Question

The time taken for the completion of \(90 \%\) of a first-order reaction is 't' min. What is the time (in seconds) taken for the completion of \(99 \%\) of the reaction? (a) \(2 t\) (b) \(t / 30\) (c) \(120 t\) (d) \(60 \mathrm{t}\)

Short answer

(c) 120t.

Step-by-step solution

Understanding First-Order Reactions

In a first-order reaction, the rate of the reaction is directly proportional to the concentration of a single reactant. The formula for the time taken for a certain percentage of a first-order reaction to be complete is given by:\[ t = \frac{2.303}{k} \log \left(\frac{[A]_0}{[A]}\right)\]where \([A]_0\) is the initial concentration, \([A]\) is the concentration at time \(t\), and \(k\) is the rate constant.

Finding Time for 90% Completion

Given that 90% of the reaction is complete in time \(t\), this implies that 10% of the reactant remains. Thus, we can set up the equation:\[ t = \frac{2.303}{k} \log \left(\frac{[A]_0}{0.1[A]_0}\right)\]Simplifying the log fraction:\[ t = \frac{2.303}{k} \log (10)\]

Finding Time for 99% Completion

For 99% completion, 1% of the reactant remains. Thus, the equation becomes:\[ t_{99} = \frac{2.303}{k} \log \left(\frac{[A]_0}{0.01[A]_0}\right)\]Simplifying this:\[ t_{99} = \frac{2.303}{k} \log (100)\]

Relating 90% and 99% Completion Times

We know that the log of products can be simplified: \(\log(100) = 2\log(10)\).Therefore:\[ t_{99} = \frac{2.303}{k} \times 2 \log (10)\]Thus, \(t_{99} = 2t\).

Converting Time from Minutes to Seconds

Since the original time 't' for 90% completion is given in minutes, to find the equivalent time for 99% completion in seconds, we multiply \(t_{99}\) by 60:\[ t_{99} = 2t \times 60 = 120t\]

Key concepts

Rate Constant

In the realm of chemical kinetics, understanding the concept of the rate constant is essential, especially when dealing with first-order reactions. For a first-order reaction, the rate constant, denoted as \(k\), is a proportionality factor that relates the rate of the reaction to the concentration of the reactant. This means the reaction rate is directly proportional to the concentration of one reactant. The formula that demonstrates this relationship is:\[ \text{Rate} = k[A] \]where \([A]\) is the concentration of the reactant at a given time.The rate constant allows chemists to determine how fast a reaction proceeds. It is invariant to the concentration, which means no matter how much reactant is present, \(k\) remains the same at a constant temperature. This characteristic makes it a powerful tool for predicting reaction behavior over time. Additionally, the units of \(k\) vary depending on the order of the reaction. For first-order reactions, \(k\) has units of \(s^{-1}\). This reflects that the rate depends only on time, not on the concentration of reactants.When addressing an exercise like the one provided, understanding the rate constant's role is vital because it influences how we calculate the time needed for the reaction to reach a certain completion level. You'd need to manipulate the formula to derive other important time-related parameters like half-life or completion times.

Reaction Completion Time

Reaction completion time refers to the duration required for a reaction to reach a particular extent of completion. For a first-order reaction, this concept ties closely to the mathematical formulation of the reaction itself. When discussing the completion time in percentage terms, the formula\[ t = \frac{2.303}{k} \log \left(\frac{[A]_0}{[A]}\right) \]becomes handy. This equation helps in determining the time, \(t\), for a given percentage of the reactant to remain. For example, in the context of the provided exercise, understanding that 90% reaction completion means only 10% of the original reactant is left is crucial.- **Key considerations:** - Completing 90% of the reaction implies only 10% reactants remain. - Completing 99% of the reaction implies only 1% reactants remain.This mathematical relationship is pivotal in calculating how long it will take for these percentages to be realized, given a consistent rate constant. Thus, it helps to link theoretical knowledge with practical real-world situations, where knowing completion time is as critical as any other reaction parameter.

Concentration of Reactants

The concentration of reactants plays a crucial role in the study and understanding of chemical reactions, particularly first-order reactions. In these reactions, the rate depends on the concentration of a single reactant, making it an essential parameter to monitor and measure.When we talk about concentration in the context of reaction kinetics, it usually refers to the amount of reactant present initially and at various points during the reaction. This is often represented as \([A]_0\) for the initial concentration, and \([A]\) for the concentration at some time \(t\). The change in concentration over time is key to determining how the reaction progresses.In practical exercises, such as the one described, it's necessary to comprehend that:- **Initial concentration \([A]_0\):** This is the concentration before the reaction begins, providing a baseline.- **Remaining concentration \([A]\):** As the reaction proceeds, this value decreases.Understanding these concentration dynamics is integral to applying the formula \( t = \frac{2.303}{k} \log \left(\frac{[A]_0}{[A]}\right) \). This formula models the logarithmic decrease in reactant concentration over time, which is characteristic of first-order kinetics. Therefore, tracking how reactant concentration dwindles enables us to calculate how long it will take for a reaction to reach a specific completion point. From initial concentrations to determining time required for different completion percentages, grasping the concentration changes in first-order reactions is instrumental in both academic exercises and real-world chemical applications.