Troe Reaction
=============

Troe (fall-off) reaction rate constant equations take the form:

.. math::

   \frac{k_0[\mathrm{M}]}{1 + \frac{k_0[\mathrm{M}]}{k_{\infty}}} F_C^{\left(1 + \frac{1}{N} [\log_{10}(\frac{k_0[\mathrm{M}]}{k_{\infty}})]^2\right)^{-1}}

where:

- :math:`k_0` is the low-pressure limiting rate constant (:math:`(\mbox{mol}\,\mathrm{m}^{-3})^{-(n-1)}\,\mathrm{s}^{-1}`),
- :math:`k_{\infty}` is the high-pressure limiting rate constant (:math:`(\mbox{mol}\,\mathrm{m}^{-3})^{-(n-1)}\,\mathrm{s}^{-1}`),
- :math:`[\mathrm{M}]` is the density of air (:math:`\mathrm{mol}\,\mathrm{m}^{-3}`),
- :math:`F_C` and :math:`N` are parameters that determine the shape of the fall-off curve,
  and are typically 0.6 and 1.0, respectively :cite:`Finlayson-Pitts2000` :cite:`Gipson`.

:math:`k_0` and :math:`k_{\infty}` are calculated as `Arrhenius` rate constants with :math:`D=300` and :math:`E=0`.

Input data for Troe reactions have the following format:

.. tab-set::

    .. tab-item:: YAML

        .. code-block:: yaml

            type: TROE
            name: foo-troe
            k0_A: 5.6e-12
            k0_B: -1.8
            k0_C: 180.0
            kinf_A: 3.4e-12
            kinf_B: -1.6
            kinf_C: 104.1
            Fc: 0.7
            N: 0.9
            gas phase: gas
            reactants:
              - species name: foo
              - species name: bar
                coefficient: 2
            products:
              - species name: baz
              - species name: qux
                coefficient: 0.65


    .. tab-item:: JSON

        .. code-block:: json

            {
                "type": "TROE",
                "k0_A": 5.6e-12,
                "k0_B": -1.8,
                "k0_C": 180.0,
                "kinf_A": 3.4e-12,
                "kinf_B": -1.6,
                "kinf_C": 104.1,
                "Fc": 0.7,
                "N": 0.9,
                "gas phase": "gas",
                "reactants": [
                    {
                        "species name": "foo"
                    },
                    {
                        "species name": "bar",
                        "coefficient": 2
                    }
                ],
                "products": [
                    {
                        "species name": "baz"
                    },
                    {
                        "species name": "qux",
                        "coefficient": 0.65
                    }
                ]
            }

The key-value pairs ``reactants`` and ``products`` are required. Reactants without a ``coefficient`` value
are assumed to appear once in the reaction equation. Products without a specified ``coefficient`` are
assumed to have a coefficient of 1.0.

The ``gas phase`` key is required and must be set to the name of the phase the reaction
takes place in. The reactants and products must be present in the specified phase.

The two sets of parameters beginning with ``k0_`` and ``kinf_`` are the `Arrhenius` parameters for the
:math:`k_0` and :math:`k_{\infty}` rate constants, respectively. When not present, ``_A`` parameters are
assumed to be 1.0, ``_B`` to be 0.0, ``_C`` to be 0.0, ``Fc`` to be 0.6, and ``N`` to be 1.0.

Rate constants are in units of :math:`\mathrm{(m^{3}\ mol^{-1})^{(n-1)}\ s^{-1}}` where :math:`n` is the total number of reactants.