(a) A certain first-order reaction has a rate constant of \(2.75 \times 10^{-2} \mathrm{~s}^{-1}\) at \(20^{\circ} \mathrm{C}\). What is the value of \(k\) at \(60^{\circ} \mathrm{C}\) if \(E_{a}=75.5 \mathrm{~kJ} / \mathrm{mol} ?(\mathbf{b})\) Another first-order reaction also has a rate constant of \(2.75 \times 10^{-2} \mathrm{~s}^{-1}\) at \(20^{\circ} \mathrm{C}\). What is the value of \(k\) at \(60^{\circ} \mathrm{C}\) if \(E_{a}=125 \mathrm{~kJ} / \mathrm{mol} ?(\mathrm{c})\) What assumptions do you need to make in order to calculate answers for parts (a) and (b)?

Understanding the speed at which a reaction occurs is essential in chemistry, and this is where the concept of rate constants becomes relevant. A rate constant, represented by the symbol 'k', is a proportionality constant in the rate law that provides a quantitative measure of the speed of a chemical reaction. It varies with temperature, as described by the Arrhenius equation.

In a chemical reaction, the rate law expresses the relationship between the rate of the reaction and the concentration of the reactants. For a given reaction at a certain temperature, the rate constant is fixed and can be used to predict the reaction rate if the concentrations of the reactants are known. As the temperature changes, so does the rate constant.

The Arrhenius equation, which includes the rate constant, provides a way to calculate how 'k' changes with temperature, taking into account the activation energy and the temperature in Kelvin. This relationship is crucial because it allows chemists to understand and predict how the reaction speed will change under different temperature conditions. An increase in temperature typically leads to an increase in the rate constant, implying a faster reaction rate.

Activation energy (Ea) is a key term in understanding chemical kinetics, which refers to the minimum energy required for a reaction to occur. It is the energy barrier that must be overcome by the reactants to transform into products. The concept of activation energy is critical when discussing reaction rates and how they are affected by temperature changes.

The Arrhenius equation includes the activation energy as a factor that influences the rate constant. A higher activation energy means that the reactants need more energy to react, making the reaction slower at the same temperature. Conversely, a lower activation energy equates to a faster reaction since less energy is required for the reactants to overcome the energy barrier.

This interplay between temperature and activation energy is why reaction rates can vary significantly even if other factors remain constant. When applying the Arrhenius equation, as shown in the textbook example, we see that different activation energies result in different rate constants at a given temperature, illustrating the sensitivity of reaction rate to even small changes in these parameters.

In chemical kinetics, reactions can be categorized based on how their rate depends on the concentration of reactants. A first-order reaction is one where the reaction rate is directly proportional to the concentration of a single reactant. This means that if you were to double the concentration of the reactant, the rate of the reaction would also double.

First-order reactions are characterized by their linear relationship between the natural logarithm of reactant concentration and time. They have rate laws that can be written as: rate = k [A], where [A] is the concentration of the reactant, and k is the rate constant for the reaction.

The simplicity of first-order kinetics makes it easier to understand and calculate important variables using the Arrhenius equation. For example, in the exercise provided, the assumption that the reactions are first-order allows us to directly connect the rate constant to the reaction's temperature and activation energy without worrying about the concentration of reactants. It is assumed that the reaction rate is solely influenced by the changes in temperature and is not affected by any other potential variables such as reactant concentration or the presence of a catalyst.