Question

\(99 \%\) of a first order reaction was completed in 32 minutes when \(99.9 \%\) of the reaction wull complete: (a) \(50 \mathrm{~min}\) (b) \(46 \mathrm{~min}\) (c) \(48 \mathrm{~min}\) (d) \(49 \mathrm{~min}\)

Short answer

The time for 99.9% completion is approximately 48 minutes.

Step-by-step solution

Understanding the First Order Reaction Formula

For a first order reaction, the time taken for a certain percentage of reaction to complete can be calculated using the formula \( k = \frac{2.303}{t} \log \frac{[A]_0}{[A]} \), where \( k \) is the reaction rate constant, \( t \) is the time, \( [A]_0 \) is the initial concentration, and \( [A] \) is the concentration of reactant remaining at time \( t \) . The logarithmic nature of this equation means that equal time intervals are required for equal logarithmic decrements in concentration.

Calculating the Rate Constant \( k \) for 99% Completion

First, we calculate the rate constant \( k \) using the given time (32 minutes) for 99% completion. We know that \( [A] \) is 1% of \( [A]_0 \) because 99% has reacted. So the equation simplifies to \( k = \frac{2.303}{32} \log \frac{100}{1} \) . Now we calculate \( k \) .

Using the Rate Constant to Determine the Time for 99.9% Completion

For 99.9% completion, \( [A] \) is 0.1% of \( [A]_0 \) . We use the same rate constant \( k \) and formula to determine the time \( t \) required. So we set up the equation \( k = \frac{2.303}{t} \log \frac{1000}{1} \) and solve for \( t \) using the previously calculated value of \( k \) .

Solving for Time \( t \)

From the equation in step 3, we plug in our value of \( k \) and solve for \( t \) which will give us the time required for 99.9% of the reaction to be completed. This will require logarithmic calculation for the concentration decrement from 100% to 0.1%.

Key concepts

Chemical Kinetics

Chemical kinetics is an essential part of chemistry that studies the speed or rate at which chemical reactions occur and the factors that affect these rates. It's a dynamic field that helps us understand how different conditions like temperature, pressure, and concentration influence the reaction time. For JEE aspirants and students in general, grasping the fundamentals of chemical kinetics is crucial as it not only pops up in examinations but also lays the foundation for understanding reaction mechanisms in organic and inorganic chemistry.

When dealing with chemical kinetics, you'll often encounter the term 'reaction order', which signifies the power dependency of the rate on the concentration of reactants. In the case of a first-order reaction, the rate is directly proportional to the concentration of one reactant. This means if you were to double the concentration of the reactant, the reaction rate would also double.

Reaction Rate Constant

Understanding the reaction rate constant, often symbolized by 'k', is crucial when studying chemical kinetics. It is a proportionality factor that connects the rate of a reaction to the concentrations of the reactants raised to a power corresponding to the reaction order. For a first-order reaction, the rate constant links the reaction rate to the concentration of the reactant in a linear fashion.

In mathematical terms, the rate of a first-order reaction can be expressed as 'rate = k[A]', where '[A]' represents the concentration of the reactant. This constant is specific to each reaction and is influenced by external conditions, particularly temperature. As an intrinsic property of the reaction, the rate constant is essential for determining how long it takes a reaction to reach a certain percentage of completion—an invaluable calculation for industries that rely on time-sensitive chemical processes.

Logarithmic Relationships in Chemistry

In first-order chemical reactions, logarithmic relationships play a pivotal role in understanding how concentrations change over time. This is beautifully illustrated in the equation 'k = \( \frac{2.303}{t} \log \frac{[A]_0}{[A]} \)'. The logarithm here provides a way to relate the initial concentration of the reactant '[A]_0' to the concentration remaining '[A]' after time 't'.

This relationship is not linear but exponential, which implies that equal decrements in the logarithm of concentration correspond to equal time intervals. Understanding this concept allows chemists and students alike to predict how long it will take for a reaction to reach a specific percentage completion and is especially helpful in quality control and pharmaceutical applications where exact dosing is of utmost importance.