In short, the highest correlation coefficient (R2) value and the lower chi-square (X2) values are the best indications to fit the adsorption model and describe the adsorption process.

Eman El-Gamal Thank you for your answer!

Of course, I understand that, but in my case, the model that has the highest R2 does not have the lowest X2 value. So I'm confused as how to explain that.

When analyzing the fit of a model to experimental or numerical data, both R-squared (R2R^2) and Chi-squared (χ2\chi^2) are commonly used goodness-of-fit metrics, but they serve different purposes and interpretations.

1. R-squared (R2R^2) – Coefficient of Determination

• Purpose: Measures how well the independent variable(s) explain the variance in the dependent variable.
• Typical Use: Regression models, including linear and nonlinear fitting.
• Range: 0≤R2≤10 \leq R^2 \leq 1 (Higher values indicate better fit).
• Formula: R2=1−∑(yobs−ypred)2∑(yobs−yˉ)2R^2 = 1 - \frac{\sum (y_{\{obs}} - y_{\{pred}})^2}{\sum (y_{\{obs}} - \bar{y})^2} where: yobsy_{\{obs}} = observed values ypredy_{\{pred}} = predicted values yˉ\bar{y} = mean of observed values

Interpretation:

• An R2R^2 value close to 1 indicates a strong correlation between the predicted and observed values.
• Does not indicate whether the model is appropriate, just how well it fits the data.

2. Chi-squared (χ2\chi^2) – Goodness of Fit Test

• Purpose: Evaluates how well observed data match expected (theoretical) values, accounting for data uncertainty.
• Typical Use: Statistical hypothesis testing, curve fitting in physical sciences, and assessing fit with known distributions.
• Formula: χ2=∑(yobs−ypred)2σ2\chi^2 = \sum \frac{(y_{\text{obs}} - y_{\text{pred}})^2}{\sigma^2}where:yobsy_{\text{obs}} = observed values ypredy_{\text{pred}} = predicted values σ\sigma = standard deviation or uncertainty of observed values

Interpretation:

• Lower χ2\chi^2 values indicate better fit.
• If χ2≈\chi^2 \approx degrees of freedom (DOF), the model is considered a good fit.
• High χ2\chi^2 suggests the model may not be appropriate for the data.

Which Should You Use?

• Use R2R^2 if you're interested in explaining the variance and assessing the proportion of explained variability.
• Use χ2\chi^2 if your data include uncertainties or errors and you're interested in statistical hypothesis testing.

For Sips Isotherm Model Fitting:

• If your focus is on assessing how well the model describes experimental adsorption data in terms of variance, R2R^2 is useful.
• If experimental errors and uncertainties are considered significant, χ2\chi^2 provides a more rigorous assessment of the fit.