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Derive the rate equation using Absolute Reaction Rate Theory (Transition State Theory).

Chemistry

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Here's the derivation for the rate equation based on Absolute Reaction Rate Theory (also known as Transition State Theory):

  1. Introduction to Transition State Theory

    Absolute Reaction Rate Theory (ARRT) proposes that reactants form an activated complex (transition state) before converting into products. This activated complex is in equilibrium with the reactants, and its decomposition determines the reaction rate.

  2. Equilibrium between Reactants and Transition State

    Consider a bimolecular reaction:

    A+B⇌[AB]‡→ProductsA + B \rightleftharpoons [AB]^{\ddagger} \to Products

    The equilibrium constant K‡K^{\ddagger} for the formation of the activated complex [AB]‡[AB]^{\ddagger} from reactants A and B is given by:

    K‡=[AB]‡[A][B]K^{\ddagger} = \frac{[AB]^{\ddagger}}{[A][B]}

    From this, the concentration of the activated complex is:

    [AB]‡=K‡[A][B][AB]^{\ddagger} = K^{\ddagger}[A][B]

  3. Rate of Reaction

    The rate of reaction is proportional to the concentration of the activated complex and the frequency at which it decomposes into products. The decomposition frequency is often expressed as ν‡\nu^{\ddagger}, which is related to the vibrational frequency along the reaction coordinate.
    The rate constant kk is given by:

    k=ν‡K‡k = \nu^{\ddagger} K^{\ddagger}

    According to statistical mechanics, the frequency of decomposition ν‡\nu^{\ddagger} can be expressed as kBTh\frac{k_B T}{h}, where kBk_B is Boltzmann's constant, TT is the absolute temperature, and hh is Planck's constant.
    So, the rate constant becomes:

    k=kBThK‡k = \frac{k_B T}{h} K^{\ddagger}

  4. Relating K‡K^{\ddagger} to Thermodynamic Parameters

    The equilibrium constant K‡K^{\ddagger} can be related to the standard Gibbs free energy of activation, ΔG‡\Delta G^{\ddagger}, by the equation:

    ΔG‡=−RTln⁡K‡\Delta G^{\ddagger} = -RT \ln K^{\ddagger}

    Rearranging for K‡K^{\ddagger}:

    K‡=e−ΔG‡RTK^{\ddagger} = e^{-\frac{\Delta G^{\ddagger}}{RT}}

    Substitute this into the rate constant equation:

    k=kBThe−ΔG‡RTk = \frac{k_B T}{h} e^{-\frac{\Delta G^{\ddagger}}{RT}}

  5. Expressing ΔG‡\Delta G^{\ddagger} in terms of Enthalpy and Entropy

    The Gibbs free energy of activation can be expressed in terms of the standard enthalpy of activation (ΔH‡\Delta H^{\ddagger}) and standard entropy of activation (ΔS‡\Delta S^{\ddagger}):

    ΔG‡=ΔH‡−TΔS‡\Delta G^{\ddagger} = \Delta H^{\ddagger} - T\Delta S^{\ddagger}

    Substitute this into the rate constant equation:

    k=kBThe−(ΔH‡−TΔS‡)RTk = \frac{k_B T}{h} e^{-\frac{(\Delta H^{\ddagger} - T\Delta S^{\ddagger})}{RT}} k=kBThe−ΔH‡RTeTΔS‡RTk = \frac{k_B T}{h} e^{-\frac{\Delta H^{\ddagger}}{RT}} e^{\frac{T\Delta S^{\ddagger}}{RT}} k=kBThe−ΔH‡RTeΔS‡Rk = \frac{k_B T}{h} e^{-\frac{\Delta H^{\ddagger}}{RT}} e^{\frac{\Delta S^{\ddagger}}{R}}

  6. Final Rate Equation

    The rate of reaction is given by:

    Rate=k[A][B]Rate = k[A][B]

    Substituting the expression for kk:

    Rate=kBTheΔS‡Re−ΔH‡RT[A][B]Rate = \frac{k_B T}{h} e^{\frac{\Delta S^{\ddagger}}{R}} e^{-\frac{\Delta H^{\ddagger}}{RT}} [A][B]

    This is the Eyring equation, which is the rate equation derived from Absolute Reaction Rate Theory.

    The final rate equation for absolute reaction rate theory is:

    k=kBTheΔS‡Re−ΔH‡RT\boxed{k = \frac{k_B T}{h} e^{\frac{\Delta S^{\ddagger}}{R}} e^{-\frac{\Delta H^{\ddagger}}{RT}}}
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