Formula Sheet: Chemical Reactions

Reaction Stoichiometry

General Stoichiometric Equation

For a general reaction:

\[aA + bB \rightarrow cC + dD\]

Where:

• a, b, c, d = stoichiometric coefficients (dimensionless)
• A, B = reactants
• C, D = products

Limiting Reactant

The limiting reactant is determined by comparing molar ratios:

\[\text{Limiting reactant} = \min\left(\frac{n_A}{a}, \frac{n_B}{b}\right)\]

Where:

• n_A, n_B = moles of reactants A and B (mol)
• a, b = stoichiometric coefficients

Percent Excess Reactant

\[\text{Percent Excess} = \frac{n_{\text{actual}} - n_{\text{stoichiometric}}}{n_{\text{stoichiometric}}} \times 100\%\]

Where:

• n_actual = actual moles of reactant fed (mol)
• n_{stoichiometric} = stoichiometric moles required (mol)

Conversion and Yield

Fractional Conversion:

\[X_A = \frac{n_{A,\text{in}} - n_{A,\text{out}}}{n_{A,\text{in}}} = \frac{\text{moles reacted}}{\text{moles fed}}\]

Where:

• X_A = fractional conversion of reactant A (dimensionless, 0 to 1)
• n_A,in = moles of A entering reactor (mol)
• n_A,out = moles of A leaving reactor (mol)

Percent Yield:

\[\text{Yield} = \frac{n_{\text{product,actual}}}{n_{\text{product,theoretical}}} \times 100\%\]

Selectivity:

\[S_{C/D} = \frac{n_C}{n_D} = \frac{\text{moles of desired product C}}{\text{moles of undesired product D}}\]

Reaction Rates and Kinetics

Rate of Reaction Definition

For the reaction: \(aA + bB \rightarrow cC + dD\)

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

Where:

• r = rate of reaction (mol/L·s or mol/L·min)
• C_i = concentration of species i (mol/L)
• t = time (s or min)

Note: Negative sign indicates consumption; positive indicates formation.

Rate Laws

General Form:

\[r = k C_A^{\alpha} C_B^{\beta}\]

Where:

• k = rate constant (units vary with reaction order)
• α, β = partial orders with respect to A and B (dimensionless)
• Overall order n = α + β

Elementary Reactions

For elementary reactions, the rate law follows directly from stoichiometry:

\[aA + bB \rightarrow \text{products}\] \[r = k C_A^a C_B^b\]

Reaction Order and Rate Constant Units

Units of rate constant k for different orders:

• Zero order (n = 0): mol/(L·s) or mol/(L·min)
• First order (n = 1): s^-1 or min^-1
• Second order (n = 2): L/(mol·s) or L/(mol·min)
• n-th order: (L/mol)^n-1/s or (L/mol)^n-1/min

Integrated Rate Laws

Zero Order Reaction

Rate law: \(r = k\)

Integrated form:

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

Half-life:

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

Where:

• C_A = concentration at time t (mol/L)
• C_A0 = initial concentration (mol/L)
• k = rate constant (mol/L·s)
• t = time (s)

First Order Reaction

Rate law: \(r = kC_A\)

Integrated form:

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

Conversion form:

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

Half-life:

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

Note: Half-life is independent of initial concentration for first-order reactions.

Second Order Reaction (Single Reactant)

Rate law: \(r = kC_A^2\)

Integrated form:

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

Conversion form:

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

Half-life:

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

Second Order Reaction (Two Reactants)

For \(A + B \rightarrow \text{products}\) with rate law: \(r = kC_A C_B\)

When C_A0 ≠ C_B0:

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

When C_A0 = C_B0:

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

n-th Order Reaction

For 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 of Reaction Rates

Arrhenius Equation

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

Where:

• k = rate constant (units vary)
• 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) or 1.987 cal/(mol·K)
• T = absolute temperature (K)

Logarithmic form:

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

Two-Temperature Form of Arrhenius Equation

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

Where:

• k₁, k₂ = rate constants at temperatures T₁ and T₂
• T₁, T₂ = absolute temperatures (K)

Determining Activation Energy from Data

Plot \(\ln k\) vs. \(1/T\) (Arrhenius plot):

• Slope = -E_a/R
• Intercept = ln A

Reversible Reactions and Equilibrium

Reversible Reaction Rate

For: \(A + B \rightleftharpoons C + D\)

\[r_{\text{net}} = r_f - r_r = k_f C_A C_B - k_r C_C C_D\]

Where:

• r_f = forward reaction rate
• r_r = reverse reaction rate
• k_f = forward rate constant
• k_r = reverse rate constant

Equilibrium Constant Relationship

At equilibrium, \(r_{\text{net}} = 0\):

\[K_c = \frac{k_f}{k_r} = \frac{C_C^c C_D^d}{C_A^a C_B^b}\]

Where:

• K_c = equilibrium constant based on concentration
• Concentrations are equilibrium values

Van't Hoff Equation

Temperature dependence of equilibrium constant:

\[\frac{d \ln K}{dT} = \frac{\Delta H_{rxn}^{\circ}}{RT^2}\]

Integrated form:

\[\ln\frac{K_2}{K_1} = -\frac{\Delta H_{rxn}^{\circ}}{R}\left(\frac{1}{T_2} - \frac{1}{T_1}\right)\]

Where:

• ΔH_rxn° = standard enthalpy of reaction (J/mol)
• K₁, K₂ = equilibrium constants at T₁ and T₂

Reactor Design Equations

Batch Reactor

General mole balance:

\[\frac{dN_A}{dt} = r_A V\]

For constant volume:

\[\frac{dC_A}{dt} = r_A\]

Design equation (time required for conversion X_A):

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

For constant volume batch reactor:

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

Where:

• N_A = moles of A (mol)
• N_A0 = initial moles of A (mol)
• C_A = concentration of A (mol/L)
• C_A0 = initial concentration of A (mol/L)
• r_A = rate of reaction of A (negative for consumption)
• V = reactor volume (L)
• t = time (s or min)

Continuous Stirred Tank Reactor (CSTR)

General mole balance at steady state:

\[F_{A0} - F_A + r_A V = 0\]

Design equation:

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

Alternative form:

\[V = \frac{F_{A0} - F_A}{-r_A} = \frac{Q(C_{A0} - C_A)}{-r_A}\]

Space time:

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

Where:

• F_A0 = molar flow rate of A entering (mol/s)
• F_A = molar flow rate of A leaving (mol/s)
• Q = volumetric flow rate (L/s)
• τ = space time (s)
• -r_A = rate evaluated at exit conditions for CSTR

Plug Flow Reactor (PFR)

Differential mole balance:

\[\frac{dF_A}{dV} = r_A\]

Design equation:

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

For constant volumetric flow:

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

Space time:

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

Where:

• V = reactor volume (L)
• -r_A = rate as function of conversion (varies along reactor length)

Relationship Between Concentration and Conversion

For liquid phase (constant density):

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

For gas phase (variable density):

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

Where:

• ε_A = fractional volume change = \(\frac{V_{total,X_A=1} - V_{total,X_A=0}}{V_{total,X_A=0}}\)
• P, P₀ = pressure at position and inlet (atm or Pa)
• T, T₀ = temperature at position and inlet (K)

For gas phase with stoichiometry: \(aA \rightarrow bB\)

\[\epsilon_A = \frac{b - a}{a} y_{A0}\]

Where:

• y_A0 = mole fraction of A in feed

Multiple CSTR in Series

For n CSTRs of equal volume in series:

\[V_{\text{total}} = \sum_{i=1}^n V_i = F_{A0} \sum_{i=1}^n \frac{X_{Ai} - X_{A,i-1}}{-r_{Ai}}\]

Where:

• X_Ai = conversion at exit of reactor i
• -r_Ai = rate evaluated at exit conditions of reactor i

Specific Rate Laws in Reactor Design

First Order Reaction in Batch Reactor

\[t = \frac{1}{k} \ln\frac{C_{A0}}{C_A} = \frac{1}{k} \ln\frac{1}{1-X_A}\]

First Order Reaction in CSTR

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

First Order Reaction in PFR

\[\tau = \frac{V}{Q} = \frac{1}{k} \ln\frac{1}{1-X_A}\]

Note: For first-order reactions, batch reactor and PFR have identical design equations when comparing space time to batch time.

Second Order Reaction in Batch Reactor

For \(r_A = -kC_A^2\):

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

Second Order Reaction in CSTR

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

Second Order Reaction in PFR

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

Complex Reactions

Parallel Reactions

For reactions occurring simultaneously:

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

Instantaneous selectivity:

\[S_{B/C} = \frac{r_B}{r_C} = \frac{k_1 C_A^{\alpha_1}}{k_2 C_A^{\alpha_2}}\]

Overall selectivity:

\[S_{B/C} = \frac{C_B - C_{B0}}{C_C - C_{C0}}\]

Series (Consecutive) Reactions

For: \(A \xrightarrow{k_1} B \xrightarrow{k_2} C\)

Batch reactor concentrations:

\[C_A = C_{A0} e^{-k_1 t}\] \[C_B = C_{A0} \frac{k_1}{k_2 - k_1} (e^{-k_1 t} - e^{-k_2 t})\]

Time for maximum intermediate B concentration:

\[t_{\max} = \frac{1}{k_2 - k_1} \ln\frac{k_2}{k_1}\]

Maximum concentration of B:

\[C_{B,\max} = C_{A0} \left(\frac{k_1}{k_2}\right)^{k_2/(k_2-k_1)}\]

Reversible Reactions in Reactors

For: \(A \rightleftharpoons B\) with \(r_A = -k_f C_A + k_r C_B\)

Equilibrium conversion:

\[X_{A,eq} = \frac{K_c}{1 + K_c}\]

Where: \(K_c = \frac{k_f}{k_r}\)

Maximum conversion is limited by equilibrium.

Non-Isothermal Reactor Design

Energy Balance for Batch Reactor

General energy balance:

\[\rho V C_p \frac{dT}{dt} = -\Delta H_{rxn} V r_A + Q\]

Where:

• ρ = density (kg/L)
• C_p = heat capacity (J/kg·K)
• ΔH_rxn = heat of reaction (J/mol), negative for exothermic
• Q = heat transfer rate (J/s or W)

Energy Balance for CSTR

Steady-state energy balance:

\[Q \rho C_p (T - T_0) = -\Delta H_{rxn} F_{A0} X_A + \dot{Q}\]

Or:

\[\sum F_{i0} C_{p,i} (T - T_0) = -\Delta H_{rxn}(T) F_{A0} X_A + \dot{Q}\]

Where:

• Q̇ = heat transfer rate to/from reactor (J/s)
• T₀ = inlet temperature (K)
• T = reactor temperature (K)

Energy Balance for PFR

Differential energy balance:

\[\sum F_i C_{p,i} \frac{dT}{dV} = -\Delta H_{rxn}(T) r_A + \frac{\dot{Q}}{V}\]

Or with heat exchange:

\[F_{A0} \sum \Theta_i C_{p,i} \frac{dT}{dX_A} = -\Delta H_{rxn}(T) + \frac{Ua(T_a - T)}{F_{A0}}\]

Where:

• Θ_i = \(F_{i0}/F_{A0}\) = ratio of inlet molar flow rates
• U = overall heat transfer coefficient (W/m²·K)
• a = heat transfer area per unit volume (m²/m³)
• T_a = coolant/heating medium temperature (K)

Heat of Reaction Temperature Dependence

\[\Delta H_{rxn}(T) = \Delta H_{rxn}(T_0) + \int_{T_0}^T \Delta C_p dT\]

Where:

\[\Delta C_p = \sum_{\text{products}} \nu_i C_{p,i} - \sum_{\text{reactants}} \nu_j C_{p,j}\]

• ν_i = stoichiometric coefficients

Adiabatic Reactor Temperature

For adiabatic operation (Q̇ = 0):

\[T = T_0 + \frac{(-\Delta H_{rxn}(T_0)) X_A}{\sum \Theta_i C_{p,i}}\]

Adiabatic temperature rise:

\[\Delta T_{ad} = \frac{(-\Delta H_{rxn}) C_{A0}}{\rho C_p}\]

Catalysis and Heterogeneous Reactions

Langmuir-Hinshelwood Kinetics

Single reactant adsorption:

\[r = \frac{k K_A P_A}{1 + K_A P_A}\]

Two reactants (Langmuir-Hinshelwood):

\[r = \frac{k K_A K_B P_A P_B}{(1 + K_A P_A + K_B P_B)^2}\]

Where:

• k = rate constant
• K_A, K_B = adsorption equilibrium constants (1/atm or 1/Pa)
• P_A, P_B = partial pressures (atm or Pa)

Catalyst Effectiveness Factor

Effectiveness factor:

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

For first-order reaction in spherical catalyst pellet:

\[\eta = \frac{3}{\phi_s^2}(\phi_s \coth \phi_s - 1)\]

Where φ_s is the Thiele modulus.

Thiele Modulus

For spherical pellet:

\[\phi_s = \frac{R}{3}\sqrt{\frac{k \rho_c}{D_e}}\]

For slab geometry:

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

Where:

• R = particle radius (m)
• L = half-thickness of slab (m)
• k = rate constant (s^-1 for first order)
• ρ_c = catalyst density (kg/m³)
• D_e = effective diffusivity (m²/s)

Simplified effectiveness factor for large φ:

\[\eta \approx \frac{1}{\phi_s} \quad \text{for } \phi_s > 3\]

Weisz-Prater Criterion

To check for internal diffusion limitations:

\[C_{WP} = \frac{-r_A^{\text{obs}} \rho_c R^2}{D_e C_{As}}\]

Where:

• C_WP = Weisz-Prater criterion (dimensionless)
• r_A^obs = observed rate per mass of catalyst
• C_As = concentration at catalyst surface

Interpretation:

• C_WP < 1:="" no="" diffusion="">
• C_WP >> 1: Strong diffusion limitation

Enzyme Kinetics

Michaelis-Menten Equation

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

Where:

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

Note: K_M is the substrate concentration at which \(r = V_{\max}/2\)

Lineweaver-Burk Plot

Linearized form of Michaelis-Menten:

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

Plot 1/r vs. 1/C_S:

• Slope = K_M/V_max
• Intercept = 1/V_max

Enzyme Inhibition

Competitive inhibition:

\[r = \frac{V_{\max} C_S}{K_M \left(1 + \frac{C_I}{K_I}\right) + C_S}\]

Non-competitive inhibition:

\[r = \frac{V_{\max} C_S}{\left(K_M + C_S\right)\left(1 + \frac{C_I}{K_I}\right)}\]

Where:

• C_I = inhibitor concentration (mol/L)
• K_I = inhibition constant (mol/L)

Residence Time Distribution

Exit Age Distribution Function

E(t) curve:

\[E(t) = \frac{C(t)}{\int_0^{\infty} C(t) dt}\]

Where:

• E(t) = residence time distribution function (1/time)
• C(t) = tracer concentration at reactor exit

Normalization:

\[\int_0^{\infty} E(t) dt = 1\]

Mean Residence Time

\[\bar{t} = \int_0^{\infty} t E(t) dt = \frac{\int_0^{\infty} t C(t) dt}{\int_0^{\infty} C(t) dt}\]

For ideal reactors:

• PFR: All molecules have same residence time = τ
• CSTR: \(E(t) = \frac{1}{\tau} e^{-t/\tau}\)

Variance of RTD

\[\sigma^2 = \int_0^{\infty} (t - \bar{t})^2 E(t) dt = \int_0^{\infty} t^2 E(t) dt - \bar{t}^2\]

Dimensionless variance:

\[\sigma_{\theta}^2 = \frac{\sigma^2}{\bar{t}^2}\]

For ideal CSTR: σ²_θ = 1

For ideal PFR: σ²_θ = 0

Cumulative Distribution Function

\[F(t) = \int_0^t E(t') dt'\]

F(t) represents the fraction of material that has been in the reactor for time less than t.

Reaction Thermodynamics

Standard Heat of Reaction

\[\Delta H_{rxn}^{\circ} = \sum_{\text{products}} \nu_i \Delta H_{f,i}^{\circ} - \sum_{\text{reactants}} \nu_j \Delta H_{f,j}^{\circ}\]

Where:

• ΔH°_f,i = standard heat of formation of species i (kJ/mol)
• ν_i = stoichiometric coefficient

Sign convention:

• ΔH_rxn < 0:="" exothermic="" reaction="" (heat="">
• ΔH_rxn > 0: Endothermic reaction (heat absorbed)

Standard Gibbs Free Energy of Reaction

\[\Delta G_{rxn}^{\circ} = \sum_{\text{products}} \nu_i \Delta G_{f,i}^{\circ} - \sum_{\text{reactants}} \nu_j \Delta G_{f,j}^{\circ}\]

Relationship to equilibrium constant:

\[\Delta G_{rxn}^{\circ} = -RT \ln K\]

Where:

• K = equilibrium constant (dimensionless for activities, has units for concentrations)
• R = 8.314 J/(mol·K)
• T = temperature (K)

Spontaneity criterion:

• ΔG < 0:="" reaction="" is="">
• ΔG = 0: System at equilibrium
• ΔG > 0: Reaction is non-spontaneous

Relationship Between K_p and K_c

For gas phase reactions:

\[K_p = K_c (RT)^{\Delta n}\]

Where:

• K_p = equilibrium constant in terms of partial pressures
• K_c = equilibrium constant in terms of concentrations
• Δn = (moles of gaseous products) - (moles of gaseous reactants)
• R = 0.08206 L·atm/(mol·K) when using atm and L

Mass Transfer Effects in Reactions

External Mass Transfer

Rate of mass transfer to catalyst surface:

\[r_{mt} = k_c a_s (C_A - C_{As})\]

Where:

• k_c = mass transfer coefficient (m/s)
• a_s = external surface area per unit volume (m²/m³)
• C_A = bulk concentration (mol/m³)
• C_As = concentration at catalyst surface (mol/m³)

Overall Reaction Rate with Mass Transfer

At steady state, mass transfer rate equals reaction rate:

\[k_c a_s (C_A - C_{As}) = \eta k C_{As}^n\]

Where:

• η = effectiveness factor
• k = intrinsic rate constant
• n = reaction order

Damköhler Number

Ratio of reaction rate to mass transfer rate:

\[Da = \frac{\text{reaction rate}}{\text{mass transfer rate}} = \frac{k}{k_c a_s}\]

Interpretation:

• Da < 1:="" reaction-limited="" (kinetics="">
• Da >> 1: Mass transfer-limited

Special Topics

Autocatalytic Reactions

For: \(A + B \rightarrow 2B\) (B is product and catalyst)

\[r = k C_A C_B\]

Maximum rate occurs at:

\[C_A = C_B = \frac{C_{A0}}{2}\]

Chain Reactions

Initiation: Formation of free radicals
Propagation: Chain carriers react and regenerate
Termination: Removal of chain carriers

Steady-state approximation for intermediates:

\[\frac{dC_I}{dt} = 0\]

Where:

• C_I = concentration of intermediate (free radical)

Polymerization Kinetics

Degree of polymerization (DP):

\[DP = \frac{\text{moles of monomer consumed}}{\text{moles of polymer formed}}\]

Number average molecular weight:

\[\bar{M}_n = \frac{\sum N_i M_i}{\sum N_i}\]

Weight average molecular weight:

\[\bar{M}_w = \frac{\sum w_i M_i}{\sum w_i} = \frac{\sum N_i M_i^2}{\sum N_i M_i}\]

Where:

• N_i = number of molecules of molecular weight M_i
• w_i = weight fraction

Deactivating Catalysts

Activity:

\[a(t) = \frac{r(t)}{r_0}\]

Simple deactivation:

\[\frac{da}{dt} = -k_d a^m\]

Where:

• a(t) = catalyst activity at time t (dimensionless, 0 to 1)
• r₀ = initial rate
• k_d = deactivation rate constant
• m = order of deactivation

For first-order deactivation (m = 1):

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