Question

For a first order reaction, which of the following are not correct? a. \(t_{3 / 8}=2 t_{3 / 4}\) b. \(t_{3 / 4}=2 t_{1 / 2}\) c. \(t_{15 / 6}=4 t_{1 / 2}\) d. \(t_{15 / 16}=3 t_{3 / 4}\)

Short answer

Options (a), (c), and (d) are not correct for a first-order reaction.

Step-by-step solution

Understand First-Order Reaction Kinetics

For a first-order reaction, the rate of reaction is proportional to the concentration of a single reactant. The time to reach a certain fraction of completion is related to the half-life of the reaction, which is constant.

Analyze Option (a)

Option (a) claims \(t_{3 / 8}=2 t_{3 / 4}\). For a first-order reaction, the time required to reduce concentration from 3/8 to zero would not be twice the time to reduce from 3/4 because the time relationships depend on logarithmic decay. This statement is incorrect.

Analyze Option (b)

Option (b) is \(t_{3 / 4}=2 t_{1 / 2}\). For first-order reactions, \(t_{3/4}\) should indeed be twice \(t_{1/2}\) because reducing concentration by a further half from 1/2 to 1/4 requires the same time as the first half-life. This statement is correct.

Analyze Option (c)

Option (c) is \(t_{15 / 6}=4 t_{1 / 2}\). The notation seems incorrect; let's correct it to \(t_{5/6}\). It is incorrect for a first-order reaction, as \(t_{5/6}\) does not equal four half-lives. The accurate conversion involves using the equation for decay time based on natural logarithms.

Analyze Option (d)

Option (d) states \(t_{15 / 16}=3 t_{3 / 4}\). Reducing the concentration from \(3/4\) to \(15/16\) does not follow a straightforward multiple of \(t_{3/4}\), and the exact reduction would require applying the integrated rate law. This statement is incorrect.

Key concepts

Reaction Kinetics

Understanding reaction kinetics is essential when studying chemical reactions. Reaction kinetics describes how the rate of a chemical reaction depends on the concentration of reactants and the specific rate constant. For a first-order reaction, the rate is directly proportional to the concentration of a single reactant. This means if the concentration of the reactant is doubled, the rate of the reaction will also double. By understanding reaction kinetics, you can predict how fast a reaction will proceed under certain conditions.
This concept helps in designing chemical processes and in understanding various natural phenomena.

• The rate of reaction is typically measured in moles per liter per second (M/L/s).
• Reaction mechanisms provide insight into the steps involved in transforming reactants into products.

Grasping these basics can demystify why some reactions happen quickly while others take longer, paving the way for more advanced studies in chemistry.

Half-Life

Half-life is a crucial concept in first-order reactions. It is the time required for half of the reactant to be consumed in a chemical reaction. In other words, it is the time taken for the concentration of a reactant to decrease by half. A special feature of first-order reactions is that the half-life is constant, meaning it doesn't change as the concentration decreases.
Understanding half-life is particularly useful in fields such as pharmacology, where you need to know how quickly a drug is metabolized. It is also important in radioactive decay calculations.

• The formula for half-life ( t_{1/2} ) in a first-order reaction is given as t_{1/2} = rac{0.693}{k} , where k is the rate constant.
• This constant time period creates straightforward experimental and theoretical analysis of decay processes.

By familiarizing oneself with half-life, one can predict how much time is required for a given fraction of a reactive species to break down, which is invaluable in many scientific applications.

Logarithmic Decay

Logarithmic decay is an intriguing phenomenon observed in first-order reactions. It describes how the concentration of reactants decreases over time in a manner that's exponential and not linear. In these reactions, as time progresses, the concentration decreases similarly to an exponential decay curve. This type of decay follows the equation [A] = [A]_0e^{-kt} , where [A] is the concentration at time t , [A]_0 is the initial concentration, and k is the rate constant.
This logarithmic relationship is important when determining how long a reaction will take to reach a certain completion stage.

• The rate of decay indicates how quickly the concentration of reactants changes.
• This decay pattern allows chemists to predict the concentrations of reactants at any given time during the reaction.

Understanding logarithmic decay can help you grasp why reactions slow down as they near completion and how to quantify these dynamics in chemical processes.

Integrated Rate Law

The integrated rate law is a mathematical expression that provides a relationship between the concentration of reactants and time for first-order reactions. This is particularly useful for calculating concentrations at various time points during a reaction. The integrated rate law for first-order kinetics can be expressed as ext{ln} [A] = ext{ln} [A]_0 - kt .
This equation shows a linear relationship between the natural log of concentration and time, making it straightforward to plot and analyze experimental data.

• The slope of a plot of ext{ln} [A] versus time gives the rate constant k .
• The intercept corresponds to ext{ln} [A]_0 , the natural log of the initial concentration.

Using the integrated rate law allows scientists to gain accurate predictions about the behavior of a reaction under varying conditions, essential for both academic studies and practical applications like engineering and pharmacology.