Question

What are the units of the rate constant for (a) a firstorder reaction, (b) a second-order reaction, and (c) a zero-order reaction?

Short answer

The units of the rate constant k are (a) s^{-1} for a first-order reaction, (b) M^{-1}s^{-1} for a second-order reaction, and (c) M/s for a zero-order reaction.

Step-by-step solution

Understand the Rate Law

The rate of a chemical reaction can be expressed by the rate law, which for a reaction with a single reactant can be written as rate = k[A]^n, where rate is the reaction rate, k is the rate constant, [A] is the concentration of reactant A, and n is the order of the reaction.

Determine the Units for a First-Order Reaction

For a first-order reaction (n=1), the rate law is rate = k[A]. The units are derived from the equation rate (M/s) = k[A] (M), where M is molarity (mol/L) and s is seconds. Therefore, the units for k in a first-order reaction are s^{-1}.

Determine the Units for a Second-Order Reaction

For a second-order reaction (n=2), the rate law is rate = k[A]^2. The units are derived from the equation rate (M/s) = k[A]^2 (M^2), leading to the units for k being M^{-1}s^{-1}.

Determine the Units for a Zero-Order Reaction

For a zero-order reaction (n=0), the rate law simplifies to rate = k. The rate has units of M/s (since it is the change in concentration over time), and since n=0, there are no concentration units to consider, resulting in k having units of M/s.

Key concepts

Understanding Chemical Kinetics

Chemical kinetics is the study of the speed or rate at which chemical reactions occur and the factors that affect this rate. It is a crucial aspect of chemistry because it helps us understand how fast reactions take place, which is essential in various practical applications such as pharmaceuticals, environmental engineering, and the development of new materials.

At the core of chemical kinetics is the rate law, which relates the rate of a reaction to the concentration of the reactants and the rate constant, denoted by 'k'. This constant is a measure of the speed of the reaction and is influenced by temperature, physical state of the reactants, and the presence of catalysts.

The order of the reaction indicates the dependency of the rate on the reactant concentrations. Understanding the units of the rate constant for various reaction orders is essential for analyzing reaction rates quantitatively.

Diving into First-Order Reactions

First-order reactions are characterized by their linear relationship between the rate of reaction and the concentration of one reactant. Mathematically, this is expressed in the rate law as rate = k[A], where 'A' represents the reactant and 'k' is the rate constant.

The units of rate are typically given as molarity per second (M/s). Since the concentration [A] is in molarity (M), the rate constant for a first-order reaction must cancel out the units of concentration, leading to units of s^{-1}. This signifies that the rate of reaction changes by a factor of e for each second that passes, where e is the base of the natural logarithm.

Understanding the units of k for first-order reactions aids in determining reaction half-lives and predicting how long it will take for a reactant's concentration to change over time. This concept is widely applied in processes like radioactive decay and pharmacokinetics.

Exploring Second-Order Reactions

Second-order reactions depend on the concentrations of two reactants or the square of the concentration of one reactant. The rate law for a second-order reaction with one reactant is written as rate = k[A]^2. Here, the concentration [A] is squared in the rate law.

The units for rate remain M/s, but because we square the concentration of A (M^2), the rate constant 'k' must have units that will cancel out M^2. As a result, the units for k in a second-order reaction are M^{-1}s^{-1}.

The significance of these units is that the rate constant value reflects how the rate will change with the inverse of the concentration of A (per molarity) per second. Mastery of these calculations is necessary for chemists when studying reactions such as bimolecular collisions and enzyme kinetics.

Zero-Order Reactions Simplified

Zero-order reactions are unique because their rate is independent of the concentration of the reactants. In other words, the rate of reaction remains constant regardless of the concentration levels. The rate law for a zero-order reaction is rate = k.

The units of rate are M/s, and since there is no concentration term in the rate law, the rate constant 'k' directly has the units of M/s. Zero-order kinetics often occur in situations where a reaction is catalyzed by a substance that is present in a fixed amount and is fully occupied by the substrate, such as in some enzyme-catalyzed reactions.

Recognizing that the units of the rate constant are tied to the order of the reaction not only assists in solving problems focused on chemical kinetics but also in designing controlled experiments and interpreting experimental data in the field of chemistry.