Question

For a first order reaction, the half life period is independent of a. Intial concentration b. cube root of initial concentration c. first power of final concentration d. square root of final concentration

Short answer

The half-life period is independent of all options: a, b, c, and d.

Step-by-step solution

Understanding Half-Life for First Order Reactions

In first-order reactions, the half-life is defined as the time required for half of the reactant to be consumed. The formula for half-life \( t_{1/2} \) for a first-order reaction is \( t_{1/2} = \frac{0.693}{k} \), where \( k \) is the rate constant. Importantly, this formula indicates that the half-life of a first-order reaction is constant, depending only on \( k \) and not on the concentrations of reactants or products.

Analyzing Independence from Concentrations

Given that the half-life \( t_{1/2} \) for a first-order reaction depends only on the rate constant \( k \), it implies that the half-life is independent of any form of reactant concentrations - whether initial or final. Thus, this includes independence from initial concentration, the cube root of initial concentration, and the powers of final concentration.

Evaluating the Options

The question asks for independence of the half-life period for a first-order reaction. We must identify which among the given options does not influence \( t_{1/2} \): (a) Initial concentration (b) Cube root of initial concentration (c) First power of final concentration (d) Square root of final concentration. Based on our understanding in the previous steps, for first-order reactions, the half-life \( t_{1/2} \) is independent of all provided options since they all involve reactant concentrations.

Key concepts

Reaction Kinetics

Reaction kinetics is the study of the rates at which chemical reactions occur. It helps us understand how different conditions affect the speed of reactions. When it comes to first-order reactions, these are uniquely characterized by the fact that the reaction rate is directly proportional to the concentration of one reactant. This means that if the concentration of the reactant doubles, the reaction rate doubles too.

In practical terms, first-order reactions can be expressed mathematically by the equation:

• \[ ext{Rate} = k[A] \]

where \( k \) is the rate constant and \( [A] \) is the concentration of the reactant. These types of reactions are quite common and include processes like radioactive decay and certain enzyme-catalyzed reactions.

• Understanding reaction kinetics is crucial for controlling how fast reactions proceed.
• It allows us to optimize conditions in industrial processes and laboratory experiments.

Half-Life Period

When discussing first-order reactions, the half-life period is a key concept. The half-life, denoted as \( t_{1/2} \), is the time taken for half of the reactant to be consumed in the reaction. For first-order reactions, the half-life is particularly noteworthy because it is constant and does not depend on how much of the reactant is present.

According to the formula:

• \[ t_{1/2} = \frac{0.693}{k} \]

this constancy arises because the only variable that affects the half-life is the rate constant \( k \). This is different from reactions of other orders where the half-life could vary with concentration changes. The constancy of the half-life in first-order reactions makes them predictable and easier to analyze.

• Half-life helps in determining how long a reaction will take to reach completion.
• It is especially important in areas such as pharmacology and environmental science.

Rate Constant

The rate constant, symbolized by \( k \), is a crucial part of understanding first-order reactions. It is a measure of how quickly a reaction proceeds and is determined experimentally. The larger the value of the rate constant, the faster the reaction.

In the context of first-order reactions, \( k \) has a major influence on the half-life, as shown in the formula:

• \[ t_{1/2} = \frac{0.693}{k} \]

Here, you can see that the rate constant is inversely proportional to the half-life. This means that with a higher \( k \), the time taken for half the reactant to disappear is shorter.
Understanding \( k \) is essential because:

• It helps predict how rapidly a reaction will proceed under set conditions.
• It is used to calculate other kinetics-related parameters, giving a clearer picture of the reaction dynamics.

Concentration Independence

In first-order reactions, an important principle is that the rate of reaction and its half-life are independent of the concentration of the reactants. This concept is somewhat counter-intuitive because, in many other reaction types, changes in concentration directly influence these parameters.

This independence is a result of the logarithmic relationship between concentration and time in first-order kinetics. Simply put, even as the concentration of the reactant decreases, the rate of reaction remains proportionally consistent, leading to a steady half-life.
Key aspects of concentration independence include:

• It simplifies the mathematical analysis and prediction of reaction progress.
• It allows for easy calculations over extended times and varying conditions in fields like pharmacokinetics and radiochemistry.

Understanding this property helps in accurately modeling reactions that follow first-order kinetics, regardless of changes in concentration over time.