Problem 217 - A Complete Resource Book in Chemistry for JEE Main

Question

The half life period of a first order chemical reaction is \(6.93\) minutes. The time required for the completion of \(99 \%\) of the chemical reaction will be \((\log 2=0.301):\) (a) \(23.03\) minutes (b) \(46.06\) minutes (c) \(460.6\) minutes (d) \(230.3\) minutes

Step-by-Step Solution

Answer

The time required for 99% reaction completion is 46.06 minutes, answer (b).

Step 1: Understand Half-Life Formula for First Order Reaction

In a first order reaction, the half-life period is given by the formula \( t_{1/2} = \frac{0.693}{k} \), where \( k \) is the rate constant. Here, the half-life is given as \(6.93\) minutes.

Step 2: Calculate the Rate Constant \(k\)

Rearrange the half-life formula to solve for \( k \):\[ k = \frac{0.693}{t_{1/2}} = \frac{0.693}{6.93} \].Calculating gives \( k \approx 0.1 \text{ min}^{-1} \).

Step 3: Use First Order Reaction Completion Formula

For a first order reaction, the time \( t \) required to complete \( n \% \) of the reaction is given by \( t = \frac{2.303}{k} \log \left( \frac{100}{100-n} \right) \). Here, \( n = 99 \% \).

Step 4: Substitute Values into Completion Formula

Substitute \( n = 99 \% \) and \( k = 0.1 \text{ min}^{-1} \) into the formula:\[t = \frac{2.303}{0.1} \log \left( \frac{100}{1} \right) = 23.03 \log(100) \].Since \( \log(100) = 2 \), we have \( t = 23.03 \times 2 \).

Step 5: Calculate Time for 99% Completion

Thus, \( t = 23.03 \times 2 = 46.06 \text{ minutes} \). This means that the time required to complete 99% of the reaction is 46.06 minutes.

Key Concepts

First Order Reaction

A first order reaction is a type of chemical reaction where the rate of reaction is directly proportional to the concentration of one reactant. This means that as the concentration of the reactant decreases, the rate at which the reaction proceeds also decreases. However, it is not affected by changes in the concentration of other reactants. In mathematical terms, this can be represented as: \[rate = k imes [A]\] where:

• \(k\) is the rate constant, a unique number for each reaction occurring at a certain temperature.
• \([A]\) denotes the concentration of the reactant.

First order reactions are particularly important because many reactions, including radioactivity and some chemical decompositions, follow this kinetic pattern. Understanding how these reactions behave over time is crucial for predicting how fast they will occur under various conditions.

Half-life

Half-life is a central concept in kinetics, especially for first order reactions. It is defined as the time required for the concentration of a reactant to decrease to half of its initial concentration. The unique feature of first order reactions is that their half-life is constant and does not depend on the initial concentration. For a first order reaction, the formula for half-life \(t_{1/2}\) is: \[t_{1/2} = \frac{0.693}{k}\] Here, \(k\) is the rate constant. The constant \(0.693\) is derived from the natural logarithm of 2, reflecting the exponential decay pattern of the reaction. This property makes it very useful in practical applications, such as determining the duration needed for a substance to significantly reduce in quantity without calculating its concentration at each point in time.

Rate Constant

The rate constant \(k\) is a fundamental factor in chemical kinetics that quantifies the speed of a chemical reaction. For first order reactions, it holds a special place because it remains unaffected by reactant concentration and solely depends on temperature. The unit of the rate constant for first order reactions is typically expressed as \(\text{min}^{-1}\), indicating how much of the reactant converts to product per minute. There are several influencing factors for \(k\), but temperature is the most significant one, as outlined by the Arrhenius equation. The equation explicates how higher temperatures generally increase the rate constant, speeding up the reaction. Understanding \(k\) allows chemists to decipher and predict the dynamics of a reaction, which is used in everything from environmental science to pharmaceuticals.

Reaction Completion Time

Determining the reaction completion time, especially for a particular percentage like 99%, requires a good grasp of first order kinetics. Completion time is essentially the duration required for a specific fraction of the reaction to go through. For first order reactions, the time \(t\) needed to reach a certain completion is given by: \[t = \frac{2.303}{k} \log \left( \frac{100}{100-n} \right)\] where:

• \(n\)% is the desired completion percentage of the reaction.
• \(k\) is the rate constant.

By applying this formula, we can precisely calculate how long it takes for a reaction to reach a certain point of completion, such as 99%. This method provides a systematic way to determine reaction longevity, offering insights into how reactions will proceed over time. This is critical in fields like industrial synthesis, where optimization of reaction conditions is essential for cost-effectiveness.