91Ó°ÊÓ

In first-order kinetics, the reaction rate constant (k) is a crucial parameter. It helps predict how fast a chemical reaction will proceed. To determine this constant, we use the relationship of concentration and time. For a first-order reaction like \(\mathrm{A} \rightarrow \mathrm{B}\), the formula is:

• \[ k = \frac{1}{t} \ln \left(\frac{[A]_0}{[A]}\right) \]

Here, \(k\) is the rate constant, \(t\) is the time, \([A]_0\) is the initial concentration of reactant A, and \([A]\) is the concentration of A after time \(t\).
For our exercise, we calculated \(k\) using known values: \(t = 1\) hour, \([A]_0 = 0.8\) moles, and \([A] = 0.2\) moles, which gave us \(k = \ln(4)\). This value allows us to predict how long the second reaction should take.

Mole conversion is a fundamental concept in chemical reactions. It involves determining how many moles of one substance react with or produce another substance. In our reaction \( \mathrm{A} \to \mathrm{B}\), we perform a mole conversion to find out how many moles of A convert to B.

• Initially, 0.8 moles of \(A\) are present. 0.6 moles convert to \(B\), leaving 0.2 moles of \(A\).
• For the second scenario, 0.9 moles of \(A\) are used. 0.675 moles are converted to \(B\), leaving 0.225 moles of \(A\) remaining.

By understanding the change in moles, we can calculate remaining reactants or predict product formation in a reaction. It's a crucial step before applying the rate constant formula.

The calculation of time required for a specific amount of reactant to convert into product is central to kinetics. In first-order reactions, once the reaction rate constant \(k\) is known, we can calculate the time for any conversion using:

• \[ t = \frac{1}{k} \ln \left(\frac{[A]_0}{[A]}\right) \]

This formula rearranges the rate law to solve for \(t\). By substituting the initial and remaining concentrations of A for the second scenario, we predict the time specific to that condition.
The constant \(k\) calculated from initial reaction conditions (\[k = \ln(4)\]) is used again to confirm that the time for 0.9 moles to convert 0.675 moles of B is indeed 1 hour. This demonstrates the consistency and predictive power of kinetic equations.

When tackling IIT JEE physical chemistry problems, having a strong grasp of kinetics is vital. These exams often test not only formulas but also understanding.
In this exercise, we used several significant concepts:

• Understanding rates of reaction through reaction constants.
• Converting moles to relate to reactants and products.
• Calculating the time from rate constants and mole conversions.

It's crucial to remember these steps and concepts:

• Grasp initial information and conditions clearly.
• Utilize formulas appropriately.
• Apply logical problem-solving skills to interpret the results.

Competence in these areas will aid in tackling diverse problems under exam conditions effectively, especially for the IIT JEE, which emphasizes precision and comprehension in physical chemistry.