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