Modeling Adsorption Isotherms with AdsorpR
📦 Introduction
The AdsorpR package provides functions
for modeling four classical adsorption isotherms:
- Langmuir (monolayer adsorption)
- Freundlich (heterogeneous surfaces)
- BET (multilayer adsorption)
- Temkin (linear decrease of adsorption energy)
These models are commonly used in environmental and chemical engineering studies to describe sorption mechanisms of contaminants onto solid adsorbents.
🧪 Sample Dataset
We demonstrate model usage using a simple example dataset
representing equilibrium concentration (Ce) and the amount
adsorbed (Qe):
📐 Langmuir Isotherm
result_l <- langmuir_model(Ce, Qe)
print(result_l[1:2]) # Qmax and KL
#> $`Langmuir Qmax (mg/g)`
#> Ce
#> 8.669989
#>
#> $`Langmuir KL (L/mg)`
#> Ce
#> 0.1040153
print(result_l$`Model Summary`)
#>
#> Call:
#> lm(formula = Ce_by_Qe ~ Ce)
#>
#> Residuals:
#> 1 2 3 4 5
#> 0.025780 -0.006227 -0.026329 -0.031780 0.038556
#>
#> Coefficients:
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept) 1.10888 0.03778 29.35 8.69e-05 ***
#> Ce 0.11534 0.01139 10.12 0.00205 **
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Residual standard error: 0.03602 on 3 degrees of freedom
#> Multiple R-squared: 0.9716, Adjusted R-squared: 0.9621
#> F-statistic: 102.5 on 1 and 3 DF, p-value: 0.002052
result_l$Plot
#> `geom_smooth()` using formula = 'y ~ x'
📐 Freundlich Isotherm
result_f <- freundlich_model(Ce, Qe)
print(result_f[1:2]) # Kf and n
#> $`Freundlich Kf`
#> (Intercept)
#> 0.8265344
#>
#> $`Freundlich n`
#> log_Ce
#> 1.228491
print(result_f$`Model Summary`)
#>
#> Call:
#> lm(formula = log_Qe ~ log_Ce)
#>
#> Residuals:
#> 1 2 3 4 5
#> -0.014171 0.013790 0.016579 0.007632 -0.023829
#>
#> Coefficients:
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept) -0.08274 0.01818 -4.551 0.019877 *
#> log_Ce 0.81401 0.03759 21.653 0.000216 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Residual standard error: 0.02075 on 3 degrees of freedom
#> Multiple R-squared: 0.9936, Adjusted R-squared: 0.9915
#> F-statistic: 468.8 on 1 and 3 DF, p-value: 0.0002156
result_f$Plot
#> `geom_smooth()` using formula = 'y ~ x'
📐 BET Isotherm
result_b <- bet_model(Ce, Qe)
print(result_b[1:2]) # Qm and Cb
#> $`BET Qm (mg/g)`
#> BET_x
#> 0.346352
#>
#> $`BET Cb`
#> BET_x
#> -3.023631
print(result_b$`Model Summary`)
#>
#> Call:
#> lm(formula = BET_y ~ BET_x)
#>
#> Residuals:
#> 1 2 3 4 5
#> 0.53410 -0.06129 -0.56939 -0.81374 0.91032
#>
#> Coefficients:
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept) -0.9549 0.8783 -1.087 0.3565
#> BET_x 3.8421 1.4566 2.638 0.0778 .
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Residual standard error: 0.8375 on 3 degrees of freedom
#> Multiple R-squared: 0.6987, Adjusted R-squared: 0.5983
#> F-statistic: 6.958 on 1 and 3 DF, p-value: 0.0778
result_b$Plot
#> `geom_smooth()` using formula = 'y ~ x'
📐 Temkin Isotherm
result_t <- temkin_model(Ce, Qe)
print(result_t[1:2]) # A and B
#> $`Temkin A`
#> (Intercept)
#> 1.712047
#>
#> $`Temkin B`
#> ln_Ce
#> 1.324248
print(result_t$`Model Summary`)
#>
#> Call:
#> lm(formula = adjusted_Qe ~ ln_Ce)
#>
#> Residuals:
#> 1 2 3 4 5
#> 217.9 -321.9 -165.7 129.2 140.4
#>
#> Coefficients:
#> Estimate Std. Error t value Pr(>|t|)
#> (Intercept) 1764.1 234.5 7.522 0.004870 **
#> ln_Ce 3280.9 210.6 15.577 0.000575 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Residual standard error: 267.7 on 3 degrees of freedom
#> Multiple R-squared: 0.9878, Adjusted R-squared: 0.9837
#> F-statistic: 242.7 on 1 and 3 DF, p-value: 0.0005749
result_t$Plot
#> `geom_smooth()` using formula = 'y ~ x'
🔁 Non-linear Isotherm Modeling
Non-linear Langmuir
Ce <- c(1, 2, 4, 6, 8, 10)
Qe <- c(0.9, 1.6, 2.3, 2.7, 2.9, 3.0)
result <- nonlinear_langmuir(Ce, Qe)
print(result$`Langmuir Qmax (mg/g)`)
#> Qmax
#> 4.000269
print(result$`Langmuir KL (L/mg)`)
#> KL
#> 0.3262891
print(result$AIC)
#> [1] -12.78784
print(result$`Pseudo R2`)
#> [1] 0.9955326
print(result$Plot)
#> `geom_smooth()` using formula = 'y ~ x'
Non-linear Freundlich
Ce <- c(0.5, 1, 2, 4, 6, 8)
Qe <- c(0.3, 0.8, 1.6, 2.4, 2.9, 3.2)
result <- nonlinear_freundlich(Ce, Qe)
print(result$`Freundlich Kf`)
#> Kf
#> 0.9214474
print(result$`Freundlich n`)
#> n
#> 1.598388
print(result$AIC)
#> [1] 3.105419
print(result$`Pseudo R2`)
#> [1] 0.9680789
print(result$Plot)
#> `geom_smooth()` using formula = 'y ~ x'
#> Warning in simpleLoess(y, x, w, span, degree = degree, parametric = parametric,
#> : Chernobyl! trL>n 6
#> Warning in simpleLoess(y, x, w, span, degree = degree, parametric = parametric,
#> : Chernobyl! trL>n 6
#> Warning in sqrt(sum.squares/one.delta): NaNs produced
Non-linear BET
Ce <- c(1, 2.5, 4, 5.5, 7)
Qe <- c(0.4, 1.0, 1.7, 2.3, 2.7)
result <- nonlinear_bet(Ce, Qe)
#> Warning in nls(Qe ~ (Qm * Cb * Ce)/((Cs - Ce) * (1 + (Cb - 1) * Ce/Cs)), : step
#> factor 5.82077e-11 reduced below 'minFactor' of 1e-10
print(result$`BET Qm (mg/g)`)
#> Qm
#> 0.2899103
print(result$`BET Cb`)
#> Cb
#> 836531218564
print(result$AIC)
#> [1] 18.29629
print(result$`Pseudo R2`)
#> [1] 0.02393608
print(result$Plot)
#> `geom_smooth()` using formula = 'y ~ x'
#> Warning in simpleLoess(y, x, w, span, degree = degree, parametric = parametric,
#> : span too small. fewer data values than degrees of freedom.
#> Warning in simpleLoess(y, x, w, span, degree = degree, parametric = parametric,
#> : pseudoinverse used at 0.97
#> Warning in simpleLoess(y, x, w, span, degree = degree, parametric = parametric,
#> : neighborhood radius 3.03
#> Warning in simpleLoess(y, x, w, span, degree = degree, parametric = parametric,
#> : reciprocal condition number 0
#> Warning in simpleLoess(y, x, w, span, degree = degree, parametric = parametric,
#> : There are other near singularities as well. 9.1809
Non-linear Temkin
Ce <- c(0.5, 1.5, 3, 4.5, 6)
Qe <- c(0.7, 1.3, 2.0, 2.4, 2.7)
result <- nonlinear_temkin(Ce, Qe)
print(result$`Temkin A`)
#> AT
#> 4.126122
print(result$`Temkin B`)
#> bT
#> 0.8129749
print(result$AIC)
#> [1] -2.230427
print(result$`Pseudo R2`)
#> [1] 0.9788445
print(result$Plot)
#> `geom_smooth()` using formula = 'y ~ x'
#> Warning in simpleLoess(y, x, w, span, degree = degree, parametric = parametric,
#> : span too small. fewer data values than degrees of freedom.
#> Warning in simpleLoess(y, x, w, span, degree = degree, parametric = parametric,
#> : pseudoinverse used at 0.4725
#> Warning in simpleLoess(y, x, w, span, degree = degree, parametric = parametric,
#> : neighborhood radius 2.5275
#> Warning in simpleLoess(y, x, w, span, degree = degree, parametric = parametric,
#> : reciprocal condition number 0
#> Warning in simpleLoess(y, x, w, span, degree = degree, parametric = parametric,
#> : There are other near singularities as well. 9.1658
📝 Conclusion
The AdsorpR package offers a clean and structured way to
fit adsorption isotherm models and visualize results using
ggplot2. It is useful for:
- Environmental monitoring studies
- Sorption modeling of nutrients and pollutants in soil
- Research on water/soil remediation