Understanding First-Order Reaction Kinetics and Half-Life

Chemical kinetics is the study of reaction rates. For a chemical reaction, the rate constant (\(k\)) is a proportionality constant that relates the reaction rate to the concentrations of reactants. The order of a reaction describes how the rate depends on the concentration of reactants. A first-order reaction is one where the reaction rate is directly proportional to the concentration of one reactant.

Calculating Rate Constant from Half-Life for a First-Order Reaction

The half-life (\(t_{1/2}\)) of a reaction is the time required for the concentration of a reactant to decrease to half of its initial value. For a first-order reaction, the half-life is constant and independent of the initial concentration of the reactant. There is a specific relationship between the half-life (\(t_{1/2}\)) and the rate constant (\(k\)) for a first-order reaction. This relationship is given by the formula:

\(t_{1/2} = \frac{0.693}{k}\)

We are given that the first-order reaction has a half-life of 693 seconds. We need to find its rate constant (\(k\)). We can rearrange the formula to solve for \(k\):

\(k = \frac{0.693}{t_{1/2}}\)

Step-by-Step Calculation

Given:

• Half-life, \(t_{1/2} = 693\) seconds

We need to find the rate constant, \(k\).

Using the formula for a first-order reaction:

\(k = \frac{0.693}{t_{1/2}}\)

Substitute the given value of \(t_{1/2}\):

\(k = \frac{0.693}{693 \text{ seconds}}\)

Now, perform the calculation:

\(k = 0.001 \text{ sec}^{-1}\)

The unit for the rate constant of a first-order reaction is typically time\(\text{⁻¹}\) (e.g., s\(\text{⁻¹}\), min\(\text{⁻¹}\), hr\(\text{⁻¹}\)). In this case, since the half-life is in seconds, the rate constant will have units of seconds\(\text{⁻¹}\).

Comparing with Options

The calculated rate constant is 0.001 sec\(\text{⁻¹}\). Let's look at the options provided:

1. 0.001 sec\(\text{⁻¹}\)
2. 1 sec\(\text{⁻¹}\)
3. 0.001 sec\(\text{⁻¹}\)
4. 0.1 sec\(\text{⁻¹}\)

Our calculated value matches option 1 and option 3.

Revision Table: First-Order Reaction Key Formulas

Additional Information on Reaction Order and Half-Life

The relationship between half-life and rate constant depends on the order of the reaction. While the half-life of a first-order reaction is constant, this is not true for other reaction orders.

• Zero-Order Reaction: The half-life of a zero-order reaction is dependent on the initial concentration. The formula is \(t_{1/2} = \frac{[\text{A}]_0}{2k}\).
• Second-Order Reaction: The half-life of a second-order reaction is also dependent on the initial concentration. The formula is \(t_{1/2} = \frac{1}{k[\text{A}]_0}\).

This makes the first-order reaction's constant half-life a unique and important characteristic.