PE Exam Exam > PE Exam Notes > Chemical Engineering for PE > Cheatsheet: Kinetics
Cheatsheet: Kinetics
1. Reaction Rate Fundamentals
1.1 Rate Definitions
| Term | Definition |
|---|---|
| Reaction Rate (r) | Change in concentration per unit time; r = -dCA/dt for reactant A |
| Rate Law | Mathematical expression relating reaction rate to reactant concentrations: r = kCAαCBβ |
| Rate Constant (k) | Proportionality constant in rate law; temperature dependent |
| Reaction Order | Sum of exponents in rate law (α + β); can be fractional, zero, or negative |
1.2 Rate Expression Forms
| Basis | Expression |
|---|---|
| Concentration | r = dC/dt (mol/L·s) |
| Conversion | rA = -dNA/dt = -FA0dX/dt |
| Extent of Reaction | r = dξ/dt where ξ is extent of reaction |
| Per Unit Volume | r = (1/V)(dNA/dt) |
2. Reaction Order and Rate Laws
2.1 Elementary Reaction Orders
| Order | Rate Law |
|---|---|
| Zero Order | r = k; CA = CA0 - kt; t1/2 = CA0/2k |
| First Order | r = kCA; ln(CA/CA0) = -kt; t1/2 = ln(2)/k = 0.693/k |
| Second Order (Type 1) | r = kCA2; 1/CA = 1/CA0 + kt; t1/2 = 1/(kCA0) |
| Second Order (Type 2) | r = kCACB; ln(CA/CB) = ln(CA0/CB0) + k(CB0 - CA0)t |
| nth Order | r = kCAn; 1/CAn-1 = 1/CA0n-1 + (n-1)kt |
2.2 Determining Reaction Order
- Differential Method: Plot ln(r) vs ln(CA); slope = n
- Integral Method: Test integrated forms; linear plot indicates correct order
- Half-Life Method: Plot ln(t1/2) vs ln(CA0); slope = 1-n
- Initial Rate Method: Vary CA0, measure initial rates; r0 ∝ CA0n
3. Temperature Dependence
3.1 Arrhenius Equation
| Form | Equation |
|---|---|
| Exponential | k = A·exp(-Ea/RT) |
| Logarithmic | ln(k) = ln(A) - Ea/RT |
| Two Temperatures | ln(k2/k1) = (Ea/R)(1/T1 - 1/T2) |
- A = pre-exponential factor or frequency factor (same units as k)
- Ea = activation energy (J/mol or cal/mol)
- R = gas constant (8.314 J/mol·K or 1.987 cal/mol·K)
- T = absolute temperature (K)
- Plot ln(k) vs 1/T: slope = -Ea/R, intercept = ln(A)
3.2 Modified Arrhenius Equations
- Modified Form: k = ATnexp(-Ea/RT)
- Van't Hoff Equation (equilibrium): d(ln Keq)/dT = ΔH°/RT2
- Temperature Sensitivity: Rule of thumb - rate doubles for every 10°C increase
4. Complex Reactions
4.1 Reversible Reactions
| Type | Rate Expression |
|---|---|
| First Order (both) | A ⇌ B; r = k1CA - k-1CB |
| Equilibrium | Keq = k1/k-1 = CB,eq/CA,eq |
| Net Rate at Equilibrium | rnet = 0; forward rate = reverse rate |
4.2 Parallel Reactions
- A → B (k1) and A → C (k2)
- rA = -(k1 + k2)CA
- Selectivity: SB/C = CB/CC = k1/k2 (first order)
- Yield: YB = CB/(CA0 - CA)
4.3 Series (Consecutive) Reactions
- A → B → C with k1, k2
- rA = -k1CA
- rB = k1CA - k2CB
- rC = k2CB
- Maximum CB at time: tmax = ln(k1/k2)/(k1 - k2)
5. Reaction Mechanisms
5.1 Elementary Reactions
| Molecularity | Description |
|---|---|
| Unimolecular | A → Products; rate = kCA |
| Bimolecular | A + B → Products; rate = kCACB |
| Termolecular | A + B + C → Products; rate = kCACBCC (rare) |
5.2 Pseudo-Order Reactions
- Pseudo-First Order: r = kCACB where CB >> CA; r = k'CA with k' = kCB
- Applies when one reactant is in large excess or is catalyst
5.3 Steady-State Approximation (SSA)
- Assumption: dCI/dt ≈ 0 for reactive intermediates
- Rate of formation = rate of consumption for intermediates
- Used to derive rate laws from multi-step mechanisms
5.4 Rate-Determining Step (RDS)
- Slowest step in mechanism controls overall rate
- Pre-equilibrium assumption: fast steps before RDS at equilibrium
- Overall rate law derived from RDS using equilibrium expressions
6. Chain Reactions
6.1 Chain Reaction Steps
| Step | Description |
|---|---|
| Initiation | Formation of reactive intermediates (radicals); slow step |
| Propagation | Intermediate reacts with stable molecule producing another intermediate and product |
| Branching | One intermediate produces multiple intermediates; accelerates reaction |
| Termination | Intermediates combine to form stable products; removes intermediates |
6.2 Chain Length
- Chain Length (ν) = rate of propagation/rate of initiation
- Long chain approximation: rate of initiation = rate of termination
- Rate of product formation >> rate of initiation
7. Catalysis
7.1 Catalyst Fundamentals
- Catalyst increases rate without being consumed; provides alternate pathway
- Lowers activation energy; does not change thermodynamics or equilibrium
- Increases both forward and reverse rates equally
- Homogeneous catalysis: catalyst in same phase as reactants
- Heterogeneous catalysis: catalyst in different phase (solid catalyst, fluid reactants)
7.2 Enzyme Kinetics (Michaelis-Menten)
| Parameter | Expression |
|---|---|
| Rate Equation | r = VmaxCS/(KM + CS) |
| Maximum Rate | Vmax = kcatCE0 |
| Michaelis Constant | KM = (k-1 + k2)/k1 |
| Low Substrate | CS <>M; r = (Vmax/KM)CS (first order) |
| High Substrate | CS >> KM; r = Vmax (zero order) |
7.3 Lineweaver-Burk Plot
- Double reciprocal: 1/r = (KM/Vmax)(1/CS) + 1/Vmax
- Plot 1/r vs 1/CS: slope = KM/Vmax, y-intercept = 1/Vmax, x-intercept = -1/KM
7.4 Enzyme Inhibition
| Type | Effect |
|---|---|
| Competitive | KM,app = KM(1 + CI/KI); Vmax unchanged; inhibitor competes for active site |
| Uncompetitive | KM,app = KM/(1 + CI/KI); Vmax,app = Vmax/(1 + CI/KI); binds enzyme-substrate complex |
| Noncompetitive | KM unchanged; Vmax,app = Vmax/(1 + CI/KI); binds enzyme or complex equally |
8. Heterogeneous Catalysis
8.1 Surface Reaction Steps
- 1. External diffusion: reactant transport from bulk to catalyst surface
- 2. Internal diffusion: reactant transport through pores
- 3. Adsorption: reactant binds to active site
- 4. Surface reaction: chemical transformation on surface
- 5. Desorption: product releases from surface
- 6. Internal diffusion: product transport out of pores
- 7. External diffusion: product transport to bulk
8.2 Langmuir-Hinshelwood Kinetics
| Case | Rate Expression |
|---|---|
| Single Site (A → B) | r = kKAPA/(1 + KAPA + KBPB) |
| Dual Site (A + B → C) | r = kKAKBPAPB/(1 + KAPA + KBPB + KCPC)2 |
- Ki = adsorption equilibrium constant for species i
- θi = KiPi/(1 + ΣKjPj) = fractional coverage of species i
8.3 Eley-Rideal Mechanism
- One reactant adsorbs, other reacts from gas phase
- r = kKAPAPB/(1 + KAPA)
9. Mass Transfer Effects
9.1 Effectiveness Factor
| Parameter | Definition |
|---|---|
| Effectiveness Factor (η) | η = actual rate/rate without diffusion limitation = robs/rsurface |
| Thiele Modulus (φ) | φ = L√(k/De) where L = characteristic length, De = effective diffusivity |
| First Order Relation | η = tanh(φ)/φ |
| Large φ (strong limit) | η ≈ 1/φ; internal diffusion controls |
| Small φ (weak limit) | η ≈ 1; reaction controls |
9.2 External Mass Transfer
- Rate = kcas(Cbulk - Csurface) where kc = mass transfer coefficient, as = surface area per volume
- Sherwood number: Sh = kcL/D = f(Re, Sc)
- Schmidt number: Sc = μ/(ρD)
9.3 Weisz-Prater Criterion
- CWP = robsρpR2/(DeCsurface)
- CWP < 1:="" no="" diffusion="">
- CWP >> 1: strong diffusion limitations
10. Ideal Reactor Design Equations
10.1 Batch Reactor
| Property | Expression |
|---|---|
| Design Equation | dCA/dt = rA or dX/dt = -rA/CA0 |
| Integrated (general) | t = CA0∫0X dX/(-rA) |
| First Order | t = (1/k)ln(1/(1-X)) = (1/k)ln(CA0/CA) |
| Second Order | t = (1/(kCA0))[X/(1-X)] |
10.2 Continuous Stirred Tank Reactor (CSTR)
| Property | Expression |
|---|---|
| Design Equation | V/FA0 = τ = X/(-rA) |
| Space Time | τ = V/v0 where v0 = volumetric flow rate |
| First Order | τ = X/[k(1-X)] or CA = CA0/(1 + kτ) |
| Second Order | τ = X/[kCA0(1-X)2] |
10.3 Plug Flow Reactor (PFR)
| Property | Expression |
|---|---|
| Design Equation | dX/dV = -rA/FA0 or V/FA0 = ∫0X dX/(-rA) |
| First Order | τ = (1/k)ln(1/(1-X)) |
| Second Order | τ = (1/(kCA0))[X/(1-X)] |
10.4 Reactor Comparison
- For same conversion and reaction order > 0: VPFR <>batch <>CSTR
- PFR equivalent to batch for same space time and reaction time
- For nth order (n > 0): CSTR requires more volume than PFR
- For autocatalytic reactions: CSTR can outperform PFR at intermediate conversions
11. Non-Ideal Reactors
11.1 Residence Time Distribution (RTD)
| Function | Definition |
|---|---|
| E(t) | Exit age distribution; ∫0∞ E(t)dt = 1 |
| F(t) | Cumulative distribution; F(t) = ∫0t E(t)dt |
| Mean Residence Time | t̄ = ∫0∞ tE(t)dt = V/v0 |
| Variance | σ2 = ∫0∞ (t - t̄)2E(t)dt |
11.2 Tracer Methods
- Pulse Input: E(t) = C(t)/∫0∞ C(t)dt
- Step Input: F(t) = C(t)/C0; E(t) = dF(t)/dt
11.3 Reactor Models
| Model | Characteristics |
|---|---|
| Tanks-in-Series | E(t) = (tn-1/(t̄n(n-1)!))exp(-nt/t̄); σ2/t̄2 = 1/n |
| Axial Dispersion | D/uL = dispersion number; Pe = uL/D = Peclet number |
| Segregated Flow | X̄ = ∫0∞ X(t)E(t)dt where X(t) from batch equation |
| Maximum Mixedness | Assumes latest possible mixing; opposite of segregated flow |
12. Multiple Reactions and Selectivity
12.1 Selectivity Definitions
| Term | Definition |
|---|---|
| Instantaneous Selectivity | SD/U = rD/rU (desired/undesired) |
| Overall Selectivity | S̄D/U = moles D formed/moles U formed |
| Yield | YD = moles D formed/moles A reacted |
| Fractional Yield | ΦD = moles D formed/maximum possible moles D |
12.2 Optimizing Selectivity
| Reaction System | Strategy |
|---|---|
| Parallel: A→D (α1), A→U (α2) | If α1 > α2: high CA; if α1 <>2: low CA |
| Series: A→D→U | Low conversion, high CA, PFR preferred |
| A + B→D (α1,β1), A+B→U (α2,β2) | Maximize Ci if αi or βi larger for desired reaction |
12.3 Reactor Choice for Selectivity
- PFR: better for series reactions to minimize over-reaction; maintains concentration gradients
- CSTR: better when low reactant concentration favors desired product
- Recycle: intermediate between PFR and CSTR; adjust recycle ratio
13. Non-Isothermal Reactors
13.1 Energy Balance
| Reactor Type | Energy Balance |
|---|---|
| Batch | ρV Ĉp(dT/dt) = (-ΔHrx)VrA + Q̇ |
| CSTR | FA0X(-ΔHrx) = ΣFiĈpi(T - T0) - Q̇ |
| PFR | FA0(dX/dV)(-ΔHrx) = ΣFiĈpi(dT/dV) + Ua(T - Ta) |
13.2 Adiabatic Operation
- Q̇ = 0; energy balance simplifies to X = (ΣFiĈpi/FA0(-ΔHrx))(T - T0)
- Adiabatic temperature rise: ΔTad = (-ΔHrx)CA0/(ρĈp)
- For exothermic: T increases with X; for endothermic: T decreases with X
13.3 Multiple Steady States (CSTR)
- Exothermic reactions in CSTR can have multiple steady states
- Heat generation curve: Q̇gen = FA0X(-ΔHrx)
- Heat removal curve: Q̇rem = ΣFiĈpi(T - T0) + UA(T - Ta)
- Intersections represent steady states; stability requires (dQ̇rem/dT) > (dQ̇gen/dT)
14. Data Analysis and Parameter Estimation
14.1 Differential Method
- Measure concentration vs time; differentiate to get rates
- Plot ln(-rA) vs ln(CA); slope gives reaction order
- Numerical differentiation: polynomial fit then differentiate
14.2 Integral Method
- Assume rate law form; integrate and test linearity
- Zero order: CA vs t linear
- First order: ln(CA) vs t linear
- Second order: 1/CA vs t linear
- Best fit indicates correct order
14.3 Half-Life Method
- Measure t1/2 at different CA0
- t1/2 ∝ CA01-n
- Plot ln(t1/2) vs ln(CA0); slope = 1 - n
14.4 Regression Analysis
- Linearize rate equation when possible
- Use least squares regression: minimize Σ(ycalc - yexp)2
- Non-linear regression for complex rate laws
- Check residuals for systematic errors
15. Important Dimensionless Numbers
15.1 Reactor Performance
| Number | Definition |
|---|---|
| Damköhler Number (Da) | Da = kτ = reaction rate/mass transfer rate |
| Peclet Number (Pe) | Pe = uL/D = convection/dispersion |
| Bodenstein Number (Bo) | Bo = Pe for reactors |
15.2 Mass Transfer
| Number | Definition |
|---|---|
| Sherwood Number (Sh) | Sh = kcL/D = convective mass transfer/diffusive mass transfer |
| Schmidt Number (Sc) | Sc = μ/(ρD) = momentum diffusivity/mass diffusivity |
| Reynolds Number (Re) | Re = ρuL/μ = inertial forces/viscous forces |
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