Chemical Reaction Engineering: cram pack

Formula sheets, the key method steps for every topic and 8 practice problems, with full solutions at the back.

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1Formulas and methods

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)

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

  1. Write Kc with products over reactants, each raised to the power of its coefficient.
  2. Set up an ICE table: Initial, Change (in steps of x times the coefficients), Equilibrium.
  3. Put the equilibrium expressions into Kc and solve for x.
  4. 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
t (min)X (%)t = ?30%

A zero-order reaction has k = 0.025 mol/(L·min) and starts at 2 mol/L. How long does it take to reach 30% conversion? Find t.

k = 0.025 mol/(L·min)CA0 = 2 mol/LX = 30 %
t = min
2.Reaction kineticsHard
1/Tln kT = 80 °Cln 0.001ln A = ?Ea​ = 90 kJ/mol

A first-order reaction has a rate constant of 0.001 s⁻¹ at 80 °C and an activation energy of 90 kJ/mol. Find the pre-exponential factor A in the Arrhenius equation.

k = 0.001 s⁻¹T = 80 °CEa = 90 kJ/molR = 8.314 J/(mol·K)
A = s⁻¹
3.Reactor sizing (CSTR & PFR)Medium
QX = 90%same Q, k and X as the CSTR

A first-order liquid-phase reaction must reach 90% conversion. How many times larger must a CSTR be than a PFR for the same feed rate, rate constant and conversion? Find the ratio VCSTR/VPFR.

X = 90%first order, same k and Q
VCSTR/VPFR =
4.Reactor sizing (CSTR & PFR)Hard
Q = 10 L/minX = ?V = 500 L

A first-order liquid-phase reaction (k = 0.08 min⁻¹) runs in a 500 L plug-flow reactor (PFR) fed at 10 L/min. The density is constant and the reactor is at steady state. Find the conversion X in percent.

V = 500 LQ = 10 L/mink = 0.08 min⁻¹
X = %
5.Chemical equilibriumMedium
H2​I2​HII0.250.250C−x−x+2xE0.25 − x0.25 − x2x

0.25 mol/L of H₂ and 0.25 mol/L of I₂ are put in a container and react: H₂ + I₂ ⇌ 2HI, with Kc = 64. Find the equilibrium concentration [H2].

[H₂]₀ = [I₂]₀ = 0.25 mol/LKc = 64
[H2] = mol/L
6.Chemical equilibriumHard
PCl5​PCl3​Cl2​I0.800C−x+x+xE0.8 − xxx

PCl₅ decomposes: PCl₅ ⇌ PCl₃ + Cl₂, with Kc = 0.05 at the reaction temperature. A container starts with 0.8 mol/L of PCl₅ only. Find the equilibrium concentration [PCl5].

[PCl₅]₀ = 0.8 mol/LKc = 0.05
[PCl5] = mol/L
7.Mass balanceMedium
MixerStream 1350 kg/h · 40% saltStream 2ṁ₂ = ? · 15% saltProduct35% salt

How much of stream 2 must be added each hour to stream 1 so that the blended product contains 35% salt? Find ṁ2.

ṁ1 = 350 kg/hx1 = 40 %x2 = 15 %x3 = 35 %
ṁ2 = kg/h
8.Mass balanceHard
EvaporatorFeed F4500 kg/h · 15% sugarProduct P55% sugarVapour W (pure water)W = ? kg/h

An evaporator concentrates a sugar solution by boiling off pure water. Find the water evaporated W in kg/h.

F = 4500 kg/hxF = 15 %xP = 55 %
W = kg/h

3Solutions

1.Reaction kinetics

Answer: t = 24 min

Integrated rate laws: zero order: C = C₀ − kt (straight line, t½ = C₀/(2k)); first order: ln(C₀/C) = kt (t½ = ln 2/k); second order: 1/C − 1/C₀ = kt (t½ = 1/(kC₀)). Conversion X means C = C₀(1 − X).

Amount converted

CA0·X = 2·0.3 = 0.6 mol/L

Time

t = CA0·X/k = 0.6/0.025 = 24 min

Answer

t = 24 min
2.Reaction kinetics

Answer: A = 2.053 × 10¹⁰ s⁻¹

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).

Absolute temperature

T = 80 + 273.15 = 353.1 K

Solve Arrhenius for A

Ea/(RT) = 90000/(8.314·353.1) = 30.65
A = k·e30.65 = 0.001·2.053 × 10¹³ = 2.053 × 10¹⁰ s⁻¹

Answer

A = 2.053 × 10¹⁰ s⁻¹
3.Reactor sizing (CSTR & PFR)

Answer: VCSTR/VPFR = 3.909

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.

k·τ for each reactor

CSTR: k·τ = X/(1 − X) = 0.9/0.1 = 9
PFR: k·τ = ln(1/(1 − X)) = ln(10) = 2.303

Ratio of volumes

VCSTR/VPFR = 9/2.303 = 3.909

Answer

VCSTR/VPFR = 3.909

The gap grows quickly at high conversion.

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 = 500/10 = 50 min, kτ = 0.08·50 = 4

Conversion

X = 1 − e−kτ = 1 − e−4 = 98.17%

Answer

X = 98.17%
5.Chemical equilibrium

Answer: [H2] = 0.05 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

64 = (2x)²/(0.25 − x)²

Take the square root

8 = 2x/(0.25 − x) → x = 0.25·8/(2 + 8) = 0.2 mol/L

Equilibrium concentration

[H₂] = 0.25 − x = 0.05 mol/L

Answer

[H2] = 0.05 mol/L
6.Chemical equilibrium

Answer: [PCl5] = 0.6234 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

0.05 = x²/(0.8 − x) → x² + 0.05·x − 0.04 = 0

Solve the quadratic

x = (−0.05 + √(0.05² + 4·0.04))/2 = 0.1766 mol/L

Equilibrium concentration

[PCl₅] = 0.8 − 0.1766 = 0.6234 mol/L

Answer

[PCl5] = 0.6234 mol/L
7.Mass balance

Answer: ṁ2 = 87.5 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) = 350·(40 − 35)/(35 − 15) = 87.5 kg/h

Answer

ṁ2 = 87.5 kg/h
8.Mass balance

Answer: W = 3273 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 = 4500·15/55 = 1227 kg/h

Total balance

W = F − P = 4500 − 1227 = 3273 kg/h

Answer

W = 3273 kg/h

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