(a) Two reactions have identical values for \(E_{a} .\) Does this ensure that they will have the same rate constant if run at the same temperature? Explain. (b) Two similar reactions have the same rate constant at \(25^{\circ} \mathrm{C}\), but at \(35^{\circ} \mathrm{C}\) one of the reactions has a larger rate constant than the other. Account for these observations.

Activation energy (Ea) is the minimum energy required for a chemical reaction to proceed. The rate of the reaction depends on the activation energy and the temperature at which the reaction is run. The Arrhenius equation relates the rate constant (k) with activation energy and temperature as shown below: \[k = Ae^{-\frac{Ea}{RT}}\] Here: - k = rate constant - A = pre-exponential factor or frequency factor (constant for any reaction) - Ea = activation energy - R = gas constant (8.314 J/mol.K) - T = temperature in Kelvin Now, let's answer the questions mentioned in the exercise.

The question asks if two reactions with identical values of Ea will have the same rate constant if run at the same temperature. To answer this, let's apply our knowledge of the Arrhenius equation. For two reactions with the same Ea and temperature (T), the rate constants (k1 and k2) will be given by the following expressions: \[k_1 = A_1e^{-\frac{Ea}{RT}}\] \[k_2 = A_2e^{-\frac{Ea}{RT}}\] If A1 and A2 (the frequency factors) are also the same for both reactions, then k1 and k2 will be equal. However, if A1 and A2 are different, k1 and k2 will also be different. Therefore, having identical values for Ea alone does not ensure that the rate constants will be the same for the two reactions.

The question states that two similar reactions have the same rate constant at \(25^{\circ}\mathrm{C}\), but at \(35^{\circ}\mathrm{C}\), one of the reactions has a larger rate constant than the other. We'll need to explain these observations. At \(25^{\circ}\mathrm{C}\), the rate constants (k1 and k2) for the reactions are the same, which means: \[k_1 = A_1e^{-\frac{Ea_1}{R(298)}}\] \[k_2 = A_2e^{-\frac{Ea_2}{R(298)}}\] Here, 298 K is the temperature in Kelvin corresponding to \(25^{\circ}\mathrm{C}\). Since k1 = k2, \[A_1e^{-\frac{Ea_1}{R(298)}} = A_2e^{-\frac{Ea_2}{R(298)}}\] At \(35^{\circ}\mathrm{C}\), one reaction has a larger rate constant than the other. This means either the activation energy is different, or the frequency factors are different for the two reactions at this higher temperature. To verify this, we can look at the Arrhenius equation again: \[k_1 = A_1e^{-\frac{Ea_1}{R(308)}}\] \[k_2 = A_2e^{-\frac{Ea_2}{R(308)}}\] Here, 308 K is the temperature in Kelvin corresponding to \(35^{\circ}\mathrm{C}\). At this temperature, we know k1 ≠ k2, so \[A_1e^{-\frac{Ea_1}{R(308)}} ≠ A_2e^{-\frac{Ea_2}{R(308)}}\] The difference in rate constants could arise from differences in activation energy or frequency factors at different temperatures. The temperature dependence of frequency factors may also contribute to the observed difference in rate constants. In summary, having identical activation energies does not ensure the same rate constant for two reactions run at the same temperature, and it is possible for two similar reactions with the same rate constant at one temperature to have different rate constants at another temperature. This can be attributed to differences in their activation energies or frequency factors at different temperatures.