˆ i + s∆itR;ν
(45)
As follows from eq 45, the above mentioned area is included between two functions determining the tolerance limits, on the confidence level (1 - R). If any experimental point (xi, yi) is beyond the tolerance limits, it may be neglected in the calculations which allows to correct the numerical values of the surface phase capacity ns and the separation coefficient R12.
Experimental Verification has the distribution of Student t with ν ) n - 2 degrees of freedom. The confidence interval for the “real” expected value has the form
yˆ i - syˆ itR;ν < Ei(Y) < yˆ i + syˆ itR;ν
(40)
f+(x) ) yˆ i + syˆ itR;ν
(41)
The function
can be called the upper confidence curve. Correspondingly, the function
f-(x) ) yˆ i - syˆ itR;ν
(42)
is called the bottom confidence curve. The area between the confidence curves is called the confidence interval for the expected value E(Y) on the confidence level (1 R), from the data (xi, yi), i ) 1, ..., n. Another problem is determination of the tolerance interval for the values departing from the regression line.10 It enables elimination of experimental points with a large experimental error. The average deviation of each value yi from the mean Ei(Y) is of interest for us. The variance of the difference ∆i ) yi
On the basis of the above theory, we have analyzed over 100 experimental systems taken from the literature.3 However, taking into account the volume of this work, we will present the results of this analysis including several systems only. The main information concerning the systems studied in terms of our approach is included in Table 3.12-20 The suitable results are summarized in the following tables: Table 4 for Everett eq 2, Table 5 for Schiessler-Ro¨we eq 3, Table 6 for Sˇ isˇkova-Erdo¨s eq 4, and Table 7 for generalized Everett eq 5. These tables include the number of the experimental system, the value of the correlation coefficient R, the value of the surface phase capacity, the value of the separation factor R12, and the sum of deviation squares, respectively. The next (12) Sircar, S.; Novosad, J.; Myers, A. L. Fundamentals 1972, 11, 249. (13) Blackburn, A.; Kipling, J. J.; Tester, D. A. J. Chem. Soc. 1957, 2373. (14) Kipling, J. J.; Peakall, D. P. J. Chem. Soc. 1956, 4828. (15) Kipling, J. J.; Peakall, D. B. J. Chem. Soc. 1957, 4054. (16) Brown, Ch. E.; Everett, D. H.; Morgan, Ch. J. Trans. Faraday. Soc. 1975, 71, 883. (17) Os´cik, J.; Goworek, J. Pol. J. Chem. 1978, 52, 1781. (18) Goworek, J. Colloids Surf. 1990, 47, 169. (19) Boro´wko, M.; Goworek, J.; Jaroniec, M. Monatsh. Chem 1982, 113, 669. (20) Madan, B. L.; Sandle, N. K.; Tyagi, J. S. Curr. Sci. 1975, 44, 879.
Surface Phase Capacity Determination
Langmuir, Vol. 13, No. 13, 1997 3469
Table 4. Parameters for the Linear Form of Everett Eq 2 significance test for
no. of adsorption system
R
ns/mmol g-1
R12
Q ˆ
R
AE
BE
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
0.996 0.998 0.997 0.995 0.835 0.555 0.987 0.981 0.949 0.890 0.945 0.986 0.977 0.995 0.999
3.121 0.419 1.457 2.390 0.679 3.170 1.120 3.309 0.949 0.725 1.349 1.474 1.454 0.551 0.378
6.685 8.540 8.448 3.882 -5.458 1.576 61.767 8.365 2.802 3.630 18.398 -11.706 -26.759 4.221 5.411
0.000 505 0.026 934 0.002 537 0.001 104 0.947 256 0.133 916 0.018 953 0.005 413 0.072 495 0.503 165 0.064 461 0.010 191 0.018 086 0.017 854 0.008 385
+ + + + + + + + + + + + + +
+ + + + + + + + + + + + + +
+ + + + + + + + + +
Table 5. Parameters for the Linear Form of Schiessler-Ro1 we Eq 3 significance test for
no. of adsorption system
R
ns/mmol g-1
R12
Q ˆ
R
AE
BE
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
0.996 0.979 0.993 0.997 0.013 0.966 0.664 0.877 0.994 0.993 0.905 -0.487 0.399 0.994 0.998
2.783 0.358 1.318 2.609 1.115 -2.522 1.081 2.613 1.297 1.193 1.100 1.839 1.736 0.621 0.360
9.079 14.019 12.252 3.406 941.04 0.523 155.798 28.657 2.098 2.153 91.772 -97.835 627.987 3.523 6.047
0.003 972 2.980 000 0.021 188 0.010 734 1.778 173 3.268 000 0.044 271 0.060 564 0.321 732 1.475 069 0.347 677 0.066 312 0.069 150 0.277 093 0.088 422
+ + + + + + + + + + + +
+ + + + + + + + + + + +
+ + + + + + + + + + + + + +
Table 6. Parameters for the Linear Form of S ˇ isˇ kova-Erdo1 s Eq 4 significance test for
no. of adsorption system
R
ns/mmol g-1
R12
Q ˆ
R
AE
BE
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
-0.980 -0.955 -0.975 -0.988 -0.135 -0.775 -0.614 -0.788 -0.949 -0.889 -0.726 0.514 -0.388 -0.979 -0.994
2.884 0.383 1.350 2.453 1.307 1.825 1.090 2.729 1.035 0.875 1.146 1.902 1.767 0.566 0.368
8.341 11.922 11.383 3.719 111.122 2.312 145.836 23.918 2.557 2.876 86.227 -108.69 609.03 4.028 5.766
0.105 688 0.009 393 0.032 161 0.031 855 1.373 104 0.262 176 0.063 429 1.952 750 0.020 049 0.062 400 0.481 177 0.904 771 0.455 437 0.002 632 0.000 339
+ + + + + + + + + + +
+ + + + + + + + + + +
+ + + + + + + + + + + + + + +
column presents the results of the significance test referring to the correlation coefficient R. Existence and deficiency of the correlation between the variables X and Y are denoted by the symbols “+” and “-”, respectively. The last two columns present the results of the significance test of the model parameters A and B from eq 6, respectively. If the parameters A and B are essential (A * 0, B * 0), then this fact is denoted by “+”. Otherwise (A ) 0, B ) 0), the symbol “-” is used. For the generalized Everett equation, in Table 7, the parameter r is placed. This parameter has been calculated according to the formula
r ) (M1F2/M2F1)2/3
(46)
Figure 2. Adsorption of benzene (1) + cyclohexane (2) on the silica gel at 333 K (system 1, Table 4). (A) Experimental excess adsorption isotherm (circles) compared with the theoretical curve (solid line) calculated using eq 2. (B) The best-fit regression straight solid line calculated according to eq 9, 95% confidence interval placed between dotted lines and 95% tolerance interval placed between dashed lines. Table 7. Parameters for the Linear Form of Generalized Everett Eq 5 no. of adsorption system
r
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
0.878 0.878 0.884 0.897 1.364 0.924 0.756 0.924 1.323 1.323 1.125 0.730 1.046 1.293 1.293
significance test for
R
s / n1,0 mmol g-1
R12
Q ˆ
R
AE
BE
0.996 0.998 0.997 0.995 0.835 0.555 0.987 0.981 0.949 0.890 0.945 0.986 0.977 0.995 0.999
3.200 0.427 1.483 2.489 0.708 3.699 1.126 3.347 0.836 0.663 1.341 1.433 1.457 0.514 0.360
5.870 7.498 7.468 3.482 -7.445 1.456 46.696 7.729 3.707 4.802 20.698 -8.545 -27.900 5.458 6.997
0.00505 0.026934 0.002537 0.001104 0.947259 0.133916 0.018953 0.005413 0.072495 0.503164 0.064461 0.010191 0.018086 0.017854 0.008385
+ + + + + + + + + + + + + +
+ + + + + + + + + + + + + +
+ + + + + + + + + +
where Mi denotes the molecular weight of the ith component, but Fi is its density. Tables 4-7 do not include all the numerical results of our analysis. For illustrative purposes we discuss the detailed results for some selected systems. First of all,
3470 Langmuir, Vol. 13, No. 13, 1997
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Figure 3. Adsorption of benzene (1) + cyclohexane (2) on spheron at 293 K (system 2, Table 6). (A) Experimental excess adsorption isotherm (circles) compared with the theoretical curves calculated using eq 4 for all the experimental points (the solid line) and after rejection of the three points (open circles) from beyond the 95% confidence interval (the dashed line). (B) Labeling as in Figure 2B.
Figure 4. Adsorption of dichloroethane (1) + benzene (2) on boehmite at 293 K (system 6, Table 4). (A) Experimental excess adsorption isotherm (circles) compared with the theoretical curves calculated using eq 2 for all the experimental points (the solid line) and after rejection of the one point (open circle) from beyond the 95% confidence interval (the dashed line). (B) Labeling as in figure 2B.
we present the example of the statistical analysis dealing with the calculations carried out in terms of the Everett equation for the system benzene (1) + cyclohexane (2) + silica gel at 333 K (system 1, Table 4). For this system, the following values of parameters were obtained: aE ) ˆ ) 0.000 505, ns ) 3.121 mmol g-1, R12 0.320, bE ) 0.056, Q ) 6.685, R ) 0.996, and tR ) 31.184. For R ) 0.05 and ν ) 7, t0.05;7 ) 2.365, and as a consequence of this, W ) (-∞, -2.365 > ∪ < 2.365, +∞), then tR ∈ W and the subsistence of correlation between the variables X and Y is observed. The other statistical parameters have the values sy2ˆ ) 0.000 072, sa(E) ) 0.010 274, sb(E) ) 0.005 024, 2 sab(E) ) -0.000 043, sns ) 0.100093, sR12 ) 0.665 442, ta(E) ) 31.185, tb(E) ) 11.216, but ta(E) ∈ W, then AE * 0; moreover, tb(E) ∈ W, so BE * 0. The confidence intervals for the coefficients A and B of the regression straight line and confidence interval for the adsorption parameter ns are the following: 0.296 < AE < 0.345, 0.045 < BE < 0.068, 2.901 mmol g-1 < ns < 3.378 mmol g-1. Figure 2A shows the excess adsorption isotherms for the system under consideration. The circles denote the experimental points, but the solid line presents the
theoretical curve calculated by means of eq 2 for ns ) 3.121 mmol g-1 and R12 ) 6.685. For the same system we present in Figure 2B the bestfit straight solid line calculated according to eq 9, 95% confidence interval for E(Y) placed between the dotted lines and 95% tolerance interval placed between the dashed lines. The circles denote the experimental points. The results of an analogous analysis carried out in terms of the Sˇ isˇkova-Erdo¨s equation for the system benzene (1) + cyclohexane (2) + spheron at 293 K (system 2, Table 6) ˆ ) are the following: aSEr ) -0.084, bSEr ) 0.351, Q 0.009 393, ns ) 0.383 mmol g-1, R12 ) 11.922, R ) -0.955, tR ) -10.213. For R ) 0.05 and ν ) 10, t0.05;10 ) 2.228 and as a consequence of this W ) (-∞, -2.228 > ∪ < 2.228, +∞), then tR ∈ W, and the subsistence of correlation between the variables X and Y is observed. The other statistical parameters have the values Sy2ˆ ) 2 0.000 939, sa(SEr) ) 0.008 213, sb(SEr) ) 0.011 384, sab(SEr) ) -0.000 059, sns ) 0.014 826, sR12 ) 1.167 310, ta(SEr) ) -10.213, tb(SEr) ) 30.794, but ta(SEr) ∈ W, then ASEr * 0; moreover, tb(SEr) ∈ W, so BSEr * 0. The confidence intervals for the coefficients A and B of the regression line and confidence interval for the adsorp-
Surface Phase Capacity Determination
Figure 5. Adsorption of ethanol (1) + benzene (2) on titania gel at 333 K (system 7, Table 6). (A) Experimental excess adsorption isotherm (circles) compared with the theoretical curves calculated using eq 4 for all the experimental points (the solid line) and after rejection of the two points (open circles) from beyond the 95% confidence interval (the dashed line). (B) Labeling as in figure 2B.
tion parameter R12 are the following: -0.102 < ASEr < -0.066, 0.325 < BSEr < 0.376, 9.787 < R12 < 15.248. Figure 3B presents the best-fit straight solid line calculated according to eq 9, also 95% confidence interval for E(Y) placed between the dotted lines and 95% tolerance interval included between the dashed lines. The circles denote the experimental points. In this figure, we can observe that a few experimental points lie beyond the limits of 95% confidence interval. For the illustrative purposes, we rejected these points (open circles) in recalculations of the excess adsorption isotherm. Figure 3A displays two theoretical excess isotherms: the solid line deals with all the experimental points but the dashed line deals with the corrected number of the experimental points (3 pointssopen circlessare rejected). Now we focus on the example which illustrates the influence of the insignificance of the coefficients A and B of the regression straight line. For the illustrative purposes, the following experimental system in terms of the Everett equation is considered: dichloroethane (1) + benzene (2) + boehmite, at 293 K (system 6, Table 4). Due to insignificance of the AE, the error sns is singularly substantial. The lack of the correlation means that the
Langmuir, Vol. 13, No. 13, 1997 3471
Figure 6. Adsorption of benzene (1) + ethanol (2) on graphon at 298 K (system 10, Table 4). (A) Experimental excess adsorption isotherm (circles) compared with the theoretical curves calculated using eq 2 for all the experimental points (the solid line) and after rejection of the one point (open circle) from beyond the 95% tolerance interval (the dashed line). (B) The best-fit regression straight solid line calculated according to eq 9 for all the experimental points and the best-fit dotted straight line calculated after rejection of the one point (open circle) from beyond the 95% tolerance interval. The former 95% tolerance interval is placed between dashed lines, but the corrected 95% tolerance interval is placed between dashed dotted lines (see also Table 8).
values of parameters ns and R12 have no physical meaning. Probably the reason for the lack of this correlation results from the fact that one experimental point lies so far from the best-fit straight line (open circle in Figure 4B). After rejection of this point from beyond 95% confidence interval, we obtain the better agreement with the experimental points (Figure 4A). The new values of parameters for the linear form of Everett eq 2 obtained after rejection of the point from beyond 95% confidence interval (open circle) are the following: R ) 0.993, ns ) 1.608 mmol g-1, R12) 2.813, Q ˆ ) 0.002 254, and the significance test for R, AE, and BE is positive, i.e., denoted by “+”. Next, let us consider the Sˇ isˇkova-Erdo¨s eq 4. For the system ethanol (1) + benzene (2) + titania gel, at 333 K (system 7, Table 6), we can observe the insignificance of the coefficient ASEr and the lack of correlation. In consequence, the error sR12 is substantial and the values of parameters ns and R12 are nonphysical.
3472 Langmuir, Vol. 13, No. 13, 1997
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Figure 7. Adsorption of methyl acetate (1) + benzene (2) on the silica gel at 293 K (system 8, Table 7). (A) Experimental excess adsorption isotherm (circles) compared with the theoretical curves calculated using eq 5 for all the experimental points (the solid line) and after rejection of the one point (open circle) from beyond the 95% tolerance interval (the dashed line). (B) Labeling as in Figure 6B (see also Table 9).
Figure 8. Adsorption of methyl propyl ketone (1) + benzene (2) on silica gel at 293 K (system 11, Table 6). (A) Experimental excess adsorption isotherm (circles) compared with the theoretical curves calculated using eq 4 for all the experimental points (the solid line) and after rejection of the one point (open circle) from beyond the 95% tolerance interval (the dashed line). (B) Labeling as in Figure 6B (see also Table 10).
In Figure 5B, we can observe great dispersion of the experimental points around the best-fit straight line and two points lie beyond the limits of 95% confidence interval. In spite of this, the theoretical excess adsorption isotherm (solid line) is in a relatively good agreement with the experimental points (Figure 5A). New values of the parameters for the linear form of the Sˇ isˇkova-Erdo¨s eq 2 obtained after rejection of the two points from beyond 95% confidence interval (open circles) are the following: R ) -0.730, ns ) 1.037 mmol g-1, R12 ) 224.230, Q ˆ ) 0.012 934, and the significance test for R, AE, and BE is positive, i.e., denoted by “+”. This example shows that the absolute analysis of the experimental data has to contain both the linear equations and the excess adsorption isotherms. The similar results are observed for the Schiessler-Ro¨we eq 3 (see Table 5) and for the generalized Everett eq 5 (Table 7). The experimental points (xi, yi) i ) 1, 2, ..., n from beyond the tolerance interval for the linear equations in Table 1 can be eliminated from calculations.10 They are probably burdened by the experimental errors. Obviously, we can obtain different values of the parameters Aj and Bj, j ) E, GE, SR, and SEr and owing to that more reliable values of ns and R12 are obtained.
Table 8. Results of the Numerical Analysis Obtained by Means of Everett Eq 2a code of correction A B
R
ns/ mmol g-1
R12
Q ˆ
0.890 0.946
0.725 0.945
3.630 2.699
0.503 165 0.106 016
significance test for R AE B E + +
+ +
+ +
a Results for the system benzene (1) + ethanol (2) on graphon at 298 K16 (system 10 from Table 3); (A) for all the experimental points and (B) after rejection of the point from beyond the tolerance interval. See also Figure 6.
For the illustrative purposes, we present the results of our numerical analysis in Table 8 (Everett equation), Table 9 (generalized Everett equation), and Table 10 (Sˇ isˇkovaErdo¨s equation). But in Figures 6, 7, and 8, we present the excess adsorption isotherms calculated for all the experimental points (solid lines) and after elimination of the point from beyond the tolerance interval (dashed lines). Both the numerical results from Tables 8-10 and Figures 6-8 indicate applicability of our theoretical approach in obtaining more realistic parameters for the solid-liquid adsorption systems.
Surface Phase Capacity Determination
Langmuir, Vol. 13, No. 13, 1997 3473
Table 9. Results of the Numerical Analysis Obtained by Means of Generalized Everett Eq 5a code of correction A B
r
s n1,0 / mmol g-1
R
0.924 0.981 0.924 0.996
3.347 2.998
significance test for R12
Q ˆ
R AE BE
7.729 0.005 413 + + 11.000 0.001 081 + +
+ +
a Results for the system methyl acetate (1) + benzene (2) on silica gel at 298 K15 (system 8 from Table 3); (A) for all the experimental points and (B) after rejection of the point from beyond the tolerance interval. See also Figure 7.
Table 10. Results of the Numerical Analysis Obtained by Means of S ˇ isˇ kova-Erdo1 s Eq 4a code of correction
R
ns/ mmol g-1
A B
-0.726 -0.879
1.146 1.043
R12
Q ˆ
significance test for R AE BE
86.227 0.481 177 + 110.251 0.090 305 +
+ +
+ +
a For the system methyl propyl ketone (1) + benzene (2) on silica gel at 298 K17 (system 11 from Table 3); (A) for all the experimental points and (B) after rejection of the point from beyond the tolerance interval. See also Figure 8.
Conclusions Reliability of the adsorption parameters ns and R12 obtained in terms of a given linear equation depends on the results of the statistical analysis. The sum of the
squares of the remainders Q ˆ and the remainder variance sy2ˆ gives a full measure of the error referring to the experimental system considered in terms of the linear regression model. The value Q ˆ and the remainder variance sy2ˆ are closer and closer to zero, the fit of the theoretical straight line to the experimental data is better. Consequently, the values of the surface phase capacities are more reliable. The measure of the model linearity is the value of the correlation parameter R. The value |R| is close to unity, the best approximation of the experimental data by regression straight line is observed. Determination of the significance test for the coefficient R gives information about the correlation between the variables X and Y. Without this correlation, the evaluated adsorption parameters have no physical sense. Determination of the significance test for the coefficients Aj and Bj (j ) E, GE, SR, SEr) gives the information to what degree these coefficients are indeed different from zero. Insignificance of these parameters is the reason for nonphysical values of ns and R12. Moreover, the standard deviations for these parameters should be close to zero. Determination of the tolerance and confidence intervals for the experimental points departing from the regression straight lines gives possibility for eliminating the experimental points from beyond these intervals and for correcting the values of the adsorption parameters characterizing the solid-liquid adsorption systems. LA9700634
