Reaction kinetics
- First order
- ln(CA/CA0) = −kt
- Half-life (first order)
- t½ = ln 2/k
- Second order
- 1/CA = 1/CA0 + kt
- Arrhenius
- k = A·e−Ea/(RT)
- Two temperatures
- ln(k2/k1) = (Ea/R)(1/T1 − 1/T2)
Chemical Reaction Engineering: cram pack
Formula sheets, the key method steps for every topic and 8 practice problems, with full solutions at the back.
1Formulas and methods
Reactor sizing (CSTR & PFR)
- Space time
- τ = V/Q
- CSTR, first order
- τ = X/(k·(1 − X)), CA = CA0/(1 + kτ)
- PFR, first order
- τ = ln(1/(1 − X))/k, CA = CA0·e−kτ
- Conversion from volume
- CSTR: X = kτ/(1 + kτ), PFR: X = 1 − e−kτ
- N equal CSTRs in series
- X = 1 − 1/(1 + kτi)N
Chemical equilibrium
- Equilibrium constant
- Kc = [C]c[D]d/([A]a[B]b)
- ICE table
- Initial, Change (×coefficient), Equilibrium
- Quadratic
- x = (−b + √(b² − 4ac))/(2a)
Method
- Write Kc with products over reactants, each raised to the power of its coefficient.
- Set up an ICE table: Initial, Change (in steps of x times the coefficients), Equilibrium.
- Put the equilibrium expressions into Kc and solve for x.
- Check whether x is small enough for an approximation; otherwise solve the quadratic.
Mass balance
- Steady state
- in = out
- Total balance
- ṁ1 + ṁ2 = ṁ3
- Component balance
- ṁ1x1 + ṁ2x2 = ṁ3x3
- Evaporator
- F·xF = P·xP, W = F − P
2Practice problems
1.Reaction kineticsMedium
A first-order reaction in a batch reactor has k = 0.3 min⁻¹. How long does it take to reach 90% conversion? Find t.
k = 0.3 min⁻¹X = 90 %
t = min
2.Reaction kineticsHard
Two experiments measure the initial rate of A → products. With CA0 = 0.4 mol/L the rate is 0.005 mol/(L·s); with CA0 = 0.8 mol/L it is 0.02 mol/(L·s). Find the reaction order n with respect to A.
CA0,1 = 0.4 mol/Lr1 = 0.005 mol/(L·s)CA0,2 = 0.8 mol/Lr2 = 0.02 mol/(L·s)
n =
3.Reactor sizing (CSTR & PFR)Medium
A first-order liquid-phase reaction A → B with k = 0.1 min⁻¹ is carried out in a plug-flow reactor (PFR) fed at 50 L/min. The density is constant and the reactor is at steady state. Find the reactor volume V in litres for 80% conversion of A.
Q = 50 L/mink = 0.1 min⁻¹X = 80%
V = L
4.Reactor sizing (CSTR & PFR)Hard
A first-order liquid-phase reaction (k = 0.1 min⁻¹) runs in a 200 L plug-flow reactor (PFR) fed at 5 L/min. The density is constant and the reactor is at steady state. Find the conversion X in percent.
V = 200 LQ = 5 L/mink = 0.1 min⁻¹
X = %
5.Chemical equilibriumMedium
0.5 mol/L of H₂ and 0.5 mol/L of I₂ are put in a container and react: H₂ + I₂ ⇌ 2HI, with Kc = 4. Find the equilibrium concentration [HI].
[H₂]₀ = [I₂]₀ = 0.5 mol/LKc = 4
[HI] = mol/L
6.Chemical equilibriumHard
PCl₅ decomposes: PCl₅ ⇌ PCl₃ + Cl₂, with Kc = 0.5 at the reaction temperature. A container starts with 0.25 mol/L of PCl₅ only. Find the equilibrium concentration [PCl5].
[PCl₅]₀ = 0.25 mol/LKc = 0.5
[PCl5] = mol/L
7.Mass balanceMedium
How much of stream 2 must be added each hour to stream 1 so that the blended product contains 35% salt? Find ṁ2.
ṁ1 = 1000 kg/hx1 = 15 %x2 = 40 %x3 = 35 %
ṁ2 = kg/h
8.Mass balanceHard
An evaporator concentrates a NaOH solution by boiling off pure water. Find the product flow P in kg/h.
F = 600 kg/hxF = 14 %xP = 30 %
P = kg/h
3Solutions
1.Reaction kinetics
Answer: t = 7.675 min
Reaction rates: for a first-order reaction −dC/dt = kC, so C = C₀·e−kt and the half-life ln 2/k does not depend on concentration. For a second-order reaction 1/C = 1/C₀ + kt. The rate constant rises with temperature following Arrhenius: k = A·e−Ea/(RT).
Conversion and concentration
C/C0 = 1 − X = 0.1
Solve for the time
t = −ln(1 − X)/k = −ln(0.1)/0.3 = 7.675 min
Answer
t = 7.675 min
2.Reaction kinetics
Answer: n = 2
Method of initial rates: for r = k·Cⁿ, comparing two runs that differ only in the starting concentration removes k: r₂/r₁ = (C₂/C₁)ⁿ. Doubling C doubles the rate for first order and quadruples it for second order.
Ratios
r2/r1 = 0.02/0.005 = 4
C2/C1 = 0.8/0.4 = 2
Order
n = ln(4)/ln(2) = 2
Answer
n = 2
3.Reactor sizing (CSTR & PFR)
Answer: V = 804.7 L
Ideal reactors, first-order reaction (−rA = k·CA) at steady state, with space time τ = V/Q. A CSTR is perfectly mixed, so the whole volume reacts at the low outlet concentration: τ = X/(k·(1 − X)) and CA = CA0/(1 + k·τ). In a PFR each slice of fluid reacts like a small batch reactor: τ = ln(1/(1 − X))/k and CA = CA0·e−k·τ. For the same conversion a CSTR always needs more volume than a PFR.
Space time
τ = ln(1/(1 − 0.8))/0.1 = 1.609/0.1 = 16.09 min
Volume
V = Q·τ = 50·16.09 = 804.7 L
Answer
V = 804.7 L
4.Reactor sizing (CSTR & PFR)
Answer: X = 98.17 %
Ideal reactors, first-order reaction (−rA = k·CA) at steady state, with space time τ = V/Q. A CSTR is perfectly mixed, so the whole volume reacts at the low outlet concentration: τ = X/(k·(1 − X)) and CA = CA0/(1 + k·τ). In a PFR each slice of fluid reacts like a small batch reactor: τ = ln(1/(1 − X))/k and CA = CA0·e−k·τ. For the same conversion a CSTR always needs more volume than a PFR.
Space time
τ = V/Q = 200/5 = 40 min, kτ = 0.1·40 = 4
Conversion
X = 1 − e−kτ = 1 − e−4 = 98.17%
Answer
X = 98.17%
5.Chemical equilibrium
Answer: [HI] = 0.5 mol/L
Chemical equilibrium: for aA + bB ⇌ cC + dD, the equilibrium constant is Kc = [C]c[D]d/([A]a[B]b), using concentrations at equilibrium. An ICE table (Initial, Change, Equilibrium) links the starting amounts to the equilibrium amounts through one unknown x, scaled by the coefficients.
Equilibrium expression
4 = (2x)²/(0.5 − x)²
Take the square root
2 = 2x/(0.5 − x) → x = 0.5·2/(2 + 2) = 0.25 mol/L
Equilibrium concentration
[HI] = 2x = 0.5 mol/L
Answer
[HI] = 0.5 mol/L
6.Chemical equilibrium
Answer: [PCl5] = 0.06699 mol/L
Chemical equilibrium: for aA + bB ⇌ cC + dD, the equilibrium constant is Kc = [C]c[D]d/([A]a[B]b), using concentrations at equilibrium. An ICE table (Initial, Change, Equilibrium) links the starting amounts to the equilibrium amounts through one unknown x, scaled by the coefficients.
Equilibrium expression
0.5 = x²/(0.25 − x) → x² + 0.5·x − 0.125 = 0
Solve the quadratic
x = (−0.5 + √(0.5² + 4·0.125))/2 = 0.183 mol/L
Equilibrium concentration
[PCl₅] = 0.25 − 0.183 = 0.06699 mol/L
Answer
[PCl5] = 0.06699 mol/L
7.Mass balance
Answer: ṁ2 = 4000 kg/h
Steady-state mass balance: in = out for the total and for each component. Blending problems combine both.
Balances
Total: ṁ3 = ṁ1 + ṁ2
salt: ṁ1x1 + ṁ2x2 = (ṁ1 + ṁ2)x3
Solve for ṁ2
ṁ2 = ṁ1(x1 − x3)/(x3 − x2) = 1000·(15 − 35)/(35 − 40) = 4000 kg/h
Answer
ṁ2 = 4000 kg/h
8.Mass balance
Answer: P = 280 kg/h
Tie component: a component that passes through unchanged in one stream (here the solute) links the flows directly. Then the total balance gives the rest.
Solute balance
F·xF = P·xP → P = 600·14/30 = 280 kg/h
Answer
P = 280 kg/h
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