m physical modeling / IICSM

PHYSICS / ENGINEERING / COMPUTING

Chemical kinetics models

From reaction rates to integrated laws, Arrhenius behavior, competing pathways and ideal reactors, with twenty worked examples.

Subject library · 51 guides · derivations & worked examples

Fundamental physics

Space and flight

Materials and transport

Systems and mathematics

Marine and road vehicles

Electrical and computing

Companion site

Matter pathway: atom → solid → liquid → gas → plasma. Quantum mechanics and quantum field theory provide foundations across the pathway; they are not additional phases. This is a connected modeling route, not a universal heating curve. Actual phases depend on pressure, composition, and kinetics.

1. Rates and orders

Definitions & inputs. cA molar concentration,k rate constant,r reaction rate,νi stoichiometric coefficient.

  1. Stoichiometry relates species rates while experiments establish the rate law.

    dci/dt=νir,r=kcAmdc_i/dt=\nu_i r,\quad r=kc_A^m

  2. Separate variables and integrate a first-order consumption law.

    dcA/dt=−kcA⇒cA=cA0e−ktdc_A/dt=-kc_A\Rightarrow c_A=c_{A0}e^{-kt}

  3. Set concentration to one half of its initial value.

    t1/2=ln⁡2/kt_{1/2}=\ln2/k

Interpretation. Rate-constant units depend on reaction order and concentration units.

↑ Return to definitions and contents

2. Integrated higher-order laws

Definitions & inputs. c initial/current concentration,t time,k rate coefficient.

  1. Integrate zero-order loss only until the reactant is depleted.

    dc/dt=−k⇒c=c0−ktdc/dt=-k\Rightarrow c=c_0-kt

  2. Integrate the second-order law by separation.

    dc/dt=−kc2⇒1/c−1/c0=ktdc/dt=-kc^2\Rightarrow1/c-1/c_0=kt

  3. Substitute c=c0/2; second-order half-life depends on initial concentration.

    t1/2=1/(kc0)t_{1/2}=1/(kc_0)

Interpretation. Distinguish the species-consumption coefficient from a reaction-extent rate with stoichiometric factors.

↑ Return to definitions and contents

3. Temperature and pathways

Definitions & inputs. Ea activation energy,R molar gas constant,T absolute temperature,A prefactor,k1,k2 pathway rates.

  1. The exponential models thermally activated kinetics.

    k=Ae−Ea/(RT)k=Ae^{-E_a/(RT)}

  2. Eliminate a temperature-independent prefactor between two temperatures.

    ln⁡(k2/k1)=Ea(1/T1−1/T2)/R\ln(k_2/k_1)=E_a(1/T_1-1/T_2)/R

  3. Add parallel depletion rates and integrate the product branch at complete conversion.

    cA=cA0e−(k1+k2)t,Y1=k1/(k1+k2)c_A=c_{A0}e^{-(k_1+k_2)t},\quad Y_1=k_1/(k_1+k_2)

Interpretation. Temperature-dependent prefactors and changing mechanisms can invalidate a single activation energy.

↑ Return to definitions and contents

4. Ideal reactors

Definitions & inputs. V reactor volume,q volume flow,τ=V/q residence time,X conversion,c0 inlet concentration.

  1. A steady stirred tank balances inlet, outlet and uniform reaction.

    q(c0−c)=Vkcq(c_0-c)=Vkc

  2. Substitute c=c0(1−X) into the balance.

    XCSTR=kτ/(1+kτ)X_{CSTR}=k\tau/(1+k\tau)

  3. Integrate the material-element first-order loss along ideal plug flow.

    XPFR=1−e−kτX_{PFR}=1-e^{-k\tau}

Interpretation. Mixing, diffusion, heat release and residence-time distributions alter actual reactor behavior.

↑ Return to definitions and contents

Graphical worked example

First-order loss: c0=2 mol/L, k=0.1 s⁻¹ at constant volume and temperature. X axis: Reaction time (s). Y axis: Reactant concentration (mol/L).
First-order loss: c0=2 mol/L, k=0.1 s⁻¹ at constant volume and temperature. Related worked calculation · Download SVG · Plot data

Twenty worked examples

Open a problem to see its defined inputs, assumptions, equation, numerical substitution, result, and interpretation. Values are illustrative analytical exercises.

Example 01. First-order initial rate

Definitions & inputs. k=.2 s⁻¹,cA=3 mol/L.

  1. Choose the governing model and isolate the requested quantity.

    r=kcAr=kc_A

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    r=0.2(3)r=0.2(3)

  3. Evaluate the expression; the result uses the units shown.

    Result=0.6 mol L−1s−1\mathrm{Result}=0.6\ {\rm mol\,L}^{-1}{\rm s}^{-1}

Interpretation. The consumption coefficient is defined for A directly.

↑ Return to definitions and contents
Example 02. First-order concentration

Definitions & inputs. c0=2 mol/L,k=.1 s⁻¹,t=10 s.

  1. Choose the governing model and isolate the requested quantity.

    c=c0e−ktc=c_0e^{-kt}

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    c=2e−1c=2e^{-1}

  3. Evaluate the expression; the result uses the units shown.

    Result=0.7357589 mol L−1\mathrm{Result}=0.7357589\ {\rm mol\,L}^{-1}

Interpretation. Isothermal constant-volume conditions keep k constant.

↑ Return to definitions and contents
Example 03. First-order half-life

Definitions & inputs. k=.1 s⁻¹.

  1. Choose the governing model and isolate the requested quantity.

    t1/2=ln⁡2/kt_{1/2}=\ln2/k

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    t1/2=ln⁡2/0.1t_{1/2}=\ln2/0.1

  3. Evaluate the expression; the result uses the units shown.

    Result=6.931472 s\mathrm{Result}=6.931472\ {\rm s}

Interpretation. It does not depend on starting concentration.

↑ Return to definitions and contents
Example 04. Time to 90% conversion

Definitions & inputs. First-order k=.2 s⁻¹.

  1. Choose the governing model and isolate the requested quantity.

    t=−ln⁡(1−X)/kt=-\ln(1-X)/k

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    t=−ln⁡0.1/0.2t=-\ln0.1/0.2

  3. Evaluate the expression; the result uses the units shown.

    Result=11.51293 s\mathrm{Result}=11.51293\ {\rm s}

Interpretation. Conversion X is a fraction, not a percentage in the equation.

↑ Return to definitions and contents
Example 05. Measured first-order constant

Definitions & inputs. Concentration drops from 1 to .4 mol/L in 5 s.

  1. Choose the governing model and isolate the requested quantity.

    k=ln⁡(c0/c)/tk=\ln(c_0/c)/t

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    k=ln⁡(1/0.4)/5k=\ln(1/0.4)/5

  3. Evaluate the expression; the result uses the units shown.

    Result=0.1832581 s−1\mathrm{Result}=0.1832581\ {\rm s}^{-1}

Interpretation. More measurements are needed to establish first-order behavior.

↑ Return to definitions and contents
Example 06. Zero-order concentration

Definitions & inputs. c0=1 mol/L,k=.02 mol/(L s),t=20 s.

  1. Choose the governing model and isolate the requested quantity.

    c=c0−ktc=c_0-kt

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    c=1−0.02(20)c=1-0.02(20)

  3. Evaluate the expression; the result uses the units shown.

    Result=0.6 mol L−1\mathrm{Result}=0.6\ {\rm mol\,L}^{-1}

Interpretation. The law cannot continue into negative concentration.

↑ Return to definitions and contents
Example 07. Zero-order depletion time

Definitions & inputs. c0=1 mol/L,k=.02 mol/(L s).

  1. Choose the governing model and isolate the requested quantity.

    td=c0/kt_d=c_0/k

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    td=1/0.02t_d=1/0.02

  3. Evaluate the expression; the result uses the units shown.

    Result=50 s\mathrm{Result}=50\ {\rm s}

Interpretation. Real kinetics may change as reactant becomes scarce.

↑ Return to definitions and contents
Example 08. Second-order concentration

Definitions & inputs. −dc/dt=kc²,k=.5 L/(mol s),c0=1 mol/L,t=2 s.

  1. Choose the governing model and isolate the requested quantity.

    c=c0/(1+kc0t)c=c_0/(1+kc_0t)

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    c=1/[1+0.5(1)(2)]c=1/[1+0.5(1)(2)]

  3. Evaluate the expression; the result uses the units shown.

    Result=0.5 mol L−1\mathrm{Result}=0.5\ {\rm mol\,L}^{-1}

Interpretation. k is the species-consumption coefficient as explicitly defined.

↑ Return to definitions and contents
Example 09. Second-order half-life

Definitions & inputs. k=.5 L/(mol s),c0=2 mol/L.

  1. Choose the governing model and isolate the requested quantity.

    t1/2=1/(kc0)t_{1/2}=1/(kc_0)

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    t1/2=1/[0.5(2)]t_{1/2}=1/[0.5(2)]

  3. Evaluate the expression; the result uses the units shown.

    Result=1 s\mathrm{Result}=1\ {\rm s}

Interpretation. Doubling c0 halves the half-life.

↑ Return to definitions and contents
Example 10. Rate change on doubling concentration

Definitions & inputs. Empirical order m=2.

  1. Choose the governing model and isolate the requested quantity.

    r2/r1=(c2/c1)mr_2/r_1=(c_2/c_1)^m

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    r2/r1=22r_2/r_1=2^2

  3. Evaluate the expression; the result uses the units shown.

    Result=4 \mathrm{Result}=4\ {}

Interpretation. A reaction’s written stoichiometry alone does not prove this order.

↑ Return to definitions and contents
Example 11. Arrhenius rate ratio

Definitions & inputs. Ea=50 kJ/mol,T1=300 K,T2=320 K,R=8.314462618.

  1. Choose the governing model and isolate the requested quantity.

    k2/k1=e(Ea/R)(1/T1−1/T2)k_2/k_1=e^{(E_a/R)(1/T_1-1/T_2)}

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    k2/k1=e(50000/8.314462618)(1/300−1/320)k_2/k_1=e^{(50000/8.314462618)(1/300-1/320)}

  3. Evaluate the expression; the result uses the units shown.

    Result=3.500259 \mathrm{Result}=3.500259\ {}

Interpretation. The prefactor and mechanism are assumed unchanged.

↑ Return to definitions and contents
Example 12. Activation energy estimate

Definitions & inputs. Rate doubles from 300 to 310 K.

  1. Choose the governing model and isolate the requested quantity.

    Ea=Rln⁡(k2/k1)/(1/T1−1/T2)E_a=R\ln(k_2/k_1)/(1/T_1-1/T_2)

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    Ea=8.314462618ln⁡2/(1/300−1/310)E_a=8.314462618\ln2/(1/300-1/310)

  3. Evaluate the expression; the result uses the units shown.

    Result=53.59726 kJ mol−1\mathrm{Result}=53.59726\ {\rm kJ\,mol}^{-1}

Interpretation. Two points estimate an apparent activation energy, not a unique mechanism.

↑ Return to definitions and contents
Example 13. Pseudo-first-order constant

Definitions & inputs. B maintained at .1 mol/L,k2=2 L/(mol s),r=k2 cA cB.

  1. Choose the governing model and isolate the requested quantity.

    k′=k2cBk'=k_2c_B

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    k′=2(0.1)k'=2(0.1)

  3. Evaluate the expression; the result uses the units shown.

    Result=0.2 s−1\mathrm{Result}=0.2\ {\rm s}^{-1}

Interpretation. B must remain effectively constant throughout the experiment.

↑ Return to definitions and contents
Example 14. Parallel-path branch yield

Definitions & inputs. A→P k1=.3 s⁻¹,A→Q k2=.1 s⁻¹.

  1. Choose the governing model and isolate the requested quantity.

    YP=k1/(k1+k2)Y_P=k_1/(k_1+k_2)

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    YP=0.3/0.4Y_P=0.3/0.4

  3. Evaluate the expression; the result uses the units shown.

    Result=0.75 \mathrm{Result}=0.75\ {}

Interpretation. This is the final fraction through P for irreversible first-order branches.

↑ Return to definitions and contents
Example 15. Parallel-path remaining fraction

Definitions & inputs. k1=.3,k2=.1 s⁻¹,t=5 s.

  1. Choose the governing model and isolate the requested quantity.

    cA/cA0=e−(k1+k2)tc_A/c_{A0}=e^{-(k_1+k_2)t}

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    cA/cA0=e−2c_A/c_{A0}=e^{-2}

  3. Evaluate the expression; the result uses the units shown.

    Result=0.1353353 \mathrm{Result}=0.1353353\ {}

Interpretation. Product concentrations require integrating each branch.

↑ Return to definitions and contents
Example 16. Reversible equilibrium fraction

Definitions & inputs. A⇌B first-order kf=.3,kr=.1 s⁻¹,total concentration 1 mol/L.

  1. Choose the governing model and isolate the requested quantity.

    cB,eq=ctotkf/(kf+kr)c_{B,eq}=c_{tot}k_f/(k_f+k_r)

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    cB,eq=1(0.3)/0.4c_{B,eq}=1(0.3)/0.4

  3. Evaluate the expression; the result uses the units shown.

    Result=0.75 mol L−1\mathrm{Result}=0.75\ {\rm mol\,L}^{-1}

Interpretation. At equilibrium forward and reverse rates are equal, not zero.

↑ Return to definitions and contents
Example 17. Reversible relaxation time

Definitions & inputs. kf=.3,kr=.1 s⁻¹.

  1. Choose the governing model and isolate the requested quantity.

    τ=1/(kf+kr)\tau=1/(k_f+k_r)

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    τ=1/0.4\tau=1/0.4

  3. Evaluate the expression; the result uses the units shown.

    Result=2.5 s\mathrm{Result}=2.5\ {\rm s}

Interpretation. Deviations from the two-state equilibrium decay with this time constant.

↑ Return to definitions and contents
Example 18. Reactor residence time

Definitions & inputs. Volume 10 L,flow .5 L/s.

  1. Choose the governing model and isolate the requested quantity.

    τ=V/q\tau=V/q

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    τ=10/0.5\tau=10/0.5

  3. Evaluate the expression; the result uses the units shown.

    Result=20 s\mathrm{Result}=20\ {\rm s}

Interpretation. This is mean hydraulic residence time, not an identical path time in a stirred tank.

↑ Return to definitions and contents
Example 19. CSTR conversion

Definitions & inputs. First-order k=.1 s⁻¹,τ=20 s.

  1. Choose the governing model and isolate the requested quantity.

    X=kτ/(1+kτ)X=k\tau/(1+k\tau)

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    X=2/(1+2)X=2/(1+2)

  3. Evaluate the expression; the result uses the units shown.

    Result=0.6666667 \mathrm{Result}=0.6666667\ {}

Interpretation. A perfectly mixed steady reactor has outlet concentration equal to tank concentration.

↑ Return to definitions and contents
Example 20. Plug-flow conversion

Definitions & inputs. First-order k=.1 s⁻¹,τ=20 s.

  1. Choose the governing model and isolate the requested quantity.

    X=1−e−kτX=1-e^{-k\tau}

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    X=1−e−2X=1-e^{-2}

  3. Evaluate the expression; the result uses the units shown.

    Result=0.8646647 \mathrm{Result}=0.8646647\ {}

Interpretation. Ideal plug flow gives higher conversion here at the same kτ.

↑ Return to definitions and contents

Symbols and units

Each derivation and problem defines its own symbols and inputs. Symbols may be reused with different meanings in other subjects. Keep units consistent, retain sufficient precision during calculation, and apply the stated validity limits.