Reaction Kinetics

Rate laws, the Arrhenius equation, transition state theory, and catalysis

Rate Laws and Reaction Order

The rate law expresses how the rate of a chemical reaction depends on the concentration of reactants. For a general reaction $aA + bB \to \text{products}$, the rate law takes the form:

$$\text{Rate} = k[A]^m[B]^n$$

where $k$ is the rate constant, and $m$ and $n$ are the reaction orders with respect to A and B. The overall reaction order is $m + n$. These exponents must be determined experimentally and do not necessarily equal the stoichiometric coefficients.

Order	Rate Law	Integrated Form	Half-life	Linear Plot
Zero	$r = k$	$[A] = [A]_0 - kt$	$t_{1/2} = \frac{[A]_0}{2k}$	$[A]$ vs $t$
First	$r = k[A]$	$\ln[A] = \ln[A]_0 - kt$	$t_{1/2} = \frac{\ln 2}{k}$	$\ln[A]$ vs $t$
Second	$r = k[A]^2$	$\frac{1}{[A]} = \frac{1}{[A]_0} + kt$	$t_{1/2} = \frac{1}{k[A]_0}$	$1/[A]$ vs $t$

The Arrhenius Equation

Svante Arrhenius (1889) proposed that the rate constant depends exponentially on temperature:

$$k = A \exp\!\left(-\frac{E_a}{RT}\right)$$

where $A$ is the pre-exponential factor (frequency factor),$E_a$ is the activation energy,$R = 8.314$ J/(mol·K), and $T$ is the absolute temperature.

Taking the natural logarithm yields a linear form ideal for extracting $E_a$ from data:

$$\ln k = \ln A - \frac{E_a}{R}\cdot\frac{1}{T}$$

A plot of $\ln k$ versus $1/T$ (the Arrhenius plot) gives a straight line with slope $-E_a/R$ and intercept $\ln A$.

Transition State Theory

Eyring's transition state theory (TST) provides a molecular-level picture of reaction rates. The reactants pass through an activated complex (transition state) at the top of the energy barrier. The rate constant is:

$$k = \frac{k_BT}{h}\exp\!\left(-\frac{\Delta G^\ddagger}{RT}\right) = \frac{k_BT}{h}\exp\!\left(\frac{\Delta S^\ddagger}{R}\right)\exp\!\left(-\frac{\Delta H^\ddagger}{RT}\right)$$

where $k_B$ is Boltzmann's constant, $h$ is Planck's constant, and $\Delta G^\ddagger$, $\Delta H^\ddagger$, $\Delta S^\ddagger$ are the Gibbs energy, enthalpy, and entropy of activation, respectively.

Collision Theory

A simpler model treats molecules as hard spheres. The rate constant from collision theory is:

$$k = Z_{AB}\, p\, \exp\!\left(-\frac{E_a}{RT}\right)$$

where $Z_{AB}$ is the collision frequency between A and B, and $p$ is the steric factor (fraction of collisions with the correct orientation). The steric factor is typically much less than 1, reflecting the importance of molecular geometry.

Catalysis

A catalyst increases the reaction rate by providing an alternative reaction pathway with a lower activation energy. It is not consumed in the overall reaction.

Homogeneous Catalysis

Catalyst is in the same phase as the reactants. Example: acid-catalyzed ester hydrolysis in aqueous solution. The catalyst participates in forming an intermediate that decomposes to regenerate the catalyst and produce products.

Heterogeneous Catalysis

Catalyst is in a different phase (typically a solid surface). The mechanism involves: (1) adsorption of reactants, (2) surface reaction, (3) desorption of products. Industrial examples include the Haber process (Fe catalyst) and catalytic converters (Pt/Pd/Rh).

Key Principle

A catalyst lowers $E_a$ but does not change $\Delta G$ or$\Delta H$ of the overall reaction. It speeds up both the forward and reverse reactions equally, so it does not shift the equilibrium position.

Python: Arrhenius Plot & Activation Energy

Fit an Arrhenius plot from experimental rate constant data to extract the activation energy $E_a$ and pre-exponential factor $A$.

Arrhenius Plot Analysis

Python

Fits ln(k) vs 1/T to extract activation energy and pre-exponential factor from rate constant data

script.py61 lines

import numpy as np
import matplotlib
matplotlib.use('Agg')
import matplotlib.pyplot as plt
# ─────────────────────────────────────────────
# Arrhenius Plot: Extracting Activation Energy
# ─────────────────────────────────────────────
# Experimental rate constant data at various temperatures
T_data = np.array([300, 320, 340, 360, 380, 400, 420, 440, 460, 480])  # K
k_data = np.array([1.2e-4, 4.8e-4, 1.7e-3, 5.3e-3, 1.5e-2,
                    3.9e-2, 9.2e-2, 2.0e-1, 4.1e-1, 7.9e-1])  # s^-1

R = 8.314  # J/(mol*K)

# Arrhenius: ln(k) = ln(A) - E_a/(R*T)
inv_T = 1.0 / T_data
ln_k = np.log(k_data)

# Linear fit: ln(k) = slope * (1/T) + intercept
coeffs = np.polyfit(inv_T, ln_k, 1)
slope = coeffs[0]
intercept = coeffs[1]

E_a = -slope * R / 1000.0  # kJ/mol
A = np.exp(intercept)       # pre-exponential factor

print(f"Activation energy E_a = {E_a:.1f} kJ/mol")
print(f"Pre-exponential factor A = {A:.3e} s^-1")
print(f"R^2 of fit: {np.corrcoef(inv_T, ln_k)[0,1]**2:.6f}")

# Plot
inv_T_fit = np.linspace(inv_T.min() * 0.95, inv_T.max() * 1.05, 200)
ln_k_fit = slope * inv_T_fit + intercept

fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(12, 5))

# Arrhenius plot
ax1.scatter(1000 * inv_T, ln_k, color='cyan', s=80, zorder=5, label='Data')
ax1.plot(1000 * inv_T_fit, ln_k_fit, 'r--', linewidth=2,
         label=f'Fit: $E_a$ = {E_a:.1f} kJ/mol')
ax1.set_xlabel('1000/T  (1/K)', fontsize=12)
ax1.set_ylabel('ln(k)', fontsize=12)
ax1.set_title('Arrhenius Plot', fontsize=14)
ax1.legend(fontsize=11)
ax1.grid(True, alpha=0.3)

# k vs T
T_smooth = np.linspace(290, 490, 300)
k_smooth = A * np.exp(-E_a * 1000 / (R * T_smooth))
ax2.scatter(T_data, k_data, color='cyan', s=80, zorder=5, label='Data')
ax2.plot(T_smooth, k_smooth, 'r-', linewidth=2, label='Arrhenius fit')
ax2.set_xlabel('Temperature (K)', fontsize=12)
ax2.set_ylabel('Rate constant k (s$^{-1}$)', fontsize=12)
ax2.set_title('Rate Constant vs Temperature', fontsize=14)
ax2.legend(fontsize=11)
ax2.grid(True, alpha=0.3)

plt.tight_layout()
plt.savefig('output.png', dpi=150, bbox_inches='tight')
print("Plot saved.")

Click Run to execute the Python code

Code will be executed with Python 3 on the server

Fortran: Coupled Rate Equations

Numerical integration of the consecutive reaction $A \xrightarrow{k_1} B \xrightarrow{k_2} C$ using a simple Euler method. The coupled ODEs are:

$$\frac{d[A]}{dt} = -k_1[A], \quad \frac{d[B]}{dt} = k_1[A] - k_2[B], \quad \frac{d[C]}{dt} = k_2[B]$$

Coupled Rate Equations (A -> B -> C)

Fortran

Euler integration of consecutive first-order reactions with mass conservation check

coupled_kinetics.f9053 lines

program coupled_kinetics
  implicit none
  integer, parameter :: dp = selected_real_kind(15)
  integer, parameter :: N = 10000
  real(dp) :: k1, k2, dt, t_max
  real(dp) :: A_conc, B_conc, C_conc, t
  real(dp) :: dA, dB, dC
  integer :: i, print_step

! Rate constants (s^-1)
  k1 = 0.5_dp
  k2 = 0.1_dp

! Initial conditions: [A]=1, [B]=0, [C]=0
  A_conc = 1.0_dp
  B_conc = 0.0_dp
  C_conc = 0.0_dp

t_max = 20.0_dp
  dt = t_max / N
  print_step = N / 20

write(*,'(A)') '=== Consecutive Reaction A -> B -> C ==='
  write(*,'(A,F6.3,A,F6.3,A)') 'k1 = ', k1, ' s^-1,  k2 = ', k2, ' s^-1'
  write(*,'(A)') ''
  write(*,'(A8, A12, A12, A12)') 'Time', '[A]', '[B]', '[C]'
  write(*,'(A8, A12, A12, A12)') '(s)', '(mol/L)', '(mol/L)', '(mol/L)'
  write(*,'(A)') '----------------------------------------------'
  write(*,'(F8.2, F12.6, F12.6, F12.6)') 0.0_dp, A_conc, B_conc, C_conc

! Euler integration
  do i = 1, N
    t = i * dt
    dA = -k1 * A_conc * dt
    dB = (k1 * A_conc - k2 * B_conc) * dt
    dC = k2 * B_conc * dt

A_conc = A_conc + dA
    B_conc = B_conc + dB
    C_conc = C_conc + dC

if (mod(i, print_step) == 0) then
      write(*,'(F8.2, F12.6, F12.6, F12.6)') t, A_conc, B_conc, C_conc
    end if
  end do

! Analytic check for [B]_max
  ! t_max_B = ln(k1/k2)/(k1-k2)
  write(*,'(A)') ''
  write(*,'(A,F8.3,A)') 'Analytic t(B_max) = ', log(k1/k2)/(k1-k2), ' s'
  write(*,'(A,F8.3,A)') 'Mass conservation check: A+B+C = ', &
    A_conc + B_conc + C_conc, ' (should be 1.000)'
end program coupled_kinetics

Click Run to execute the Fortran code

Code will be compiled with gfortran and executed on the server

Video Lectures

Lecture 27: Reaction Rates

Goodie Bag 8: Reactions

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