The gas-phase reaction between hydrogen and iodine $$\mathrm{H}_{2}(g)+\mathrm{I}_{2}(g) \rightleftharpoons 2 \mathrm{HI}(g)$$ proceeds with a rate constant for the forward reaction at \(700^{\circ} \mathrm{C}\) of \(138 \mathrm{~L} / \mathrm{mol} \cdot \mathrm{s}\) and an activation energy of \(165 \mathrm{~kJ} / \mathrm{mol}\). (a) Calculate the activation energy of the reverse reaction given that \(\Delta H_{i}^{\circ}\) for HI is \(26.48 \mathrm{~kJ} / \mathrm{mol}\) and \(\Delta H_{i}^{\circ}\) for \(\mathrm{I}_{2}(\mathrm{~g})\) is \(62.44 \mathrm{~kJ} / \mathrm{mol}\). (b) Calculate the rate constant for the reverse reaction at \(700^{\circ} \mathrm{C}\). (Assume \(\mathrm{A}\) in the equation \(k=\mathrm{Ae}^{-\mathcal{P}_{2} / \mathrm{RT}^{\prime}}\) is the same for both forward and reverse reactions.) (c) Calculate the rate of the reverse reaction if the concentration of HI is \(0.200 M\). The reverse reaction is second-order in HI.

Chemical kinetics, the branch of physical chemistry involved in understanding the rates of chemical reactions, is crucial for the study of reaction mechanisms and the calculation of activation energy. It involves the analysis of how different experimental conditions can influence the speed of a chemical reaction and how this can be quantified through rate constants.

For a reaction such as \(\mathrm{H}_{2}(g) + \mathrm{I}_{2}(g) \rightleftharpoons 2 \mathrm{HI}(g)\), knowing the rate constant for the forward reaction allows us to predict how quickly reactants will convert to products under certain conditions. Activation energy (\(E_a\)) is a vital concept in this field, representing the minimum energy that must be overcome for reactants to transform into products.

When calculating activation energy for a reverse reaction, one must recognize the role of enthalpy change (\(\Delta H^\circ\)) in the process. This change, which can be endothermic or exothermic, will either increase or decrease the overall activation energy needed for the reverse reaction compared to the forward reaction, respectively.

The Arrhenius equation is a formula that describes how the rate constant (\(k\)) of a reaction varies with temperature and activation energy. The equation is expressed as \(k = Ae^{(-E_a)/(RT)}\), where \(A\) is the pre-exponential factor, a constant that represents the frequency of collisions with the correct orientation; \(E_a\) is the activation energy; \(R\) is the universal gas constant; and \(T\) is the temperature in Kelvin.

This equation helps chemists understand the sensitivity of reaction rates to temperature changes. For example, in the exercise, we're informed the pre-exponential factor is the same for both the forward and reverse reactions, allowing us to make comparisons and calculate the rate constant for the reverse reaction when given the activation energy and temperature. After converting Celsius to Kelvin, the equation enables precise calculation. Understanding and applying the Arrhenius equation are fundamental when dealing with temperature's effect on reaction rates.

Enthalpy change (\(\Delta H^\circ\)) in a chemical reaction refers to the heat energy absorbed or released when a reaction occurs at constant pressure. It's a state function, meaning it depends only on the initial and final states of the system, not the path taken.

The enthalpy change for a reaction can be calculated using the standard enthalpies of formation of the reactants and products. This value demonstrates whether the reaction is endothermic (heat absorbed, positive \(\Delta H^\circ\)) or exothermic (heat released, negative \(\Delta H^\circ\)).

For instance, in the provided exercise, the enthalpy change affects the activation energy of the reverse reaction. The reaction's calculated \(\Delta H^\circ\) is subtracted from the activation energy of the forward reaction to determine the activation energy for the reverse reaction according to the equation \(E_{a}^{rev} = E_{a}^{fwd} - \Delta H^\circ\). An accurate understanding of \(\Delta H^\circ\) and its implications on activation energy are essential components of reaction chemistry.