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In chemistry, first-order kinetics is a type of reaction where the rate of reaction is directly proportional to the concentration of one reactant. This means that as the concentration of the reactant changes, the rate of reaction changes at the same rate. This is a common type of kinetic reaction and helps to predict how quickly a reaction will proceed.

First-order reactions are often described using the equation:

• \( rac{d[A]}{dt} = -k[A] \)

where \( [A] \) is the concentration of the reactant, \( t \) is time, and \( k \) is the rate constant.

For first-order reactions, a useful rearrangement of this equation is:

• \( ext{ln} rac{[A_0]}{[A]} = kt \)

where \( [A_0] \) is the initial concentration and \( [A] \) is the concentration at time \( t \). This form is particularly handy for calculating how long it takes for a certain amount of a reactant to decompose.

The rate constant, denoted by \( k \), is a crucial component of the first-order kinetics equation. It reflects how fast a reaction proceeds. Specifically, the constant \( k \) is unique to each reaction and is influenced by factors such as temperature and the presence of a catalyst.

In the context of our exercise, the rate constant can be determined using the rearranged equation:

• \( k = rac{ ext{ln} rac{[A_0]}{[A]}}{t} \)

This formula allows students to insert known values—such as the concentrations and the time—to solve for \( k \). The higher the value of \( k \), the faster the reaction occurs.

Understanding \( k \) is essential for predicting reaction rates and is directly used in calculating the reaction's half-life.

Decomposition percentage is a measure of how much of a reactant has reacted compared to its initial amount. In first-order kinetics, this is often expressed as a percentage of the initial concentration remaining after a specific time.

In the given problem, we know that 75% of the compound decomposes in 60 minutes. This tells us that only 25% of the initial compound remains. Understanding decomposition percentage is vital because it enables us to determine the remaining concentration over time and apply it to the first-order kinetics formula.

• A decomposition percentage of 75% means that \( [A] = 0.25[A_0] \)

By calculating the remaining percentage, we assist in determining the rate constant and subsequent calculations necessary for finding properties like the half-life.

The half-life is the time it takes for the concentration of a reactant to decrease to half its initial value. For first-order reactions, the half-life is a constant, irrespective of the initial concentration.

The formula for calculating the half-life \( t_{1/2} \) for a first-order reaction is:

• \( t_{1/2} = rac{0.693}{k} \)

where \( k \) is the rate constant. This specific value of 0.693 comes from the natural logarithm of 2 (i.e., \( ext{ln}(2) \)). This formula is handy because once you calculate the rate constant, determining the half-life becomes straightforward.

Knowing the half-life helps in understanding how quickly a reaction involving first-order kinetics progresses, making this formula crucial for many chemical processes.