PUPAIM
This is the introduction to the package ’“PUPAIM”. This package is a
collection of physical and chemical adsorption isotherm models. This
package caters both the linear and nonlinear functions of the adsorption
isotherm models, and can give information about the parameters of the
isotherm used as well as the linear regression and the error analysis of
that isotherm. Using this package, fitting of adsorption data to
different adsorption isotherm models can be made easily.
Adsorption Plot
Ce <- c(0.01353, 0.04648, 0.13239, 0.27714, 0.41600, 0.63607, 0.80435, 1.10327, 1.58223)
Qe <- c(0.03409, 0.06025, 0.10622, 0.12842, 0.15299, 0.15379, 0.15735, 0.15735, 0.16607)
data <- data.frame(Ce,Qe)
plot(data,main = "Adsorption Data", xlab = "Ce",
ylab = "Qe")
This is an example of an adsorption plot with the equilibrium
capacity (Ce) on the x-axis and the adsorbed capacity (Qe) on the
y-axis.Depending on the adsorption isotherm model, these variables may
change as well as the line of fit.
There are several isotherm models that can be applied in adsorption
systems. These includes: (1) One-parameter isotherm model - Henry’s
linear model; (2) Two-parameter isotherm model - Langmuir isotherm
model, Freundlich isotherm model, Temkin isotherm model, Halsey isotherm
model, Elovich isotherm model, Hill-Deboer isotherm model,
Fowler-Guggenheim isotherm model, Dubinin-Radushkevich isotherm model,
Flory-Huggins isotherm model, Hill isotherm model, Hankin-Jura isotherm
model, Jovanovic isotherm model, Bruanuer-Emett-Teller (BET) isotherm
model and Kiselev isotherm model; (3) Three-parameter isotherm model -
Redlich-Peterson isotherm model, Sips isotherm model, Toth isotherm
model, Koble-Carrigan isotherm model, Kahn isotherm model,
Radke-Prausnitz isotherm model, and Jossens isotherm model; (4)
Four-parameter isotherm model - Baudu isotherm model, Weber-Van Vliet
isotherm model, Marczewski-Jaronic isotherm model, and Fritz-Schlunder
isotherm model; (5) Five-parameter isotherm model - Fritz-Schlunder
isotherm model. Adsorption isotherm models are not limited to these
since there are still more other isotherm models in existence.
If a single adsorption data is used to these isotherm models, the
corresponding data fitting as well as error analysis would different
from one to another.
Example: One-Parameter Adsorption Isotherm Model
Ce <- c(0.01353, 0.04648, 0.13239, 0.27714, 0.41600, 0.63607, 0.80435, 1.10327, 1.58223)
Qe <- c(0.03409, 0.06025, 0.10622, 0.12842, 0.15299, 0.15379, 0.15735, 0.15735, 0.16607)
data <- data.frame(Ce,Qe)
henryanalysis(Ce, Qe)
#> [1] "Henry Isotherm Analysis"
#>
#> Call:
#> lm(formula = y ~ x)
#>
#> Residuals:
#> Min 1Q Median 3Q Max
#> -0.05242 -0.02854 0.01149 0.02369 0.03866
#>
#> Coefficients:
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept) 0.08558 0.01628 5.258 0.00118 **
#> x 0.06911 0.02175 3.178 0.01554 *
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Residual standard error: 0.03263 on 7 degrees of freedom
#> Multiple R-squared: 0.5906, Adjusted R-squared: 0.5321
#> F-statistic: 10.1 on 1 and 7 DF, p-value: 0.01554
#>
#> [1] "K"
#> [1] 0.06910649
#> [1] "Akaike Information Criterion"
#> [1] -32.32572
#> [1] "Bayesian Information Criterion"
#> [1] -31.73405
#> $`Relative Mean Square Error`
#> [1] 0.02877796
#>
#> $`Mean Absolute Error`
#> [1] 0.02539692
#>
#> $`Mean Squared Error`
#> [1] 0.000828171
#>
#> $`Relative Absolute Error`
#> [1] 0.6659385
#>
#> $`Standard Error for the Regression S`
#> [1] 5.507576e-17
Example: Two-Parameter Adsorption Isotherm Model
Ce <- c(0.01353, 0.04648, 0.13239, 0.27714, 0.41600, 0.63607, 0.80435, 1.10327, 1.58223)
Qe <- c(0.03409, 0.06025, 0.10622, 0.12842, 0.15299, 0.15379, 0.15735, 0.15735, 0.16607)
data <- data.frame(Ce,Qe)
hill.LM(Ce, Qe)
#> [1] "Hill Isotherm Linear Analysis"
#>
#> Call:
#> lm(formula = y ~ x)
#>
#> Residuals:
#> Min 1Q Median 3Q Max
#> -0.10629 -0.05076 0.01193 0.02262 0.16555
#>
#> Coefficients:
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept) 0.93956 0.03790 24.79 4.43e-08 ***
#> x 0.86213 0.04431 19.45 2.36e-07 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Residual standard error: 0.0856 on 7 degrees of freedom
#> Multiple R-squared: 0.9818, Adjusted R-squared: 0.9792
#> F-statistic: 378.5 on 1 and 7 DF, p-value: 2.365e-07
#>
#> [1] "nh"
#> [1] 0.8621345
#> [1] "Kd"
#> [1] -0.9395641
#> [1] "qh"
#> [1] 0.1785724
#> [1] "Aikake Information Criterion"
#> [1] -14.9659
#> [1] "Bayesian Information Criterion"
#> [1] -14.37422
#> $`Relative Mean Squared Error`
#> [1] 0.07549337
#>
#> $`Mean Absolute Error`
#> [1] 0.05699024
#>
#> $`Mean Squared Error`
#> [1] 0.005699248
#>
#> $`Relative Absolute Error`
#> [1] 0.1186211
#>
#> $`Standard Error for the Regression S`
#> [1] 8.707216e-16
hillanalysis(Ce, Qe)
#> [1] "Hill Isotherm Nonlinear Analysis"
#>
#> Formula: y ~ ((qh * x^nh)/(Kd + x^nh))
#>
#> Parameters:
#> Estimate Std. Error t value Pr(>|t|)
#> qh 0.178572 0.008176 21.841 6.02e-07 ***
#> nh 0.895025 0.104650 8.553 0.00014 ***
#> Kd 0.111296 0.040343 2.759 0.03291 *
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Residual standard error: 0.005496 on 6 degrees of freedom
#>
#> Algorithm "port", convergence message: relative convergence (4)
#>
#> [1] "Akaike Information Criterion"
#> [1] -63.77458
#> [1] "Bayesian Information Criterion"
#> [1] -62.98569
#> $`Root Mean Squared Error`
#> [1] 0.6358941
#>
#> $`Mean Absolute Error`
#> [1] 0.003279588
#>
#> $`Mean Squared Error`
#> [1] 2.013943e-05
#>
#> $`Relative Absolute Error`
#> [1] 0.08599483
#>
#> $`Standard Error for the Regression S`
#> [1] 0.0007138005
#>
#>
#> Call:
#> lm(formula = Qe ~ predict(fit2))
#>
#> Residuals:
#> Min 1Q Median 3Q Max
#> -0.0059912 -0.0038649 0.0000993 0.0009123 0.0094829
#>
#> Coefficients:
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept) 0.001776 0.004889 0.363 0.727
#> predict(fit2) 0.987352 0.037071 26.634 2.69e-08 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Residual standard error: 0.005041 on 7 degrees of freedom
#> Multiple R-squared: 0.9902, Adjusted R-squared: 0.9888
#> F-statistic: 709.4 on 1 and 7 DF, p-value: 2.695e-08
Example: Three-Parameter Adsorption Isotherm Model
Ce <- c(0.01353, 0.04648, 0.13239, 0.27714, 0.41600, 0.63607, 0.80435, 1.10327, 1.58223)
Qe <- c(0.03409, 0.06025, 0.10622, 0.12842, 0.15299, 0.15379, 0.15735, 0.15735, 0.16607)
data <- data.frame(Ce,Qe)
redlichpeterson.LM(Qe, Ce)
#> [1] "Redlich-Peterson Isotherm Linear Analysis"
#>
#> Call:
#> lm(formula = y ~ x)
#>
#> Residuals:
#> Min 1Q Median 3Q Max
#> -0.40185 -0.25413 0.02166 0.13093 0.59882
#>
#> Coefficients:
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept) 2.050 0.237 8.651 5.52e-05 ***
#> x -49.319 7.632 -6.462 0.000346 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Residual standard error: 0.3192 on 7 degrees of freedom
#> Multiple R-squared: 0.8564, Adjusted R-squared: 0.8359
#> F-statistic: 41.76 on 1 and 7 DF, p-value: 0.0003462
#>
#> [1] "Krep"
#> [1] 0.4878194
#> [1] "Arep"
#> [1] -24.05862
#> [1] "Nrep"
#> [1] 1.759505
#> [1] "Aikake Information Criterion"
#> [1] 8.725798
#> [1] "Bayesian Information Criterion"
#> [1] 9.317471
#> $`Relative Mean Squared Error`
#> [1] 0.2815331
#>
#> $`Mean Absolute Error`
#> [1] 0.2115007
#>
#> $`Mean Squared Error`
#> [1] 0.07926087
#>
#> $`Relative Absolute Error`
#> [1] 0.3698422
#>
#> $`Standard Error for the Regression S`
#> [1] 1.300837e-15
redlichpetersonanalysis(Qe, Ce)
#> [1] "Redlich-Peterson Isotherm Nonlinear Analysis"
#>
#> Formula: y ~ (Krp * x)/(1 + (Arp * (x^g)))
#>
#> Parameters:
#> Estimate Std. Error t value Pr(>|t|)
#> Krp 1.678 3.679 0.456 0.664
#> Arp -18.557 180.161 -0.103 0.921
#> g 1.760 5.722 0.308 0.769
#>
#> Residual standard error: 0.2448 on 6 degrees of freedom
#>
#> Algorithm "port", convergence message: false convergence (8)
#>
#> [1] "Aikake Information Criterion"
#> [1] 4.563126
#> [1] "Bayesian Information Criterion"
#> [1] 5.352024
#> $`Relative Mean Squared Error`
#> [1] 0.1999124
#>
#> $`Mean Absolute Error`
#> [1] 0.1741445
#>
#> $`Mean Squared Error`
#> [1] 0.03996497
#>
#> $`Relative Absolute error`
#> [1] 0.4127507
#>
#> $`Standard Error for the Regression S`
#> [1] 0.2612517
#>
#>
#> Call:
#> lm(formula = Qe ~ predict(fit2))
#>
#> Residuals:
#> Min 1Q Median 3Q Max
#> -0.34274 -0.09869 -0.01216 0.12168 0.23508
#>
#> Coefficients:
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept) -0.1779 0.1214 -1.465 0.18633
#> predict(fit2) 1.1595 0.1615 7.181 0.00018 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Residual standard error: 0.1961 on 7 degrees of freedom
#> Multiple R-squared: 0.8805, Adjusted R-squared: 0.8634
#> F-statistic: 51.57 on 1 and 7 DF, p-value: 0.0001804
Example: Four-Parameter Adsorption Isotherm Model
Ce <- c(0.01353, 0.04648, 0.13239, 0.27714, 0.41600, 0.63607, 0.80435, 1.10327, 1.58223)
Qe <- c(0.03409, 0.06025, 0.10622, 0.12842, 0.15299, 0.15379, 0.15735, 0.15735, 0.16607)
data <- data.frame(Ce,Qe)
FS4analysis(Ce, Qe)
#> [1] "Fritz-Schlunder Four Parameter Isotherm Nonlinear Analysis"
#>
#> Formula: y ~ (C * x^a)/(1 + D * x^b)
#>
#> Parameters:
#> Estimate Std. Error t value Pr(>|t|)
#> C 0.62499 0.44719 1.398 0.2211
#> D 2.87927 2.79846 1.029 0.3507
#> a 0.67949 0.17620 3.856 0.0119 *
#> b 0.84770 0.06944 12.208 6.52e-05 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Residual standard error: 0.005249 on 5 degrees of freedom
#>
#> Number of iterations to convergence: 11
#> Achieved convergence tolerance: 6.613e-06
#>
#> [1] "Akaike Information Criterion"
#> [1] -64.24246
#> [1] "Bayesian Information Criterion"
#> [1] -63.25634
#> $`Root Mean Squared Error`
#> [1] 0.00391273
#>
#> $`Mean Absolute Error`
#> [1] 0.003466885
#>
#> $`Mean Squared Error`
#> [1] 1.530946e-05
#>
#> $`Relative Absolute Error`
#> [1] 0.09090599
#>
#> $`Standard Error for the Regression S`
#> [1] 0.002872805
#>
#>
#> Call:
#> lm(formula = Qe ~ predict(fit3))
#>
#> Residuals:
#> Min 1Q Median 3Q Max
#> -0.004354 -0.004107 -0.001295 0.002352 0.007638
#>
#> Coefficients:
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept) 0.0006429 0.0043268 0.149 0.886
#> predict(fit3) 0.9954182 0.0328031 30.345 1.09e-08 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Residual standard error: 0.00443 on 7 degrees of freedom
#> Multiple R-squared: 0.9925, Adjusted R-squared: 0.9914
#> F-statistic: 920.8 on 1 and 7 DF, p-value: 1.089e-08
Published Journals
The PUPAIM R package is a very effective tool to fitting a certain
adsorption isotherm model. As what demonstrated by the published journal
of Mr. Lloyd de Osio, an undergraduate Chemistry student, and
Mr. Chester C. Deocaris, the maintainer of PUPAIM package, with the
title: > “FITTING HENRY’S ADSORPTION ISOTHERM MODEL IN R USING THE
PUPAIM PACKAGE” (via)