CHEM E 465 Reactor Design
Contents
Rate Law
Equilibrium constant
| Description | Equations |
|---|---|
| Equilibrium constant and concentration | $K_c = \prod_i C_i^{\nu_i}$ |
| Equilibrium constant and rate constant | $K_i = \dfrac{k_i}{k_{-i}}$ |
| van’t Hoff equation ★ $\Delta h_{\mathrm{rxn}}^\circ \not= f(T)$ |
$\dfrac{d \ln K}{dT} = \dfrac{\Delta h_{\mathrm{rxn}}^\circ}{RT^2}$ |
| T dependence of equilibrium constant | $\ln \left(\dfrac{K_2}{K_1}\right)=-\dfrac{\Delta h_{\mathrm{rxn}}^\circ}{R}\left(\dfrac{1}{T_2}-\dfrac{1}{T_1}\right)$ |
Rate constant
| Description | Equations |
|---|---|
| General reaction | $\ce{aA + bB ->[\mathit{k}] cC + dD}$ |
| Relative reaction rates | $\dfrac{-r_A}{a} = \dfrac{-r_B}{b} = \dfrac{-r_C}{c} = \dfrac{-r_D}{d}$ |
| Power law | $r_A = -k C_A^a C_B^b$ |
| Unit of rate constant | $k \ [=] \ \mathrm{M^{1-n}/s}$ |
| Arrhenius equation ★ $E_a \not= f(T)$ |
$\dfrac{d \ln k}{dT} = \dfrac{E_a}{RT^2}$ |
| Arrhenius equation | $k = A\exp\left(-\dfrac{E_a}{RT}\right)$ |
| T dependence of rate constant | $\ln \left(\dfrac{k_2}{k_1}\right)=-\dfrac{E_a}{R}\left(\dfrac{1}{T_2}-\dfrac{1}{T_1}\right)$ |
| Arrhenius plot | $\ln k = \ln A - \dfrac{E_a}{R}\dfrac{1}{T}$ |
| Note | Increase $E_a$, increase $k$’s sensitivity to T |
| Collision theory | $k \propto \sqrt{T}\exp\left(-\dfrac{E_0}{RT}\right)$ |
| Transition state theory | $k \propto T\exp\left(-\dfrac{\Delta H^*}{RT}\right)$ |
Reactor Design and Sizing
- Reactor design equation (mole balance): $r_i = f(n) = f(X)$
- Rate law: $r_i = f(C_i)$
- Stoichiometry: $C_i = f(X)$
- Combine: $r_i = f(X)$
| Description | Equations |
|---|---|
| General mole balance | $\begin{aligned}&\mathrm{in} &&- \mathrm{out} &&+ \mathrm{generation} &&= \mathrm{accumulation} \\ &\dot{n}_{i0} &&- \dot{n}_i &&+ G_i &&= \dfrac{dn_i}{dt}\end{aligned}$ |
| General generation term | $G_i = \int r_i dV$ |
| Spatially uniform generation term | $G_i = r_i V$ |
| Conversion | $X_i \equiv X = \dfrac{n_{i0} - n_{i}}{n_{i0}} = \dfrac{\dot{n}_{i0} - \dot{n}_{i}}{\dot{n}_{i0}}$ |
| Mole/flow rate in terms of conversion | $n_i = n_{i0} (1 - X) \newline \dot{n}_i = \dot{n}_{i0} (1 - X)$ |
| Heterogeneous reaction rate | $r_i = \rho_b r_i'$ |
| Reactor | Design Equations | Integral Form |
|---|---|---|
| Batch ★ Perfectly mixed |
$\begin{aligned}r_i V = \dfrac{dn_i}{dt} = -n_{i0}\dfrac{dX}{dt}\end{aligned}$ | $\begin{aligned}t = \displaystyle\int_{n_{i1}}^{n_{i0}} \dfrac{dn_i}{-r_i V} = n_{i0} \displaystyle\int_0^X \dfrac{dX}{-r_i V}\end{aligned}$ |
| CSTR ★ Perfectly mixed, steady-state |
$V = \dfrac{\dot{n}_{i0} - \dot{n}_i}{-r_i} = \dfrac{\dot{n}_{i0}X}{-r_i}$ | - |
| PFR ★ Steady state ★ Plug flow, no radial dependence |
$r_i = \dfrac{d\dot{n}_i}{dV} = -\dot{n}_{i0}\dfrac{dX}{dV}$ | $V = \displaystyle\int_{\dot{n}_i}^{\dot{n}_{i0}}\dfrac{d\dot{n}_i}{-r_i} = \dot{n}_{i0} \displaystyle\int_0^X \dfrac{dX}{-r_i}$ |
| PBR ★ Steady state |
$r_i' = \dfrac{d\dot{n}_i}{dm} = -\dot{n}_{i0}\dfrac{dX}{dm}$ | $m = \displaystyle\int_{\dot{n}_i}^{\dot{n}_{i0}}\dfrac{d\dot{n}_i}{-r_i'} = \dot{n}_{i0} \displaystyle\int_0^X \dfrac{dX}{-r_i'}$ |
| Semi-batch reactor | $r_i = \dfrac{\dfrac{dn_i}{dt} - \dot{n}_{i0}}{V(t)}$ | - |
Levenspiel plot
| Description | Equations |
|---|---|
| Levenspiel plot | $\dfrac{\dot{n}_{i0}}{-r_i}$ vs. $X$ |
| CSTR volume | $V = \dfrac{\dot{n}_{i0}X}{-r_i}$ = Area of rectangle |
| PFR volume | $V = \dot{n}_{i0} \displaystyle\int_0^X \dfrac{dX}{-r_i}$ = Area under Levenspiel plot |
| Reaction of order > 0 | $V_{\mathrm{CSTR}} > V_\mathrm{PFR}$ (Choose PFR) |
| Reaction of order < 0 | $V_{\mathrm{CSTR}} < V_\mathrm{PFR}$ (Choose CSTR) |
Reactor choice
| Reactor Type | Advantage | Disadvantage |
|---|---|---|
| Batch | - High conversion | - High labor cost and downtime - Difficult to scale up - Batch-to-batch variability |
| CSTR | - Good T control | - Hard to get high conversion |
| PFR | - Easy maintenance - High conversion per unit volume |
- Difficult for T control |
Stoichiometry
Stoichiometric table
| Species | Initial | Change | Remaining |
|---|---|---|---|
| $\ce{A}$ | $n_{{A}0}$ | $-n_{A0}X$ | $n_A = n_{A0}(1-X)$ |
| $\ce{B}$ | $n_{B0}$ | $-\frac{b}{a}n_{A0}X$ | $n_B = n_{A0}(\Theta_B - \frac{b}{a}X)$ |
| $\ce{C}$ | $n_{C0}$ | $\frac{c}{a}n_{A0}X$ | $n_C = n_{A0}(\Theta_C + \frac{c}{a}X)$ |
| $\ce{D}$ | $n_{D0}$ | $\frac{d}{a}n_{A0}X$ | $n_D = n_{A0}(\Theta_D + \frac{d}{a}X)$ |
| $\ce{I}$ | $n_{I0}$ | $0$ | $n_I = n_{I0}$ |
| Total | $n_{T0}$ | $\delta n_{A0}X$ | $n_T = n_{T0} + \delta n_{A0}X$ |
| Description | Equations |
|---|---|
| Sample reaction | $\color{blue}\ce{A + \dfrac{b}{a} B -> \dfrac{c}{a}C + \dfrac{d}{a}D}$ |
| $\dfrac{\text{total mol change}}{\text{mol A reacted}}$ | $\delta = \sum \nu_i = \dfrac{d}{a} + \dfrac{c}{a} - \dfrac{b}{a} - 1$ |
| $\dfrac{\text{mol Z initially}}{\text{mol A initially}}$ | $\Theta_Z = \dfrac{n_{Z0}}{n_{A0}} = \dfrac{\dot{n}_{Z0}}{\dot{n}_{A0}}= \dfrac{y_{Z0}}{y_{A0}}= \dfrac{C_{Z0}}{C_{A0}}$ |
| $\dfrac{\text{total mol change for } X = 1}{\text{mol feed}}$ | $\varepsilon = y_{A0}\delta = \dfrac{\dot{n}_{Tf} - \dot{n}_{T0}}{\dot{n}_{T0}}$ |
| Concentration | $C_i = \dfrac{n_i}{V} = \dfrac{\dot{n}_i}{\dot{V}}$ |
| Molar fraction | $y_i = \dfrac{\dot{n}_i}{\dot{n}_T}$ |
| Pressure ratio | $p = \dfrac{P}{P_0}$ |
| Description | Equations |
|---|---|
| Molar flow rate | $\dot{n}_i = \dot{n}_{A0} (\Theta_i - \nu_i X)$ |
| Concentration ★ Constant $PVT$ |
$C_i = C_{A0} (\Theta_i - \nu_i X)$ |
| Volumetric flow rate | $\begin{aligned}\dot{V} &= \dot{V}_0 \left(\dfrac{\dot{n}_T}{\dot{n}_{T0}}\right) \left(\dfrac{P_0}{P}\right) \left(\dfrac{T}{T_0}\right) \\ &= \dot{V}_0 (1 + \varepsilon X)\left(\dfrac{P_0}{P}\right) \left(\dfrac{T}{T_0}\right) \end{aligned}$ |
| Concentration | $\begin{aligned}C_i &= \dfrac{\dot{n}_i}{\dot{V}} \\ &= C_{A0} \left(\dfrac{\dot{n}_i}{\dot{n}_{T}}\right) \left(\dfrac{P}{P_0}\right) \left(\dfrac{T_0}{T}\right) \\ &= C_{A0} y_i p \left(\dfrac{T_0}{T}\right) \\ &= \dfrac{C_{A0} (\Theta_i + \nu_i X)}{1+\varepsilon X} \left(\dfrac{P}{P_0}\right) \left(\dfrac{T_0}{T}\right) \end{aligned}$ |
Isothermal Reactor Design
Batch reactor
| Description | Equations |
|---|---|
| Characteristic reaction time (1st order) | $t_R = \dfrac{1}{k}\ln\left(\dfrac{1}{1-X}\right)$ |
| Characteristic reaction time (2nd order) | $t_R = \dfrac{1}{k C_{A0}}\dfrac{X}{1-X}$ |
| Total time | Total = Fill + Heat + Reaction + Clean |
CSTR
| Description | Equations |
|---|---|
| Space time | $\tau = \dfrac{V}{\dot{V}} = \dfrac{C_{A0}X}{-r_A}$ |
| Damkohler number | $\begin{aligned}\mathrm{Da} &= \dfrac{-r_{A0}V}{\dot{n}_{A0}} \\ &= \dfrac{\text{rate at entrance}}{\text{molar flow at entrance}} \\ &= \dfrac{\text{reaction rate}}{\text{convective mass transport rate}}\end{aligned}$ |
| Damkohler number and conversion | $\mathrm{Da} < 0.1, X < 0.1 \newline \mathrm{Da} > 10, X > 0.9$ |
| Reaction, Reactor | Damkohler number $\mathrm{Da}_i$ | Space time $\tau$ | Conversion $X$ | Concentration $C_i$ |
|---|---|---|---|---|
| 1st order, single CSTR | $\begin{aligned}\tau k = \dfrac{X}{1-X}\end{aligned}$ | $\dfrac{X}{k(1-X)}$ | $\dfrac{\tau k}{1 + \tau k}$ | $\dfrac{C_{A0}}{1 + \tau k}$ |
| 2nd order, single CSTR | $\tau k C_{A0}$ | $\dfrac{X}{kC_{A0}(1-X)^2}$ | $\dfrac{1 + 2\mathrm{Da}_2 - \sqrt{1 + 4\mathrm{Da}_2}}{2\mathrm{Da}_2}$ | $C_{A0} \dfrac{-1 + \sqrt{1 + 4\mathrm{Da}_2}}{2\mathrm{Da}_2}$ |
| 1st order, CSTR series | - | - | $1 - \dfrac{1}{(1 + \mathrm{Da}_1)^n}$ | $\dfrac{C_{A0}}{(1 + \mathrm{Da}_1)^n}$ |
PFR
| Description | Equations |
|---|---|
| Reactor volume for 2nd order gas phase rxn | $V = \dfrac{\dot{V}_{A0}}{kC_{A0}} \left[ 2\varepsilon(1 + \varepsilon) \ln(1-X) + \varepsilon^2 X + \dfrac{(1 + \varepsilon)^2 X}{1-X} \right]$ |
| Ergun equation | $\dfrac{dP}{dz} = \dfrac{G}{\rho g_c D_P}\left(\dfrac{1 - \phi}{\phi^3}\right) \left[\dfrac{150(1-\phi)\mu}{D_P} + 1.75G\right]$ |
| Porosity (void fraction) | $\phi = \dfrac{\text{void volume}}{\text{total bed volume}}$ |
| PBR pressure drop | $\dfrac{dP}{dz} = -\beta_0 \left(\dfrac{P_0}{P}\right)\left(\dfrac{T}{T_0}\right)\left(\dfrac{\dot{n}_T}{\dot{n}_{T0}}\right)$ |
| Packed bed property | $\beta_0 = \dfrac{G(1-\phi)}{\rho_0 g_c D_P \phi^3}\left[\dfrac{150(1-\phi)\mu}{D_P} + 1.75G \right] [=] \mathrm{\dfrac{Pa}{m}}$ |
| Bulk density of catalyst | $\rho_b = \rho_c(1-\phi)$ |
| Catalyst mass | $m = (1-\phi)A_c z\rho_c = A_c z \rho_b$ |
| Ergun equation for packed bed (multiple reaction) | $\dfrac{dp}{dm} = -\dfrac{\alpha}{2p}\dfrac{T}{T_0}\dfrac{\dot{n}_T}{\dot{n}_{T0}}$ |
| $\alpha = \dfrac{2\beta_0}{A_c \rho_c (1-\phi)P_0} [=] \mathrm{kg^{-1}}$ | |
| Ergun equation for packed bed (single reaction) | $\dfrac{dp}{dm} = -\dfrac{\alpha}{2p} (1+\varepsilon X) \dfrac{T}{T_0}$ |
| System of pressure and conversion ODEs | $\begin{cases}\dfrac{dX}{dm} = F_1(X, p) & \text{Reactor design} \\ \dfrac{dp}{dm} = F_2(X, p) & \text{Ergun equation}\end{cases}$ |
| Pressure ratio ★ $\varepsilon = 0$ or $\varepsilon X \ll 1$, isothermal |
$p = \sqrt{1-\alpha m} = \sqrt{1 - \dfrac{2\beta_0}{P_0}z}$ |
| Pressure drop in pipes | $p = \sqrt{1 - \alpha_p V}$ |
| Pipe factor | $\alpha_p = \dfrac{4 f G^2}{A_c \rho_0 P_0 D}$ |
Rate law determination by data
Batch reactors
| Description | Equations |
|---|---|
| Power law | $r_A = -k C_A^\alpha C_B^\beta$ |
| Integral method | $\dfrac{dC_A}{dt} = -kC_A^\alpha$ |
| 0th order rxn | $C_A = C_{A0} - kt$ |
| 1st order rxn | $\ln\dfrac{C_{A0}}{C_A} = kt \\ \ln C_A = \ln C_{A0} - kt$ |
| 2nd order rxn | $\dfrac{1}{C_A} = kt + \dfrac{1}{C_{A0}}$ |
| Differential method | $\ln\left(-\dfrac{dC_A}{dt}\right) = \ln k_A + \alpha \ln C_A$ |
Differential reactors (PBR)
| Description | Equations |
|---|---|
| $-r_A' = \dfrac{\dot{V}_0 C_{A0} - \dot{V} C_{AP}}{\Delta m} = \dfrac{\dot{V}_{A0}X}{\Delta m} = \dfrac{\dot{V}_P}{\Delta m}$ |
Multiple reactions
Batch reactors
| Description | Equations |
|---|---|
| Parallel (competing) reactions | $\ce{A ->[\mathit{k}_1] B} \newline \ce{A ->[\mathit{k}_2] C}$ |
| Series (consecutive) reactions | $\ce{A ->[\mathit{k}_1] B ->[\mathit{k}_2] C}$ |
| Independent reactions | $\ce{A -> B + C} \newline \ce{D -> E + F}$ |
| Complex reactions | $\ce{A + B -> C + D} \newline \ce{A + C -> E} \newline \ce{E -> G}$ |
| Description | Equations |
|---|---|
| Instantaneous selectivity based on rate | $S_{D/U} = \dfrac{r_D}{r_U} = \dfrac{\text{rate of formation of D}}{\text{rate of formation of U}}$ |
| Overall selectivity based on flow rate | $\tilde{S}_{D/U} = \dfrac{\dot{n}_D}{\dot{n}_U} = \dfrac{n_D}{n_U} = \dfrac{\text{exit molar flow rate of D}}{\text{exit molar flow rate of U}}$ |
| Selectivity of CSTR | $S_{D/U} = \tilde{S}_{D/U}$ |
| Instantaneous yield based on rate | $Y_D = -\dfrac{r_D}{r_A}$ |
| Overall yield based on flow rate | $\tilde{Y}_D = \dfrac{\dot{n}_D}{\dot{n}_{A0} - \dot{n}_A} = \dfrac{n_D}{n_{A0} - n_A}$ |
| Conversion of batch and flow reactor | $X_A = \dfrac{\dot{n}_{A0} - \dot{n}_{A}}{\dot{n}_{A0}}$ |
| Conversion of semi-batch reactor | $X_A = \dfrac{C_{A0} V_0 - C_A V}{C_{A0}V}$ |
| Conversion of semi-batch reactor | $X_B = \dfrac{\dot{n}_{B0} t - C_B V}{\dot{n}_{B0} t}$ |
Parallel reactions
| Description | Equations |
|---|---|
| Concentration dependence of instantaneous selectivity | $S_{D/U} = \dfrac{r_D}{r_U} = \dfrac{k_D}{k_U}C_A^{\alpha_1 - \alpha_2}$ |
| $\alpha_1 > \alpha_2$ | $\uparrow C_A, \uparrow S_{D/U}$ |
| $\alpha_1 < \alpha_2$ | $\downarrow C_A, \uparrow S_{D/U}$ |
| Temperature dependence of instantaneous selectivity | $S_{D/U} = \dfrac{r_D}{r_U} ~ \dfrac{k_D}{k_U} = \dfrac{A_D}{A_U}\exp\left(-\dfrac{E_D - E_U}{RT}\right)$ |
| $E_D > E_U$ | $\uparrow T, \uparrow S_{D/U}$ |
| $E_D < E_U$ | $\downarrow T, \uparrow S_{D/U}$ |
Enzymatic Reactions
| Description | Equations |
|---|---|
| Pseudo-steady-state hypothesis | $r_{A*} = \sum r_{i, A*} = 0$ |
Mechanism development
| Rate law | Mechanism (rule of thumb) |
|---|---|
| Species concentration in denominator | Species collision with active intermediate |
| Constant in denominator | Reaction of spontaneous decomposition of active intermediate |
| Species concentration in numerator | Species produce active intermediate |
Michaelis-Menten kinetics
| Description | Equations |
|---|---|
| Overall reaction | $\ce{E + S -> E + P}$ |
| Reaction mechanism | $\ce{S + E <=>[\mathit{k}_1][\mathit{k}_{-1}] ES} \newline \ce{ES + W ->[\mathit{k}_{2}] P + E}$ |
| Enzyme balance | $\ce{[E_T] = [E] + [ES]} \newline \ce{[E] = [E_T] - [ES]}$ |
| Pseudo-steady-state approximation | $\begin{aligned}r_{\ce{ES}} &= k_1\ce{[S][E]} - k_{-1} \ce{[ES]} - k_2 \ce{[ES][W]} = 0 \\ r_{\ce{ES}} &= k_1\ce{[S]([E_T] - [ES])} - k_{-1} \ce{[ES]} - k_2 \ce{[ES][W]} = 0 \end{aligned} \newline \ce{[ES]} = \dfrac{k_1 \ce{[S][E_T]}}{k_1 \ce{[S]} + k_{-1} + k_2 \ce{[W]}}$ |
| Turnover number (# substrates converted to product per unit time on one enzyme at saturation) | $k_{\mathrm{cat}} = k_2 \ce{[W]}$ |
| Michaelis-Menten constant (attraction of enzyme of its substrate, [Substrate] which rate of rxn is 1/2 max) | $K_M = \dfrac{k_{\mathrm{cat}} + k_{-1}}{k_1}$ |
| Maximum rate | $V_{\max} = k_{\mathrm{cat}} \ce{[E_T]}$ |
| Michaelis-Menten equation Rate of reaction |
$\begin{aligned}r_{\ce{P}} &= k_{2} \ce{[ES][W]} \\ &= \dfrac{k_1 k_{2} \ce{[S][E_T][W]}}{k_1 \ce{[S]} + k_{-1} + k_2 \ce{[W]}} \\ &= \dfrac{k_{\mathrm{cat}} \ce{[S][E_T]}}{K_M + \ce{[S]}} \\ &= \dfrac{V_{\max} \ce{[S]}}{K_M + \ce{[S]}} \end{aligned}$ |
| Lineweaver-Burk equation | $\dfrac{1}{r_{\ce{P}}} = \dfrac{K_M}{V_{\max}}\dfrac{1}{\ce{[S]}} + \dfrac{1}{V_{\max}}$ |
| Eadie-Hofstee equation | $r_{\ce{P}} = V_{\max} - K_M\dfrac{r_{\ce{P}}}{\ce{[S]}}$ |
| Hanes-Woolf equation | $\dfrac{\ce{[S]}}{r_{\ce{P}}} = \dfrac{K_M}{V_{\max}} + \dfrac{1}{V_{\max}}\ce{[S]}$ |
Product-enzyme complex
| Description | Equations |
|---|---|
| Overall reaction | $\ce{E + S -> E + P}$ |
| Reaction mechanism | $\ce{S + E <=> ES <=> PE <=> P + E}$ |
| Briggs-Haldane equation | $r_{\ce{P}} = \dfrac{V_{\max}(\ce{[S]} - \ce{[P]} / K_C)}{\ce{[S]} + K_{\max} + K_P \ce{[P]}}$ |
Batch enzymatic reactor
| Description | Equations |
|---|---|
| Time | $\begin{aligned}t &= \dfrac{K_M}{V_{\max}}\ln\left(\dfrac{\ce{[A]_0}}{\ce{[A]}}\right) + \dfrac{\ce{[A]_0 - [A]}}{V_{\max}} \\ &= \dfrac{K_M}{V_{\max}} \ln\left(\dfrac{1}{1-X}\right) + \dfrac{\ce{[A]_0}X}{V_{\max}} \end{aligned}$ |
| Linearized form | $\dfrac{1}{t} \ln\left(\dfrac{\ce{[A]_0}}{\ce{[A]}}\right) = \dfrac{V_{\max}}{K_M} - \dfrac{\ce{[A]_0 - [A]}}{K_M t}$ |
Enzymatic inhibition
Competitive inhibition
| Description | Equations |
|---|---|
| Reaction mechanism | $\ce{E + S <=> ES} \newline \ce{ES -> E + P} \newline \ce{E + I <=> EI} \text{ (inactive)}$ |
| Reaction rate | $r_{\ce{P}} = \dfrac{V_{\max}\ce{[S]}}{\ce{[S]} + K_M \left[1 + \dfrac{\ce{[I]}}{K_I}\right]}$ |
| Lineweaver-Burk form $\uparrow K_I, \uparrow \text{slope}$ |
$\dfrac{1}{r_{\ce{P}}} = \dfrac{1}{\ce{[S]}} \left[\dfrac{K_M}{V_{\max}} \left[ 1 + \dfrac{\ce{[I]}}{K_I}\right] \right] + \dfrac{1}{V_{\max}}$ |
Uncompetitive inhibition
| Description | Equations |
|---|---|
| Reaction mechanism | $\ce{E + S <=> ES} \newline \ce{ES -> E + P} \newline \ce{ES + I <=> ESI} \text{ (inactive)}$ |
| Reaction rate | $r_{\ce{P}} = \dfrac{V_{\max}\ce{[S]}}{K_M + \ce{[S]} \left[ 1 + \dfrac{\ce{[I]}}{K_I}\right]}$ |
| Lineweaver-Burk form $\uparrow K_I, \uparrow \text{intercept}$ |
$\dfrac{1}{r_{\ce{P}}} = \dfrac{1}{\ce{[S]}} \dfrac{K_M}{V_{\max}} + \dfrac{1}{V_{\max}} \left[ 1 + \dfrac{\ce{[I]}}{K_I}\right]$ |
Noncompetitive (mixed) inhibition
| Description | Equations |
|---|---|
| Reaction mechanism | $\ce{E + S <=> ES} \newline \ce{ES -> E + P} \newline \ce{E + I <=> EI} \text{ (inactive)} \newline \ce{ES + I <=> ESI} \text{ (inactive)} \newline \ce{S + EI <=> ESI} \text{ (inactive)}$ |
| Reaction rate | $r_{\ce{P}} = \dfrac{V_{\max}\ce{[S]}}{(\ce{[S]} + K_M) \left[ 1 + \dfrac{\ce{[I]}}{K_I}\right]}$ |
| Lineweaver-Burk form $\uparrow K_I, \uparrow \text{slope}, \uparrow \text{intercept}$ |
$\dfrac{1}{r_{\ce{P}}} = \dfrac{1}{\ce{[S]}} \dfrac{K_M}{V_{\max}} \left[ 1 + \dfrac{\ce{[I]}}{K_I}\right] + \dfrac{1}{V_{\max}} \left[ 1 + \dfrac{\ce{[I]}}{K_I}\right]$ |
Catalytic Reactions
Reaction mechanisms
| Reaction | Mechanism | Rate Law |
|---|---|---|
| Adsorption | $\ce{A + ^* <=> A^*}$ | $r_{\ce{A}} = k_{\ce{A}}\left[P_{\ce{A}} \ce{[^*] - \dfrac{\ce{[A^*]}}{K_{\ce{A}}}}\right]$ |
| Desorption | $\ce{A^* <=> A + ^*}$ | $r_{\ce{D}} = k_{\ce{D}}\left[\ce{[A^*] - \dfrac{P_{\ce{A}} \ce{[^*]}}{K_{\ce{D}}}}\right]$ |
| Single site surface rxn | $\ce{A^* <=>[\mathit{k}_S][\mathit{k}_{-S}] B^*}$ | $r_{\ce{S}} = k_{\ce{S}} \left[\ce{[A^*]} - \dfrac{\ce{[B^*]}}{K_{\ce{S}}}\right]$ |
| Dual site (I) surface rxn | $\ce{A^* + ^* <=>[\mathit{k}_S][\mathit{k}_{-S}] B^* + ^*}$ | $r_{\ce{S}} = k_{\ce{S}} \left[\ce{[A^*][^*]} - \dfrac{\ce{[B^*][^*]}}{K_{\ce{S}}}\right]$ |
| Dual site (II) surface rxn | $\ce{A^* + B^* <=>[\mathit{k}_S][\mathit{k}_{-S}] C^* + D^*}$ | $r_{\ce{S}} = k_{\ce{S}} \left[\ce{[A^*][B^*]} - \dfrac{\ce{[C^*][D^*]}}{K_{\ce{S}}}\right]$ |
| Dual site (III) surface rxn | $\ce{A^* + B^{*}' <=>[\mathit{k}_S][\mathit{k}_{-S}] C^{*}' + D^*}$ | $r_{\ce{S}} = k_{\ce{S}} \left[\ce{[A^*][B^{*}']} - \dfrac{\ce{[C^{*}'][D^*]}}{K_{\ce{S}}}\right]$ |
| Eley-Rideal surface rxn | $\ce{A^* + B (g) <=>[\mathit{k}_S][\mathit{k}_{-S}] C^{*}}$ | $r_{\ce{S}} = k_{\ce{S}} \left[\ce{[A^*]} P_{\ce{B}} - \dfrac{\ce{[C^{*}]}}{K_{\ce{S}}}\right]$ |