Arrhenius

Arrhenius equation

The Arrhenius equation gives the dependence of a rate coefficient of a reaction on the absolute temperature \(T\) as

\[k(T) = A T^{n} \exp\left(-\frac{E_a}{k_B T}\right)\]

where \(A\) is the pre-exponential (frequency) factor, \(n\) is the temperature exponent, \(E_a\) is the activation energy, and \(k_B\) is the Boltzmann constant. The Arrhenius equation is used to describe the temperature dependence of the rate coefficient of a chemical reaction.

The Arrhenius equation is also used to describe the temperature dependence of the diffusion coefficient, the viscosity, and thermal conductivity of a gas:

\[ \begin{align}\begin{aligned}D(T) = D_0 \exp\left(-\frac{E_D}{R T}\right)\\\mu(T) = \mu_0 \exp\left(\frac{E_\mu}{R T}\right)\\k(T) = k_0 \exp\left(-\frac{E_k}{R T}\right)\end{aligned}\end{align} \]

where \(D_0\) is the diffusion coefficient at infinite temperature, \(E_D\) is the activation energy for diffusion, \(\mu_0\) is the viscosity at infinite temperature, \(E_\mu\) is the activation energy for viscosity, \(k_0\) is the thermal conductivity at infinite temperature, and \(E_k\) is the activation energy for thermal conductivity.

For a review of the Arrhenius equation, and a determination of the parameters for various reactions, read the following chapter.

Arrhenius reactions in CRANE

To specify an Arrhenius reaction in CRANE, simply provide the Arrhenius expression for the rate coefficient in the Chemical Reaction block of the file, for example:

[ChemicalReactions]
  [./ScalarNetwork]
    species = 'e He He* O O_2 '
    equation_constants = 'Tg Te'
    equation_values = '345 2.5'
    equation_variables = 'T0 Teg'

    reactions = 'e + He* -> He + e         : { 7.0e-10 * (Teg)^(0.5) }
                 e + O_2 -> O + O + e      : { 1.41e-9 * Te^(0.22) * exp(-12.62/Te)}'
  [../]
[]