Question

Three different reactions involve a single reactant converting to products. Reaction A has a half-life that is independent of the initial concentration of the reactant, reaction \(\mathrm{B}\) has a half-life that doubles when the initial concentration of the reactant doubles, and reaction \(\mathrm{C}\) has a half-life that doubles when the initial concentration of the reactant is halved. Which state- ment is most consistent with these observations? a. Reaction A is first order; reaction \(\mathrm{B}\) is second order; and reaction C is zero order. b. Reaction A is first order; reaction \(\mathrm{B}\) is zero order; and reaction C is zero order. c. Reaction A is zero order; reaction B is first order; and reaction C is second order. d. Reaction \(\mathrm{A}\) is second order; reaction \(\mathrm{B}\) is first order; and reaction C is zero order.

Short answer

The correct answer is a. Reaction A is first order; reaction B is second order; and reaction C is zero order.

Step-by-step solution

Analyze Reaction A

Identify the kinetic order of Reaction A. A half-life that is independent of the initial concentration indicates a first-order reaction. In first-order kinetics, the rate is directly proportional to the concentration of the reactant, and the half-life is constant regardless of the initial concentration.

Analyze Reaction B

Determine the kinetic order of Reaction B. A half-life that doubles when the initial concentration of the reactant doubles is characteristic of a second-order reaction. In second-order kinetics, the rate is proportional to the square of the concentration of the reactant, and thus the half-life is inversely proportional to the initial concentration.

Analyze Reaction C

Examine the kinetic order of Reaction C. A half-life that doubles when the initial concentration of the reactant is halved suggests a zero-order reaction. In zero-order kinetics, the rate is independent of the concentration of the reactant, and as the concentration decreases, the half-life increases.

Match Observations to Options

Compare the kinetic orders derived from observations with the options provided. Reaction A exhibits first-order kinetics, Reaction B exhibits second-order kinetics, and Reaction C exhibits zero-order kinetics.

Key concepts

First-Order Reaction

In understanding chemical kinetics, a first-order reaction is fundamental to grasp. It's one where the rate at which the reaction occurs is directly proportional to the concentration of a single reactant. This means that as you double the amount of reactant, you double the reaction rate. A characteristic feature of a first-order reaction is its constant half-life, which is the time taken for half of the reactant to be converted into product.

A practical example is the decay of radioactive substances, where the half-life remains unchanged regardless of how much substance you start with. If given that the half-life of a reaction does not change with the initial concentration of reactant, we can infer that such a reaction is first-order, just as in Reaction A from the problem statement.

Second-Order Reaction

A second-order reaction is characterized by a reaction rate that is proportional to the square of the concentration of one reactant, or to the product of the concentrations of two reactants.

One of the interesting aspects of second-order reactions is that their half-life values are directly related to the initial concentration of the reactant: as the concentration increases, the half-life decreases. This inverse relationship is helpful for identifying second-order reactions in practice. If you observe that the half-life of a reaction doubles when the initial concentration doubles, as happens with Reaction B, you're likely dealing with a second-order reaction.

Second-order reactions are common in processes where two reactant molecules collide and react, such as the dimerization of alkenes under specific conditions.

Zero-Order Reaction

Moving on to the specifics of a zero-order reaction, this is a reaction where the rate is constant and unaffected by the concentration of the reactant. That's right; in zero-order kinetics, the reaction cruises at its own pace regardless of how much reactant is available.

Another peculiarity of zero-order reactions is their changing half-life: as the reactant concentration reduces, the half-life increases. This is because the reaction rates are no longer “throttled” by the reactant's concentration—it's as if the reaction is saying, 'I'll take my time, thank you very much.' Reaction C is a perfect example of zero-order kinetics, showcasing the increasing half-life with decreasing concentration.

Reaction Rate

The reaction rate is the speed at which a chemical reaction proceeds. It's the cornerstone of chemical kinetics, the field that unpacks the complexities of how reactions happen. The rate can be influenced by several factors, including reactant concentrations, temperature, and the presence of catalysts.

To put it in everyday terms, think of reaction rate like the speed of your car. Just as you can accelerate by pressing down on the gas pedal, you can increase the reaction rate by increasing the concentration of reactants or raising the temperature. The understanding of reaction rates is crucial, as it helps chemists control and optimize reactions for industrial processes, environmental modeling, and even pharmaceutical drug development.

Half-life of Reaction

Lastly, the half-life of a reaction is an informative concept that provides insights into the duration and progress of a chemical reaction. It is the period required for the concentration of a reactant to decrease by half its initial amount. In the context of different order reactions, half-life behaves differently.

For a first-order reaction, the half-life is a constant value, independent of the initial concentration. In a second-order reaction, the half-life is inversely proportional to the initial concentration, making it a variable that decreases with increasing concentration. In zero-order reactions, the half-life increases as the initial concentration decreases. Recognizing these patterns in half-life behavior is vital for chemists to predict how long a reactant will last under different conditions and to design reactions for desired outcomes.