Question

Indicate the order of reaction consistent with each observation. a. The half-life of the reaction gets shorter as the initial concentration is increased. b. A plot of the natural log of the concentration of the reactant versus time yields a straight line. c. The half-life of the reaction gets longer as the initial concentration is increased.

Short answer

The order of reaction for observation a is second order or higher. For observation b, it is first order. For observation c, it is often zero order.

Step-by-step solution

- Understanding reaction order for Observation a

For a reaction where the half-life gets shorter with an increase in initial concentration, this suggests that the reaction rate increases significantly with the concentration. This is characteristic of reactions with an order greater than one, implying a second-order or higher reaction.

- Analyzing plot data for Observation b

When a plot of the natural logarithm (ln) of the concentration (c) of a reactant versus time (t) gives a straight line, this indicates a first-order reaction. The rate law for a first-order reaction is rate = k[c], where k is the rate constant. The equation for this relationship is ln[c] = ln[c0] - kt, where [c0] is the initial concentration.

- Understanding reaction order for Observation c

When the half-life of a reaction increases with an increase in initial concentration, this suggests that the reaction rate diminishes relative to the increase in concentration. This is characteristic of a reaction with an order less than one, most commonly a zero-order reaction, where the rate of reaction is independent of the concentration of the reactants.

Key concepts

Half-Life of Reaction

Understanding the half-life of a reaction is crucial for both chemists and students alike. The half-life, often denoted as t_1/2, is the time required for the concentration of a reactant to decrease to half its initial value. This phenomenon is not only important in chemistry, but also in fields like pharmacology, where it can determine how long a drug remains effective in the body.

For a zero-order reaction, where the rate is independent of the concentration of reactants, the half-life varies with concentration. Specifically, as the concentration increases, the half-life decreases since the rate stays constant. Conversely, for a first-order reaction, the half-life is constant and independent of the initial concentration of reactants. In second-order reactions, half-life increases as the initial concentration increases, which can be associated with reactions where the rate is more dependent on the interactions between molecules.

Rate Law

The rate law is a mathematical expression that describes the relationship between the rate of a chemical reaction and the concentration of its reactants. It is an integral part of kinetics, one of the main branches of physical chemistry. The general form of a rate law is:

\[\text{rate} = k[\text{A}]^{m}[\text{B}]^{n}\]

where \(k\) is the rate constant, \([\text{A}]\) and \([\text{B}]\) are the concentrations of reactants, and \(m\) and \(n\) represent the reaction orders with respect to each reactant. It's these exponents that tell us how the rate of reaction changes as the concentration of reactants change. Determining the rate law of a reaction usually involves a combination of experimental observation and mathematical analysis.

First-Order Reaction

A first-order reaction is one where the rate is proportional to the concentration of a single reactant raised to the first power. In other words, the rate of reaction doubles if the concentration of the reactant doubles. This type of reaction has a constant half-life which means it does not change with differing concentrations of reactant.The rate law for a first-order reaction is written as:\[\text{rate} = k[\text{A}]\]A common example of a first-order reaction is radioactive decay, where the rate at which a substance decays is proportional to the amount of the substance that remains. The linear relationship between the natural logarithm of concentration and time, as seen in a plot, confirms a first-order reaction.

Second-Order Reaction

Moving on to second-order reactions, these involve the sum of the exponents in the rate law equals two, which can arise from a single reactant being squared or from two reactants each to the first power. An increase in concentration has quite a prominent effect on the rate: if one reactant's concentration is doubled, the rate quadruples.The typical rate law for a second-order reaction involving one reactant is:\[\text{rate} = k[\text{A}]^{2}\]The half-life for a second-order reaction is inversely proportional to the initial concentration, meaning that as the initial concentration increases, the half-life decreases, unlike in a first-order reaction where the half-life remains constant regardless of concentration.

Zero-Order Reaction

Zero-order reactions are chemical reactions where the rate is independent of the concentration of the reactants. In these reactions, the rate law has no dependence on the concentration of the reactant(s); instead, the rate equals the rate constant, \(k\), and the concentration terms are raised to the power of zero. The rate law can be expressed as:\[\text{rate} = k[\text{A}]^{0} = k\]This results in a constant rate even as the concentration of the reactant changes. A common occurrence of zero-order reactions is in enzyme-catalyzed reactions at saturation point, where the enzyme is fully occupied and adding more substrate does not increase the rate of reaction. As a characteristic of zero-order kinetics, the half-life increases as the initial concentration increases, which visually can be confirmed through a straight-line plot of concentration versus time.