Question

Define half-life. Write the equation relating the half-life of a first-order reaction to the rate constant.

Short answer

Half-life is the time for a reactant to reduce to half, given by \(\text{t}_{1/2} = \frac{0.693}{k}\) for first-order reactions.

Step-by-step solution

Define Half-life

The half-life of a reaction is the time required for the concentration of a reactant to decrease to half of its initial concentration. It is a measure of how quickly a reactant is consumed in a reaction.

Identify Reaction Order

For this exercise, we need to focus on a reaction that is first-order. In first-order reactions, the rate of reaction is directly proportional to the concentration of one reactant.

First-Order Half-Life Equation

The equation relating the half-life ( ext{t}_{1/2}) of a first-order reaction to the rate constant (k) is given by:\[\text{t}_{1/2} = \frac{0.693}{k}\]This means that the half-life is inversely proportional to the rate constant.

Key concepts

First-Order Reaction

A first-order reaction is a type of chemical reaction where the reaction rate depends linearly on the concentration of a single reactant. Imagine you're observing a substance breaking down over time. In a first-order reaction, if you increase the concentration of the reactant, the rate at which the reaction occurs will also increase proportionally. This is because the reaction rate formula for a first-order reaction is:
\[ ext{Rate} = k[A] \\]where \(k\) is the rate constant and \( [A] \) is the concentration of the reactant.
To remember this easily, think about how baking a cake might work. If you have more cake mix, the batter forms faster assuming your mixing efforts are consistent. Similarly, in first-order reactions, more reactant leads to a faster reaction, up to a linear point. Other reactions might depend on multiple reactants, but first-order reactions keep it simple with just one variety.

• Directly proportional to reactant concentration
• Simple relationship between rate and concentration

Rate Constant

The concept of a rate constant, denoted by \( k \), is fundamental in understanding the speed of a reaction. In reaction kinetics, the rate constant is what determines how quickly a reaction moves to completion once it has started.
For first-order reactions, \( k \) has a specific significance. The equation relating the half-life of a first-order reaction to its rate constant is given by:
\[\text{t}_{1/2} = \frac{0.693}{k}\]This formula tells us that the half-life and the rate constant are inversely related.
This means a larger \( k \) results in a shorter half-life, implying the reactant concentration decreases faster. Think of \( k \) as a "rate dial"; a higher setting speeds up the reaction.

• Central to calculating reaction speed
• Inversely related to half-life in first-order reactions

Reaction Kinetics

Reaction kinetics is the study of reaction rates and the steps that lead to the final products. It focuses not just on how quickly reactions occur, but also on the pathway and transformations reactants undergo on their way to becoming products.
In the case of first-order reactions, kinetics simplifies to examining the rate dependent on a single reactant. For such reactions, the study is straightforward, and the rate equation can help predict how concentrations will change over time. Kinetics allows chemists to:

• Determine the reaction order
• Predict the time needed for a reaction to reach a certain extent
• Understand how different conditions affect reaction rates

By applying kinetics principles, one can visualize the pace at which reactions happen, similar to estimating travel time based on speed and distance in everyday life.