## 3 Results and Discussion
### Derived Thermodynamics Properties  
The isothermal compressibility, \(\kappa_{\rm T}\), was calculated with the modified Tait Equation (1), which was calculated by [29]:  
\[\kappa_{\rm T}(T,p)=\left(\frac{1}{\rho}\right)\left(\frac{\partial\rho}{ \partial p}\right)=\frac{C}{\left(1-C\ln\Bigl{(}\frac{B(T)+p}{B(T)+0.1} \Bigr{)}\Bigr{)}\bigl{(}B(T)+p\bigr{)}} \tag{7}\]  
We suppose that \(\rho_{\rm P}(T)=a_{0}+a_{1}T+a_{2}T^{2}+a_{3}T^{3}\), the isobaric thermal expansivity, \(\alpha_{\rm p}\), was determined  
\[\alpha_{\rm p}=-\frac{a_{1}+2a_{2}T+3a_{3}T^{2}}{a_{0}+a_{1}T+a_{2}T^{2}+a_{3 }T^{3}} \tag{8}\]  
The isothermal compressibility and isobaric thermal expansivity of the mixtures are listed in Table S2 (provided as Supplementary Materials) and illustrated in Figures 5 and 6. The isothermal compressibility and isobaric thermal expansivity of the CO\({}_{2}+n\)-decane mixtures increase with increasing CO\({}_{2}\) concentration. The excess molar volume was obtained using the following equation:  
\[V_{\rm m}^{\rm E}=\frac{x_{1}M_{1}+(1-x_{1})M_{2}}{\rho_{\rm mix}}-x_{1}\frac{ M_{1}}{\rho_{1}}-(1-x_{1})\frac{M_{2}}{\rho_{2}} \tag{9}\]  
where \(\rho_{1}\) and \(\rho_{2}\) represent the density of the pure components obtained from Refprop 10.0 [43] and this work, respectively. The excess molar volumes were calculated and are presented in Table S2 (provided as Supplementary Materials). A comparison of the excess molar volumes with the Helmholtz EoS at 343 K is shown in Figure 7. The negative values of the excess molar volume resulted from the filling effects and the dispersion interaction between CO\({}_{2}\) and the \(n\)-decane molecules. It is important to note that a deviation of 0.12% in density will generate a deviation of 0.10-0.25 cm\({}^{3}\cdot\)mol\({}^{-1}\) in excess molar volume [44]. This difference could be a result of the precision in measuring the density. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Parameters** & \(\mathbf{x_{1}=0}\) & \(\mathbf{x_{1}=0.2032}\) & \(\mathbf{x_{1}=0.4434}\) & \(\mathbf{x_{1}=0.7589}\) & \(\mathbf{x_{1}=0.8947}\) \\ \hline \(A_{0}/\text{g}\cdot\text{cm}^{-3}\) & 0.86080 & 1.03703 & 1.40990 & \(-\)0.01853 & 4.40805 \\ \(A_{1}/\text{g}\cdot\text{cm}^{-3}\cdot\text{K}^{-1}\) & \(-\)8.51553 \(\times\)\(10^{-6}\) & \(-\)1.41571 \(\times\)\(10^{-3}\) & \(-\)4.63320 \(\times\)\(10^{-3}\) & 8.91860 \(\times\)\(10^{-3}\) & \(-\)3.40563 \(\times\)\(10^{-2}\) \\ \(A_{2}/\text{g}\cdot\text{cm}^{-3}\cdot\text{K}^{-2}\) & \(-\)1.95868 \(\times\)\(10^{-6}\) & 2.03202 \(\times\)\(10^{-6}\) & 1.23117 \(\times\)\(10^{-8}\) & \(-\)2.61347 \(\times\)\(10^{-5}\) & 1.17519 \(\times\)\(10^{-4}\) \\ \(A_{3}/\text{g}\cdot\text{cm}^{-3}\cdot\text{K}^{-3}\) & 1.62432 \(\times\)\(10^{-9}\) & \(-\)2.33131 \(\times\)\(10^{-9}\) & \(-\)1.40789 \(\times\)\(10^{-10}\) & 1.73563 \(\times\)\(10^{-8}\) & \(-\)1.48211 \(\times\)\(10^{-7}\) \\ \(B_{0}/\text{MPa}\) & 344.951 & 366.422 & 327.392 & 452.091 & 271.332 \\ \(B_{1}/\text{MPa}\cdot\text{K}^{-1}\) & \(-\)1.20993 & \(-\)1.41645 & \(-\)1.32612 & \(-\)2.32403 & \(-\)1.466625 \\ \(B_{2}/\text{MPa}\cdot\text{K}^{-2}\) & 1.09394 \(\times\)\(10^{-3}\) & 1.45102 \(\times\)\(10^{-3}\) & 1.36670 \(\times\)\(10^{-3}\) & 3.00725 \(\times\)\(10^{-3}\) & 1.98299 \(\times\)\(10^{-3}\) \\ \(C\) & 9.12134 \(\times\)\(10^{-2}\) & 9.15440 \(\times\)\(10^{-2}\) & 8.94176 \(\times\)\(10^{-2}\) & 9.64531 \(\times\)\(10^{-2}\) & 9.71727 \(\times\)\(10^{-2}\) \\ \(\sigma/\text{g}\cdot\text{cm}^{-3}\) & 3.456 \(\times\)\(10^{-5}\) & 1.112 \(\times\)\(10^{-4}\) & 1.401 \(\times\)\(10^{-4}\) & 4.567 \(\times\)\(10^{-5}\) & 4.183 \(\times\)\(10^{-5}\) \\ \(AAD/\%\) & 0.028 & 0.013 & 0.017 & 0.044 & 0.042 \\ \(MAD/\%\) & 0.075 & 0.033 & 0.040 & 0.065 & 0.059 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Parameters and deviations in density correlation. Figure 5: Plot of the calculated isothermal compressibility for CO\({}_{2}\)(1) + \(n\)-decane(2) mixtures.