For a first-order reaction, (a) The degree of dissociation is equal to \(\left(1-\mathrm{e}^{-\mathrm{kt}}\right)\) (b) The pre-exponential factor in the Arrhenius equation has the dimensions of time \(\mathrm{t}^{-1}\). (c) The time taken for the completion of \(75 \%\) reation is thrice the t \(1 / 2\) of the reaction. (d) Both (a) and (b)

When discussing chemical reactions, particularly those that involve the breaking apart of molecules, the term 'degree of dissociation' becomes critical. It represents the fraction of a substance that has broken down into its constituent parts. For a first-order reaction, the formula to calculate this ratio is given by \(1-\mathrm{e}^{-\mathrm{kt}}\), where \(k\) is the rate constant, and \(t\) is the time.

The exponential term \(\mathrm{e}^{-\mathrm{kt}}\) accounts for the fraction of the substance that remains associated or un-dissociated at time \(t\). Thus, as time progresses, this value decreases, revealing that more of the substance is dissociating as expected in a reaction moving towards completion. Recognizing the degree of dissociation is crucial for students when examining reaction progress and determining the extent of a reaction at any given point in time.

The Arrhenius equation plays a pivotal role in reaction kinetics by connecting the rate constant \(k\) with the energy necessary to start a reaction and temperature. It is expressed as \(k=A\mathrm{e}^{-\frac{Ea}{RT}}\), where \(A\) is the pre-exponential factor representing the frequency of collisions that result in a reaction, \(Ea\) is the activation energy, \(R\) is the gas constant, and \(T\) is the absolute temperature in Kelvins.

Importance of Pre-exponential Factor

Not all collisions between molecules lead to a reaction. The pre-exponential factor \(A\) helps quantify those that do have the right orientation and energy. Furthermore, the dimensions of \(A\), as stated in the original exercise, are reciprocal time or time\(^{-1}\) because it relates to the frequency of effective collisions per unit of time. This characteristic allows students to better interpret the factors affecting reaction rates.

Undoubtedly, reaction kinetics form the backbone of understanding how chemical reactions occur. This branch of chemistry is concerned with reaction rates and the steps by which a reaction proceeds. The kinetics for a first-order reaction are straightforward: the rate is proportional to the concentration of a single reactant. The mathematical expression for a first-order reaction is given by \(\ln([A]_0/[A]) = kt\), linking the concentration of reactant \(A\) at time \(t\) with the rate constant \(k\).

The equation also sets up a relationship between the time it takes for a specific portion of the reactant to react and the half-life of the reaction. Studying reaction kinetics, students learn to predict how long a reaction will take to reach a certain completion percentage, such as the 75% mark mentioned in the initial problem. This predictive ability is essential in practical applications like pharmaceuticals and materials science.

The rate constant \(k\) is a crucial term in a chemical reaction's rate law. For a first-order reaction, the rate constant has the unit of reciprocal time, implying that it is a measure of how fast the reaction occurs per unit of time. It gives us the proportionality constant that connects the rate of the reaction to the concentration of the reactants.

An interesting property of the rate constant is that it changes with temperature, which is explained by the Arrhenius equation. This dependency reflects the fact that, typically, higher temperatures increase the frequency of collisions between molecules, subsequently increasing the likelihood of reactions. Therefore, understanding the rate constant is necessary to control or predict how a reaction will proceed under different temperatures or over time, which is incredibly useful in industrial chemical processes and research.