4.3 Adsorption kinetics
Adsorption kinetics of TCO on to AgNO3/silica was studied by performing an experiment in batch. In that experiment a known amount of AgNO3/silica, (0.175g) was suspended in the solution of TCO, C0=8.06 g/dm3 in n-hexane. The time history of the liquid concentration of TCO was recorded, see Figure 4. In order to prevent any external mass transfer limitation, the suspension was stirred intensively. The experimental data were analyzed using (pseudo-)first-order and (pseudo-)second-order kinetics [48]–[52], see Eq. 4.1 and 4.3, respectively:
\begin{equation} \frac{dq_{t}}{\text{dt}}={k\left(q_{\max}-q_{t}\right)}^{n}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (4.1)\nonumber \\ \end{equation}
n=1 for (pseudo-)first-order and n=2 for (pseudo-)second-order
\begin{equation} \ln\frac{\left(q_{B_{e}}-q_{B_{t}}\right)}{q_{B_{e}}}=-k^{\prime}t\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (4.2)\nonumber \\ \end{equation}\begin{equation} \frac{1}{q_{B_{t}}}=\frac{1}{k"q_{B_{e}}^{2}\text{.t}}+\frac{1}{q_{B_{e}}}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (4.3)\nonumber \\ \end{equation}
Where k’ (min-1) is the rate constant for the (pseudo-)first-order and \(k"\) (g mg-1min-1) is the rate constant for the (pseudo-)second-order kinetic models.qBe andqBt are the amount of the trans -isomer adsorbed per gram AgNO3 at equilibrium and at time t, respectively. The constants can be calculated from the intercepts and slopes of the linear plots of \(\ln{(q_{B_{e}}-q_{B_{t}})}\)versus t and \(t/q_{B_{t}}\) versus \(t\), respectively (Figure 4).
Figure 4a shows that adsorption of trans -cyclooctene proceeds rapidly during the first 30s and becomes almost constant after 1 min. The adsorption rate constants and linear regression values are collected in Table 2. The results in Figure 4 and Table 2 reveal that adsorption obeys a (pseudo-)second order rate law. Also the experimentally observed value of \(q_{B_{e}}\) is equal to 44.3 × 10-3gtrans/gAgNO3 and it can be easily observed that it is very close to \(q_{B_{e}}\) calculated from pseudo-second order kinetic model This fact suggests that the adsorption rate of trans -cyclooctene is dependent on the adsorption site availability on AgNO3 rather than thetrans -cyclooctene concentration in solution [53], [54]. Initially, there are many adsorption sites are available, however with the prolonging time the sites have been occupied, hence, limited free sites for molecule to be adsorbed on [53], [55].
Table 2 pseudo-second-order rate constant for adsorption of TCO onto AgNO3 at 23˚C