Question

In the first-order reaction, half of the reaction is com pleted in 100 seconds. The time for \(99 \%\) reaction to occur will be (a) \(664.64 \mathrm{~s}\) (b) \(646.6 \mathrm{~s}\) (c) \(660.9 \mathrm{~s}\) (d) \(654.5 \mathrm{~s}\)

Short answer

The time is approximately 664.64 seconds, so the correct option is (a).

Step-by-step solution

Understand First-Order Reaction Characteristics

In a first-order reaction, the time taken for a specific fraction of the reaction to complete can be calculated using the formula: \( t = \frac{2.303}{k} \log \frac{[A]_0}{[A]} \), where \([A]_0\) and \([A]\) are the initial and final concentrations respectively, and \(k\) is the rate constant.

Calculate Rate Constant (k)

The relation between half-life \( t_{1/2} \) and rate constant \( k \) for a first-order reaction is given by the formula: \( k = \frac{0.693}{t_{1/2}} \). Given that \( t_{1/2} = 100 \) seconds, the rate constant \( k \) can be calculated as follows: \[ k = \frac{0.693}{100} = 0.00693 \text{ s}^{-1} \].

Apply the First-Order Time Formula

We need to find the time for 99% of the reaction to occur, meaning \([A] = 0.01[A]_0\). Substituting the values into the first-order time formula: \[ t = \frac{2.303}{0.00693} \log \frac{[A]_0}{0.01[A]_0} \]. This simplifies to \[ t = \frac{2.303}{0.00693} \log 100 \].

Solve for Time (t)

The logarithm part simplifies to \( \log 100 = 2 \), thus:\[ t = \frac{2.303 \times 2}{0.00693} = \frac{4.606}{0.00693} \]. Calculating this gives \[ t \approx 664.64 \text{ seconds} \].

Choose the Correct Option

After calculating the time \( t \) as approximately 664.64 seconds, the correct choice that matches this answer is option (a) 664.64 seconds.

Key concepts

Reaction Kinetics

Reaction kinetics is the study of the rates at which chemical processes occur. Understanding these rates helps in predicting how fast a reaction will proceed, which is crucial in various fields such as chemical engineering and pharmaceuticals.
In reaction kinetics, every reaction has its own rate, which could be influenced by factors like temperature, concentration, and the presence of catalysts. For a specific reaction type, such as first-order reactions, the rate is directly proportional to the concentration of one reactant.

• This means if you double the concentration of the reactant, the rate of the reaction also doubles.
• First-order reactions often involve processes like radioactive decay or simple decomposition reactions.

Grasping reaction kinetics helps in understanding the precise control needed over a reaction to ensure efficiency and safety in industrial applications.

Half-Life Calculation

Half-life is a concept that describes the time required for half of the reactant in a reaction to be consumed. This is particularly useful in the context of first-order reactions.
For a first-order reaction, the half-life is a constant, meaning it doesn't depend on the initial concentration of the substance. The formula used to calculate the half-life is:
\[ t_{1/2} = \frac{0.693}{k} \]
where \( k \) is the rate constant. Since the half-life remains constant, it is a key parameter in understanding how quickly a reaction progresses.

• The concept of half-life is vital in various applications, including pharmacology, where it helps to regulate drug dosages.
• In the environment, the half-life helps scientists understand how long a pollutant may persist.

By mastering half-life calculations, you can predict the timings required for significant portions of the reaction to occur.

Rate Constant

The rate constant, denoted as \( k \), is a key factor in determining how fast a reaction proceeds. In a first-order reaction, the rate constant has the units of time inverse, typically seconds inverse (\( s^{-1} \)).
The value of \( k \) can be determined experimentally and is unique for each reaction at a given temperature. It is calculated using the half-life formula for first-order reactions:
\[ k = \frac{0.693}{t_{1/2}} \]
Since \( k \) influences the speed of both the half-life and the overall reaction, understanding \( k \) helps you predict the time frame of a reaction.

• A higher rate constant means a faster reaction.
• The rate constant is sensitive to factors like temperature, and knowing it allows adjustments to maintain reaction rates under different conditions.

Proficiency in calculating and utilizing the rate constant is essential for enhancing reaction efficiency and control.

First-Order Reaction Formula

The formula for a first-order reaction is essential in determining how long a reaction takes to reach a certain completion level. It's expressed as:
\[ t = \frac{2.303}{k} \log \frac{[A]_0}{[A]} \]
This formula allows you to calculate the time \( t \) it takes for a reaction starting with concentration \([A]_0\) to reach concentration \([A]\).

• The logarithmic term \( \log \frac{[A]_0}{[A]} \) represents the extent to which the concentration has decreased.
• This equation is handy in scenarios where you need to know the time required for various reaction completion percentages.

Mastering the first-order reaction formula gives you the ability to predict the temporal dynamics of a reaction, providing strategic insights for both experimental and theoretical chemical studies.