Formula Sheet: Kinetics

Reaction Rate Fundamentals

Rate of Reaction

General Rate Expression:

\[r = -\frac{1}{V}\frac{dN_A}{dt}\]

• r = reaction rate (mol/L·s or mol/m³·s)
• V = volume (L or m³)
• N_A = number of moles of species A (mol)
• t = time (s)

Rate in Terms of Concentration:

\[r = -\frac{dC_A}{dt}\]

• C_A = concentration of species A (mol/L or mol/m³)
• Negative sign indicates consumption of reactant

Rate for Stoichiometric Reactions:

For the reaction: aA + bB → cC + dD

\[-\frac{1}{a}\frac{dC_A}{dt} = -\frac{1}{b}\frac{dC_B}{dt} = \frac{1}{c}\frac{dC_C}{dt} = \frac{1}{d}\frac{dC_D}{dt}\]

• a, b, c, d = stoichiometric coefficients
• Positive for products, negative for reactants

Rate Laws and Reaction Order

General Power Law Rate Expression:

\[r = kC_A^{\alpha}C_B^{\beta}\]

• k = rate constant (units vary with reaction order)
• α = order with respect to A
• β = order with respect to B
• Overall order = α + β

Elementary Reaction Rate:

For elementary reactions, the rate law follows stoichiometry directly.

For aA + bB → products:

\[r = kC_A^a C_B^b\]

Elementary Rate Laws

Zero-Order Reactions

Rate Law:

\[r = k\]

• k = rate constant (mol/L·s)
• Rate is independent of concentration

Integrated Form:

\[C_A = C_{A0} - kt\]

• C_A0 = initial concentration (mol/L)
• C_A = concentration at time t (mol/L)

Half-Life:

\[t_{1/2} = \frac{C_{A0}}{2k}\]

• Half-life depends on initial concentration

First-Order Reactions

Rate Law:

\[r = kC_A\]

• k = rate constant (s^-1 or min^-1)

Integrated Form (Linear):

\[\ln C_A = \ln C_{A0} - kt\]

Integrated Form (Exponential):

\[C_A = C_{A0}e^{-kt}\]

Half-Life:

\[t_{1/2} = \frac{\ln 2}{k} = \frac{0.693}{k}\]

• Half-life is independent of initial concentration

Conversion Form:

\[\ln(1 - X_A) = -kt\]

• X_A = fractional conversion of A

Second-Order Reactions (Single Reactant)

Rate Law:

\[r = kC_A^2\]

• k = rate constant (L/mol·s or m³/mol·s)

Integrated Form:

\[\frac{1}{C_A} = \frac{1}{C_{A0}} + kt\]

Half-Life:

\[t_{1/2} = \frac{1}{kC_{A0}}\]

• Half-life is inversely proportional to initial concentration

Conversion Form:

\[\frac{X_A}{C_{A0}(1-X_A)} = kt\]

Second-Order Reactions (Two Reactants)

For A + B → products, with r = kC_AC_B

Equal Initial Concentrations (C_A0 = C_B0):

\[\frac{1}{C_A} = \frac{1}{C_{A0}} + kt\]

Unequal Initial Concentrations:

\[\ln\left(\frac{C_B C_{A0}}{C_A C_{B0}}\right) = (C_{B0} - C_{A0})kt\]

nth-Order Reactions

Rate Law:

\[r = kC_A^n\]

Integrated Form (n ≠ 1):

\[\frac{1}{C_A^{n-1}} = \frac{1}{C_{A0}^{n-1}} + (n-1)kt\]

Half-Life:

\[t_{1/2} = \frac{2^{n-1}-1}{(n-1)kC_{A0}^{n-1}}\]

Temperature Dependence

Arrhenius Equation

Arrhenius Law:

\[k = A e^{-E_a/RT}\]

• A = pre-exponential or frequency factor (same units as k)
• E_a = activation energy (J/mol or cal/mol)
• R = universal gas constant = 8.314 J/mol·K = 1.987 cal/mol·K
• T = absolute temperature (K)

Linearized Form:

\[\ln k = \ln A - \frac{E_a}{RT}\]

• Plot ln k vs. 1/T yields straight line with slope = -E_a/R

Two-Temperature Form:

\[\ln\left(\frac{k_2}{k_1}\right) = \frac{E_a}{R}\left(\frac{1}{T_1} - \frac{1}{T_2}\right)\]

• Used to calculate rate constant at different temperatures

Modified Arrhenius Equation

\[k = AT^n e^{-E_a/RT}\]

• n = temperature exponent (dimensionless)
• More accurate for wider temperature ranges

Reaction Mechanisms and Complex Kinetics

Reversible Reactions

For: A ⇌ B

Rate Expression:

\[r = k_f C_A - k_r C_B\]

• k_f = forward rate constant
• k_r = reverse rate constant

Equilibrium Constant:

\[K_{eq} = \frac{k_f}{k_r}\]

First-Order Reversible (A ⇌ B):

\[\ln\left(\frac{C_A - C_{Ae}}{C_{A0} - C_{Ae}}\right) = -(k_f + k_r)t\]

• C_Ae = equilibrium concentration of A

Parallel Reactions

For parallel reactions:

A → B (rate = k₁C_A)

A → C (rate = k₂C_A)

Overall Rate of A Consumption:

\[-\frac{dC_A}{dt} = (k_1 + k_2)C_A\]

Product Selectivity:

\[\frac{C_B}{C_C} = \frac{k_1}{k_2}\]

Series (Consecutive) Reactions

For: A → B → C

First-Order Series:

\[A \xrightarrow{k_1} B \xrightarrow{k_2} C\]

Concentration Profiles:

\[C_A = C_{A0}e^{-k_1 t}\] \[C_B = C_{A0}\frac{k_1}{k_2 - k_1}(e^{-k_1 t} - e^{-k_2 t})\] \[C_C = C_{A0}\left[1 + \frac{k_1 e^{-k_2 t} - k_2 e^{-k_1 t}}{k_2 - k_1}\right]\]

Time to Maximum B Concentration:

\[t_{max} = \frac{\ln(k_2/k_1)}{k_2 - k_1}\]

Enzyme Kinetics (Michaelis-Menten)

Michaelis-Menten Equation:

\[r = \frac{r_{max}C_S}{K_M + C_S}\]

• r_max = maximum reaction rate (mol/L·s)
• C_S = substrate concentration (mol/L)
• K_M = Michaelis constant (mol/L)

Alternative Form:

\[r = \frac{V_{max}C_S}{K_M + C_S}\]

• V_max = k_cat[E]₀
• [E]₀ = total enzyme concentration

Lineweaver-Burk (Double Reciprocal):

\[\frac{1}{r} = \frac{K_M}{r_{max}}\frac{1}{C_S} + \frac{1}{r_{max}}\]

• Linear plot of 1/r vs. 1/C_S
• Slope = K_M/r_max
• y-intercept = 1/r_max

Autocatalytic Reactions

For: A + B → 2B

Rate Law:

\[r = kC_A C_B\]

Integrated Form:

\[\ln\left(\frac{C_B}{C_A}\right) = \ln\left(\frac{C_{B0}}{C_{A0}}\right) + (C_{A0} + C_{B0})kt\]

Batch Reactor Design

General Batch Reactor Equation

Design Equation:

\[t = N_{A0}\int_0^{X_A} \frac{dX_A}{-r_A V}\]

• N_A0 = initial moles of A
• X_A = conversion of A
• -r_A = rate of consumption of A (mol/L·s)
• V = reactor volume

Constant Volume Batch Reactor:

\[t = C_{A0}\int_0^{X_A} \frac{dX_A}{-r_A}\]

Batch Reactor Time for nth-Order Reactions

Zero-Order:

\[t = \frac{C_{A0}X_A}{k}\]

First-Order:

\[t = -\frac{1}{k}\ln(1 - X_A)\]

Second-Order:

\[t = \frac{X_A}{kC_{A0}(1-X_A)}\]

nth-Order (n ≠ 1):

\[t = \frac{1}{k(n-1)C_{A0}^{n-1}}\left[\frac{1}{(1-X_A)^{n-1}} - 1\right]\]

Continuous Stirred-Tank Reactor (CSTR)

General CSTR Design Equation

Mole Balance:

\[V = \frac{F_{A0}X_A}{-r_A}\]

• V = reactor volume (L or m³)
• F_A0 = inlet molar flow rate of A (mol/s)
• X_A = conversion of A
• -r_A = rate evaluated at exit conditions

Space Time:

\[\tau = \frac{V}{\nu_0} = \frac{C_{A0}X_A}{-r_A}\]

• τ = space time (s)
• ν₀ = volumetric flow rate (L/s or m³/s)

Space Velocity:

\[SV = \frac{1}{\tau} = \frac{\nu_0}{V}\]

• SV = space velocity (s^-1)

CSTR Performance Equations

First-Order Reaction in CSTR:

\[\tau = \frac{C_{A0}X_A}{kC_A} = \frac{C_{A0}X_A}{kC_{A0}(1-X_A)}\] \[X_A = \frac{k\tau}{1 + k\tau}\]

Second-Order Reaction in CSTR:

\[\tau = \frac{C_{A0}X_A}{kC_A^2} = \frac{X_A}{kC_{A0}(1-X_A)^2}\]

Multiple CSTRs in Series

First-Order Reaction, n Equal-Volume CSTRs:

\[X_A = 1 - \frac{1}{(1 + k\tau_i)^n}\]

• τ_i = space time in each reactor
• n = number of reactors

Conversion in ith Reactor:

\[1 - X_{Ai} = \frac{1}{(1 + k\tau_i)^i}\]

Plug Flow Reactor (PFR)

General PFR Design Equation

Differential Form:

\[\frac{dX_A}{dV} = \frac{-r_A}{F_{A0}}\]

Integral Form:

\[V = F_{A0}\int_0^{X_A} \frac{dX_A}{-r_A}\]

Space Time:

\[\tau = \frac{V}{\nu_0} = C_{A0}\int_0^{X_A} \frac{dX_A}{-r_A}\]

PFR Performance Equations

First-Order Reaction in PFR:

\[\tau = \frac{1}{k}\ln\left(\frac{1}{1-X_A}\right)\] \[X_A = 1 - e^{-k\tau}\]

Second-Order Reaction in PFR:

\[\tau = \frac{X_A}{kC_{A0}(1-X_A)}\]

Zero-Order Reaction in PFR:

\[\tau = \frac{C_{A0}X_A}{k}\]

nth-Order Reaction in PFR (n ≠ 1):

\[\tau = \frac{1}{k(n-1)C_{A0}^{n-1}}\left[\frac{1}{(1-X_A)^{n-1}} - 1\right]\]

Reactor Comparisons

Volume Comparison for Same Conversion

For Most Reactions:

\[V_{CSTR} > V_{PFR} > V_{Batch}\]

• For isothermal operation and positive reaction order
• CSTR requires largest volume for same conversion

First-Order Reaction Volume Ratio:

\[\frac{V_{CSTR}}{V_{PFR}} = \frac{1 + k\tau}{k\tau}\]

Residence Time

Mean Residence Time:

\[\bar{t} = \frac{V}{\nu_0}\]

Residence Time Distribution (RTD) for CSTR:

\[E(t) = \frac{1}{\bar{t}}e^{-t/\bar{t}}\]

• E(t) = exit age distribution function

Non-Isothermal Reactor Design

Energy Balance for Batch Reactor

General Energy Balance:

\[N_{A0}C_{pA}\frac{dT}{dt} = (-\Delta H_{rxn})V(-r_A) + \dot{Q}\]

• C_pA = heat capacity of A (J/mol·K)
• ΔH_rxn = heat of reaction (J/mol)
• Q̇ = heat transfer rate (J/s or W)

Heat Transfer Rate:

\[\dot{Q} = UA(T_c - T)\]

• U = overall heat transfer coefficient (W/m²·K)
• A = heat transfer area (m²)
• T_c = coolant temperature (K)
• T = reactor temperature (K)

Energy Balance for CSTR

Steady-State Energy Balance:

\[F_{A0}X_A(-\Delta H_{rxn}) = \dot{Q} + F_{A0}\sum_i \Theta_i C_{pi}(T - T_0)\]

• Θ_i = ratio of inlet moles of i to inlet moles of A
• C_pi = heat capacity of species i
• T₀ = inlet temperature

Adiabatic Operation (Q̇ = 0):

\[T = T_0 + \frac{(-\Delta H_{rxn})X_A}{\sum_i \Theta_i C_{pi}}\]

Energy Balance for PFR

Differential Energy Balance:

\[\frac{dT}{dV} = \frac{UA(T_c - T) + (-r_A)(-\Delta H_{rxn})}{F_{A0}\sum_i \Theta_i C_{pi}}\]

Adiabatic PFR:

\[\frac{dT}{dX_A} = \frac{(-\Delta H_{rxn})}{\sum_i \Theta_i C_{pi}}\]

Pressure Effects and Variable Volume

Gas-Phase Reactions with Volume Change

Volume Change Parameter (ε):

\[\varepsilon_A = \frac{V_{X_A=1} - V_{X_A=0}}{V_{X_A=0}} = \frac{\delta y_{A0}}{1 + \delta y_{A0}}\]

• δ = change in total moles per mole of A reacted
• y_A0 = inlet mole fraction of A

Concentration with Volume Change:

\[C_A = C_{A0}\frac{(1-X_A)}{(1+\varepsilon_A X_A)}\frac{P}{P_0}\frac{T_0}{T}\]

Volume as Function of Conversion:

\[V = V_0(1 + \varepsilon_A X_A)\frac{P_0}{P}\frac{T}{T_0}\]

Pressure Drop in Packed Bed Reactors

Ergun Equation:

\[\frac{dP}{dz} = -\frac{G}{\rho g_c D_p}\left[\frac{150(1-\phi)\mu}{D_p \phi} + 1.75G\right]\]

• P = pressure (Pa)
• z = axial distance (m)
• G = superficial mass velocity (kg/m²·s)
• ρ = fluid density (kg/m³)
• g_c = gravitational constant
• D_p = particle diameter (m)
• φ = void fraction
• μ = viscosity (Pa·s)

Simplified Pressure Drop (Low Pressure):

\[\frac{dP}{dz} = -\frac{\beta_0}{2P}(1 + \varepsilon_A X_A)\frac{T}{T_0}\]

• β₀ = pressure drop parameter

Catalytic Reactions

Heterogeneous Catalysis

Langmuir-Hinshelwood Rate Law (Single Site):

\[-r_A = \frac{kK_A C_A}{1 + K_A C_A + K_B C_B}\]

• K_A = adsorption equilibrium constant for A
• K_B = adsorption equilibrium constant for B

General Catalytic Rate Expression:

\[-r_A = \frac{(kinetic\ term)(driving\ force)}{(adsorption\ term)^n}\]

Rate per Unit Mass of Catalyst:

\[-r'_A = \frac{-r_A}{\rho_c}\]

• ρ_c = catalyst density (kg/m³)
• Units: mol/kg_cat·s

Catalyst Effectiveness Factor

Effectiveness Factor:

\[\eta = \frac{\text{actual rate with diffusion}}{\text{rate without diffusion}}\]

Thiele Modulus (First-Order, Spherical Pellet):

\[\phi = R\sqrt{\frac{k}{D_e}}\]

• R = particle radius (m)
• k = first-order rate constant (s^-1)
• D_e = effective diffusivity (m²/s)

Effectiveness Factor for Spherical Pellet:

\[\eta = \frac{3}{\phi}\left(\frac{1}{\tanh\phi} - \frac{1}{\phi}\right)\]

Asymptotic Limits:

• For φ < 1:="" η="" ≈="" 1="" (no="" diffusion="">
• For φ >> 1: η ≈ 3/φ (strong diffusion limitations)

Catalyst Deactivation

Activity:

\[a(t) = \frac{-r_A(t)}{-r_A(t=0)}\]

• a(t) = catalyst activity at time t (dimensionless)
• a = 1 for fresh catalyst, a < 1="" for="" deactivated="">

Rate with Deactivation:

\[-r_A = k C_A^n a(t)\]

Exponential Decay Model:

\[a(t) = e^{-k_d t}\]

• k_d = deactivation rate constant (s^-1)

Chain Reactions and Polymerization

Free Radical Polymerization

Initiation:

\[I_2 \xrightarrow{k_d} 2R\cdot\] \[R\cdot + M \xrightarrow{k_i} M_1\cdot\]

Propagation:

\[M_n\cdot + M \xrightarrow{k_p} M_{n+1}\cdot\]

Termination (Combination):

\[M_n\cdot + M_m\cdot \xrightarrow{k_{tc}} M_{n+m}\]

Termination (Disproportionation):

\[M_n\cdot + M_m\cdot \xrightarrow{k_{td}} M_n + M_m\]

Rate of Polymerization:

\[R_p = k_p[M][\bar{M}\cdot]\]

• [M] = monomer concentration
• [M̄·] = total radical concentration

Steady-State Approximation:

\[[\bar{M}\cdot] = \sqrt{\frac{2fk_d[I_2]}{k_t}}\]

• f = initiator efficiency
• k_t = k_tc + k_td

Number-Average Degree of Polymerization:

\[\bar{X}_n = \frac{k_p[M]}{\sqrt{2fk_d k_t[I_2]}}\]

Mass Transfer Effects

External Mass Transfer

Mass Transfer Rate:

\[N_A = k_c A_s(C_{Ab} - C_{As})\]

• k_c = mass transfer coefficient (m/s)
• A_s = external surface area (m²)
• C_Ab = bulk concentration (mol/m³)
• C_As = surface concentration (mol/m³)

For Fast Reactions (C_As ≈ 0):

\[N_A = k_c A_s C_{Ab}\]

Overall Rate with External Mass Transfer:

\[\frac{1}{r_{obs}} = \frac{1}{k_c a} + \frac{1}{k_{rxn}}\]

• a = specific surface area (m²/m³)

Internal Mass Transfer (Pore Diffusion)

Effective Diffusivity:

\[D_e = \frac{D_{AB}\phi_p}{\tau_p}\]

• D_AB = molecular diffusivity (m²/s)
• φ_p = pellet porosity
• τ_p = tortuosity factor

Observed Rate with Diffusion:

\[r_{obs} = \eta \cdot r_{surface}\]

Conversion and Selectivity

Conversion Definitions

Fractional Conversion:

\[X_A = \frac{N_{A0} - N_A}{N_{A0}} = \frac{C_{A0} - C_A}{C_{A0}}\]

For Flow Reactors:

\[X_A = \frac{F_{A0} - F_A}{F_{A0}}\]

Yield and Selectivity

Yield:

\[Y_{D/A} = \frac{\text{moles of D formed}}{\text{moles of A fed}}\]

Selectivity (Instantaneous):

\[S_{D/U} = \frac{r_D}{r_U}\]

• D = desired product
• U = undesired product

Overall Selectivity:

\[S_{D/U} = \frac{\text{moles of D produced}}{\text{moles of U produced}}\]

For Parallel Reactions (A → D and A → U):

\[S_{D/U} = \frac{k_D C_A^{n_D}}{k_U C_A^{n_U}} = \frac{k_D}{k_U}C_A^{n_D - n_U}\]

Multiple Reactions

Competitive-Consecutive Reactions

For A → D (desired) and D → U (undesired):

Maximum Yield of D:

\[Y_{D,max} = \left(\frac{k_1}{k_2}\right)^{k_1/(k_2-k_1)}\]

• Occurs at optimal conversion or residence time

Reactor Choice for Selectivity

For Parallel Reactions:

• If n_D > n_U: use high C_A (PFR preferred)
• If n_D <>_U: use low C_A (CSTR or dilution)

For Series Reactions:

• Use PFR with optimal conversion
• Avoid high residence times if intermediate is desired

Half-Life Summary

Half-Life Expressions by Reaction Order

Zero-Order:

\[t_{1/2} = \frac{C_{A0}}{2k}\]

First-Order:

\[t_{1/2} = \frac{0.693}{k}\]

Second-Order:

\[t_{1/2} = \frac{1}{kC_{A0}}\]

nth-Order:

\[t_{1/2} = \frac{2^{n-1}-1}{(n-1)kC_{A0}^{n-1}}\]