## 4 Results and Discussion
### Adsorption Isotherm Fitting  
The experimental data of MSC-30/R1234 vf pair of this work and the reference data on MSC-30/R134a pair [29] were re-fitted with the Freundlich, Langmuir, Toth, D-R, and D-A isotherm models. All of these models describe the Type I isotherm model as recommended by the IUPAC. The root-mean-square error (RSME) method, which is also presented by [PERSON] et al. [58], is used to analyze the fitting or regression error. It is defined in the following formula:  
\[100\sqrt{\sum_{i=1}^{n}\frac{(abs(exp_{i}-model_{i}))^{2}}{N}} \tag{35}\]  
Figure 5: The transient temperature of MSC-30/R1234 vf pair at adsorption isotherm (30 \({}^{\circ}\)C).  
where \(exp_{i}\) is the experimental value at point \(i\), and \(model_{i}\) is the calculated value given by the model at point \(i\). The fitted results by the isotherm models and error values are summarized in the Supplementary Materials.  
Currently, the Toth model and the D-A model are possibly the most applied models for refrigerants/activated carbon systems and used for adsorption isotherm analysis thanks to their accuracy and simplicity. However, the Freundlich, Langmuir, and D-R models are also still being used due to their historical development and the fact that they can provide, together with the current models, some additional insight based on the relevant adsorption theories. From the perspective of classical models' accuracy, the empirical Freundlich model offers 8.53% error for MSC-30/R134a and a virtually identical 8.52% error for the MSC-30/R1234 yf pair, while the simple Langmuir model offers 6.53% error for MSC-30/R134a and 8.92% error for MSC-30/R1234 yf. In comparison with the conventional Toth model, these fitting errors are about two times higher for MSC-30/R134a, and more than five times higher for MSC-30/R1234 yf, while the actual fit error values for the Toth model are as low as 3.06% and 1.66%, respectively. The fitting of the Toth model for MSC-30/R1234 yf is shown in Figure 6.  
A relatively high fitting error is observed using D-R model with 9.82% error for MSC-30/R134a pair and 7.56% error for MSC-30/R1234 yf pair, while its limitation can be obviously seen in the fixed exponent factor. This problem is closely addressed in the development of the D-A model introducing the heterogeneity exponent factor \(n\), which becomes a fittable parameter. This substantially increases the accuracy of the model in comparison to the D-R model, although this is at the expense of an additional fitting constant. Furthermore, from Tables S1 to S4 in the Supplementary Materials, we can notice that the adsorbed phase volume correction can improve the fitting accuracy, which was also observed by others [21]. We can see that the goodness of fit in the case of MSC-30/R134a pair is decreasing in a sequence of \(\alpha\) = ND model, \(\alpha\) = 1/T, and \(\alpha\) = 0.0025, with respective error values of 4.75%, 5.44%, and 5.63%, while the D-A model without volume correction provides with a fitting error of 3.77%. On the other hand, the data also show a notable increase in accuracy for all the volume corrections of the MSC-30/R1234 yf pair; nonetheless, no significant variation between the individual correction approaches manifested, and the fitting error obtained was approximately 2.5-2.6% in every case. An example of fitting the D-A model on the MSC-30/R1234 yf pair where \(\alpha\) = 1/T is presented in Figure 7. Overall, the Toth temperature-dependent model shows better fitting errors for both pairs.  
Figure 6: Isotherm fitting results of the Toth model for MSC-30/R1234 yf.