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# Chemical Reaction Engineering ## 4 Kinetic Models ### Rate Laws * **Elementary Rate Law** $A + B \rightleftarrows C + D$ $-r_{A} = k_{A}C_{A}C_{B} - k_{-A}C_{C}C_{D}$ $k_{A}$: specific reaction rate (rate constant) $C_{i}$: concentration of species i $k_{-A}$: specifi...

# Chemical Reaction Engineering ## 4 Kinetic Models ### Rate Laws * **Elementary Rate Law** $A + B \rightleftarrows C + D$ $-r_{A} = k_{A}C_{A}C_{B} - k_{-A}C_{C}C_{D}$ $k_{A}$: specific reaction rate (rate constant) $C_{i}$: concentration of species i $k_{-A}$: specific reaction rate constant for the reverse reaction at equilibrium, $-r_{A} = 0$ $k_{A}C_{A_{e}}C_{B_{e}} = k_{-A}C_{C_{e}}C_{D_{e}}$ $\frac{k_{A}}{k_{-A}} = \frac{C_{C_{e}}C_{D_{e}}}{C_{A_{e}}C_{B_{e}}} = K_{c}$ $K_{c}$: concentration equilibrium constant * **Nonelementary Rate Laws** **Example**: $2A + B \rightarrow C$ $-r_{A} = \frac{k_{A}C_{A}C_{B}}{1 + K_{A}C_{A}}$ * **Reversible Reactions** **Example**: $A \rightleftarrows B$ $-r_{A} = k_{A}(C_{A} - \frac{C_{B}}{K_{c}})$ $K_{c} = \frac{C_{B_{e}}}{C_{A_{e}}}$ * **Irreversible Reactions** **Example**: $A \rightarrow B$ $-r_{A} = k_{A}C_{A}$ ### The Rate Constant $k = A e^{-\frac{E}{RT}}$ $k$: Rate constant $A$: Frequency factor $E$: Activation energy $R$: Gas constant $T$: Temperature $\ln{k} = \ln{A} - \frac{E}{RT}$ ### Activation Energy The activation energy is usually determined experimentally by carrying out the reaction at several different temperatures. Then, the natural logarithm of the rate constant is plotted as a function of the reciprocal of the absolute temperature. The activation energy is determined from the slope of the resulting line. $E = -R \cdot slope$ ### Reaction Order * The reaction order refers to the concentration dependence of the rate, not to the stoichiometric coefficients. * Reaction orders are determined from experimental observations. * Overall reaction order, $n$, is defined as the sum of the individual orders. $-r_{A} = k_{A}C_{A}^{\alpha}C_{B}^{\beta}$ $n = \alpha + \beta$ * Reaction orders can be integer, noninteger, or zero. Following table shows the rate laws for simple reactions: | Reaction Order | Rate Law | | :------------- | :----------------------- | | Zero-order | $-r_{A} = k$ | | First-order | $-r_{A} = kC_{A}$ | | Second-order | $-r_{A} = kC_{A}^{2}$ | | Second-order | $-r_{A} = kC_{A}C_{B}$ | ### Example For the reaction $A \rightarrow B$, the following data were obtained in a batch reactor: | Time (min) | $C_{A}$ $(mol/dm^{3})$ | | :--------- | :---------------------- | | 0 | 2 | | 1 | 1 | | 2 | 0.5 | | 3 | 0.25 | | 4 | 0.125 | Determine the reaction order and the rate constant. **Solution**: Try fitting the data to a first-order reaction: $-r_{A} = kC_{A}$ $\ln{\frac{C_{A}}{C_{A_{0}}}} = -kt$ | Time (min) | $C_{A}$ $(mol/dm^{3})$ | $\ln{C_{A}}$ | | :--------- | :---------------------- | :----------- | | 0 | 2 | 0.693 | | 1 | 1 | 0 | | 2 | 0.5 | -0.693 | | 3 | 0.25 | -1.386 | | 4 | 0.125 | -2.079 | The plot of $\ln{C_{A}}$ vs t produced a straight line, therefore, the reaction is first order. $k = -\text{slope} = 0.693 \ min^{-1}$