Solving First Order Reaction Problems: A Step-by-Step Guide

Understanding and solving problems related to first-order reactions is a crucial skill in chemical kinetics. In this guide, we will walk through the process of determining the rate constant and predicting the time required for 50% completion of a reaction, using a step-by-step approach. This method is not only educational but also highly practical, offering a clear pathway to solving similar problems.

Step-by-Step Solution: First Order Reaction Problem

The problem presented involves a first-order reaction that is 25% complete in 40 minutes. We need to determine how long it will take for the reaction to reach 50% completion. Let's break down the process using the first-order reaction kinetics equation: (k frac{2.303}{t} log left(frac{[A_0]}{[A]}right)).

Step 1: Determine the Rate Constant ((k))

The initial problem states that the reaction is 25% complete in 40 minutes. This means that 75% of the reactant remains. We can thus express the concentrations as follows:

([A_0]) 1 (initial concentration) ([A]) 0.75 (remaining concentration after 25% completion)

Substituting these values into the first-order reaction equation:

[k frac{2.303}{40} log left(frac{1}{0.75}right)]

To solve the logarithm, we have:

[log left(frac{1}{0.75}right) log(1.3333) approx 0.1249]

Substituting back into the equation for (k):

[k frac{2.303}{40} times 0.1249 approx 0.0072 text{ min}^{-1}]

The rate constant (k) is approximately (0.0072 text{ min}^{-1}).

Step 2: Calculate the Time for 50% Completion

For a first-order reaction, we can use the half-life formula to determine the time required for 50% completion:

[t_{1/2} frac{0.693}{k}]

Substituting the value of (k) into the formula:

[t_{1/2} frac{0.693}{0.0072} approx 96.25 text{ minutes}]

Therefore, the time required for the reaction to be 50% complete is approximately 96.25 minutes.

Step 3: Conclusion

In summary, 25% completion in 40 minutes yields a rate constant (k approx 0.0072 text{ min}^{-1}). The time for 50% completion is approximately 96.25 minutes.

Key Concepts and Summary

1. Rate Constant ((k)): The rate constant is calculated using the first-order reaction kinetics equation and the given concentrations and time. In this case, (k approx 0.0072 text{ min}^{-1}).

2. Half-Life ((t_{1/2})): The half-life is the time required for the reaction to reach 50% completion. Using the half-life formula, we found that (t_{1/2} approx 96.25 text{ minutes}).

By following these steps, we can easily solve similar problems involving first-order reactions, making chemical kinetics accessible and understandable.

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