Problem 218 - A Complete Resource Book in Chemistry for JEE Main

Question

The time for half life period of a certain reaction \(\mathrm{A} \longrightarrow\) products is 1 hour. When the initial concentration of the reactant ' \(A\) ', is \(2.0 \mathrm{~mol} \mathrm{~L}^{-1}\), how much time does it take for its concentration to come from \(0.50\) to \(0.25 \mathrm{~mol} \mathrm{~L}^{-1}\) if it is a zero order reaction? [2010] (a) \(4 \mathrm{~h}\) (b) \(0.5 \mathrm{~h}\) (c) \(0.25 \mathrm{~h}\) (d) \(1 \mathrm{~h}\)

Step-by-Step Solution

Answer

The time taken is 0.25 hours, thus the correct answer is (c) \(0.25 \mathrm{~h}\).

Step 1: Understanding Zero Order Reactions

In a zero-order reaction, the rate of reaction is independent of the concentration of reactants. The rate law is expressed as \( r = k \), where \( k \) is the rate constant.

Step 2: Zero Order Reaction Half-life

For a zero-order reaction, the half-life formula is \( t_{1/2} = \frac{[A]_0}{2k} \). We know \( t_{1/2} = 1 \) hour, and the initial concentration \( [A]_0 = 2.0 \mathrm{\, mol \, L^{-1}} \). We can use this information to find \( k \).

Step 3: Calculate Rate Constant \( k \)

Rearrange the half-life formula to find \( k \):\[ k = \frac{[A]_0}{2t_{1/2}} = \frac{2.0}{2 \times 1} = 1 \mathrm{\, mol \, L^{-1} \, h^{-1}} \].

Step 4: Determine Time for Concentration Change

The formula to find the time \( t \) for a zero order reaction when concentration changes is \( t = \frac{[A]_0 - [A]}{k} \). Here, \( [A]_0 = 0.50 \mathrm{\, mol \, L^{-1}} \) and \( [A] = 0.25 \mathrm{\, mol \, L^{-1}} \).

Step 5: Substitute Values and Solve for \( t \)

Substitute the known values in the formula:\[ t = \frac{0.50 - 0.25}{1} = 0.25 \mathrm{\, h} \].

Step 6: Conclusion

After calculating, the time taken for the concentration to decrease from \(0.50\) to \(0.25 \mathrm{\, mol \, L^{-1}}\) is \(0.25\) hours.

Key Concepts

Half-life

The term "half-life" refers to the time taken for the concentration of a reactant to reduce to half of its initial amount.
It provides valuable insights into the speed at which a reaction proceeds.
For a zero order reaction, the half-life is calculated using the formula \( t_{1/2} = \frac{[A]_0}{2k} \), where:

• \([A]_0\) is the initial concentration of the reactant.
• \(k\) is the rate constant.

In our example, the half-life is specified as 1 hour when the initial concentration \([A]_0\) is 2 mol/L.
This signifies that it will take 1 hour for the concentration to decrease from 2 mol/L to 1 mol/L.
Understanding half-life helps in predicting how long a reaction will take to reach a desired concentration.

Reaction Rate

The reaction rate measures how quickly the reactants are converted into products.
In zero-order reactions, the reaction rate remains constant and does not depend on the concentration of the reactant.
This makes zero-order reactions unique since most reaction rates are influenced by concentration changes.The equation for the rate of a zero-order reaction is expressed as:

• \( r = k \)

Here, \( r \) is the reaction rate, and \( k \) is the rate constant, which determines how fast the reaction proceeds.
In our specific problem, since the rate is independent of the concentration, even as the concentration decreases, the rate stays at 1 mol/L/hr until all reactants are consumed or the reaction is terminated.

Rate Constant

The rate constant \( k \) is a crucial factor in the interpretation of reaction kinetics and represents the speed of the reaction.
It is instrumental in devising the rate law and understanding the dynamics of the reaction process.
For our zero-order reaction, the rate constant was calculated using the formula derived from the half-life:

• \( k = \frac{[A]_0}{2t_{1/2}} \)

By substituting the given values, we found that \( k = 1 \) mol/L/hr.
This value specifically tells us that the reaction reduces by 1 mol of reactant per liter every hour, regardless of the amount of reactant present initially.
Having a clear value of \( k \) allows chemists to predict and analyze the behavior of the reaction over time.

Chemical Kinetics

Chemical kinetics focuses on understanding the speed and mechanisms by which chemical reactions proceed.
It encompasses various factors like temperature, concentration, and the presence of catalysts that influence reaction rates.
In the realm of kinetics, zero-order reactions bring about interesting phenomena because of their unique characteristics:

• The reaction rate is consistent over time.
• The reaction doesn't slow down as the reactants are used up.

This property can simplify the analysis but limits the influence of reactant concentration. Zero-order reactions often occur in situations such as surface reactions where the reactant is saturated on the surface or enzymatic reactions under high substrate concentrations.
Understanding these concepts prepares us to address and predict the behavior of reactions in diverse chemical systems.