Question

For a reaction of order \(n\), the integrated form of the rate equation is: \((n-1) \cdot K \cdot t\) \(=\left(C_{0}\right)^{1-n}-(C)^{1-n}\), where \(C_{0}\) and \(C\) are the values of the reactant concentration at the start and after time ' \(t\) '. What is the relationship between \(t_{3 / 4}\) and \(t_{1 / 2}\), where \(t_{3 / 4}\) is the time required for \(C\) to become \(C_{0} / 4\). (a) \(t_{3 / 4}=t_{1 / 2} \cdot\left[2^{n-1}+1\right]\) (b) \(t_{3 / 4}=t_{1 / 2}\left[2^{n-1}-1\right]\) (c) \(t_{3 / 4}=t_{1 / 2}\left[2^{n+1}-1\right]\) (d) \(t_{3 / 4}=t_{1 / 2} \cdot\left[2^{n+1}+1\right]\)

Short answer

The relationship between t_{3/4} and t_{1/2} is given by (b) t_{3 / 4}=t_{1 / 2}\left[2^{n-1}-1\right].

Step-by-step solution

Setting up the equations for t_{1/2} and t_{3/4}

First, set up the equation for the time required for the reactant concentration to drop to half its original value, t_{1/2}. When C = C_0/2, plug these values into the given equation to get an equation for t_{1/2}. Likewise, set up another equation for the time required to reduce the concentration to a quarter, t_{3/4}, by substituting C = C_0/4 into the original equation.

Simplify the t_{1/2} equation

Simplify the equation for t_{1/2} by reducing the term (C_0)^{1-n} and expressing the remaining terms in terms of C_0 and n.

Simplify the t_{3/4} equation

Similarly, simplify the equation for t_{3/4} by reducing the terms involving C_0 and expressing the result in terms of C_0 and n.

Find the relationship between t_{3/4} and t_{1/2}

Divide the equation for t_{3/4} by the equation for t_{1/2} to find a relationship between the two times. The resulting expression should match with one of the provided answer choices (a) through (d).

Key concepts

Reaction Order

Understanding the reaction order is crucial when exploring how chemical reactions proceed over time. The reaction order refers to the power to which the concentration of a reactant is raised in the rate law equation. It indicates how the rate of a reaction is affected by the concentration of that specific reactant.

In general, reaction orders can be zero, first, or second, but they can also adopt fractional or even negative values in some complex reactions. For a reaction of order zero, the rate is independent of the reactant's concentration. In a first-order reaction, the rate is directly proportional to the reactant's concentration. Meanwhile, a second-order reaction's rate is proportional to the square of the reactant's concentration. The concept of reaction order is fundamental in chemical kinetics as it helps predict how variations in concentration affect the reaction rate and thus the amount of time it takes for the reactants to transform into products.

Chemical Kinetics

Chemical kinetics is the field of chemistry that studies the rates at which reactions occur and the factors that influence these rates. The main objective is to understand the sequence of steps through which the reactants turn into products, known as the reaction mechanism.

Various factors can affect the reaction rate, including reactant concentrations, temperature, catalysts, and the physical state of the reactants. By studying kinetics, chemists are able to determine the reaction rate equation, which provides the relationship between the rate of a reaction and the concentrations of the reactants. This equation is fundamental for controlling chemical reactions in industrial processes, biological systems, and environmental changes. In the classroom, kinetic principles help students predict and control reactions effectively, playing a significant role in the design of chemical reactors and the synthesis of new materials.

Rate Equation

The rate equation, also known as the rate law, is a mathematical expression that relates the rate of a chemical reaction to the concentration of the reactants. It usually takes the form \( rate = k \cdot [A]^{m} \cdot [B]^{n} \), where \( k \) is the rate constant, \( [A] \) and \( [B] \) are the concentrations of reactants A and B, and \( m \) and \( n \) are the reaction orders with respect to A and B, respectively.

In the context of the integrated rate equation given in the exercise, we are looking at the equation after it has been integrated over time, which allows us to connect concentrations of reactants at different points in time with the elapsed time. This particular form of the rate equation is essential when we work with experimental data to determine reaction rates and orders. It also plays a central role in predicting how long a reaction will take to reach a certain level of completion, such as halfway or three quarters of the way to completion, which directly relates to the exercise at hand where we calculate \( t_{3/4} \) in relation to \( t_{1/2} \).