Question

In an experiment, the concentration of a reactant was tripled. The rate increased by a factor of \(27 .\) What is the order of the reaction with respect to that reactant?

Short answer

The order of the reaction with respect to the reactant is 3.

Step-by-step solution

Understanding the Relationship Between Concentration and Rate

The rate of a chemical reaction can be expressed as a function of the concentration of the reactants. This relationship is generally described by the rate law, which takes the form: Rate = k[A]^(n), where k is the rate constant, [A] is the concentration of reactant A, and n is the order of reaction with respect to A.

Setting Up the Rate Equation for Tripling the Concentration

If the initial concentration is [A] and the initial rate is Rate1, tripling the concentration to 3[A] will produce a new rate, Rate2. According to the rate law, we have Rate1 = k[A]^(n) and Rate2 = k(3[A])^(n).

Comparing the Two Rates

Given that Rate2 is 27 times Rate1, we can write the equation 27(Rate1) = k(3[A])^(n). Substituting Rate1 = k[A]^(n) into this equation, we get 27(k[A]^(n)) = k(3[A])^(n).

Solving for the Reaction Order

Dividing both sides by k[A]^(n), we obtain 27 = (3^n). Taking the logarithm of both sides, we get log(27) = n log(3). Since log(27) is 3 and log(3) is 1, dividing both sides by log(3) gives n = 3.

Key concepts

Rate Law

Rate law is a mathematical expression that describes the relationship between the concentration of reactants in a chemical reaction and the rate at which the reaction occurs. It is a fundamental component of chemical kinetics, the branch of physical chemistry that studies how and at what rate chemical reactions proceed.

The basic form of a rate law is given by the equation: \( Rate = k[A]^n \), where:

• \( k \) is the rate constant, a value that depends on the specific reaction and the temperature.
• \( [A] \) represents the concentration of reactant A.
• \( n \) is the reaction order with respect to reactant A, which indicates how the rate is affected by changes in concentration.

The reaction order can be an integer (0, 1, 2,...) or a fraction and must be determined experimentally. It is a critical piece of information since it tells us how a change in concentration will impact the rate of the reaction.

Chemical Kinetics

Chemical kinetics provides insights into the speed of chemical reactions and the steps that take place during these reactions. It encompasses the study of reaction rates, the rate laws that describe those rates, and the mechanisms by which reactions proceed.

Within this field, scientists investigate factors affecting reaction rates such as:

• Concentration of reactants,
• Temperature,
• Presence of a catalyst,
• Surface area of solid reactants or catalysts,
• Light,

Each of these factors plays a critical role in how quickly a reaction can occur. A higher concentration of reactants typically increases the rate of reaction, as does higher temperature and the presence of a catalyst. Kinetics is important for understanding processes in fields from industrial chemistry to biochemistry, where controlling the speed of reactions is often crucial.

Concentration and Reaction Rate

The concentration of reactants is a central factor in determining the rate of a chemical reaction. According to the rate law, as the concentration increases, the reaction rate also increases, but the degree of this increase depends on the reaction order with respect to each reactant.

In the given exercise, the relation between the concentration of a reactant and the reaction rate was explored. When the concentration was tripled, the rate increased by a factor of 27, indicating a third-order reaction with respect to that reactant, as confirmed by solving the rate equation. This third-order relationship implies that the rate of reaction is proportional to the cube of the concentration of the reactant. It is important to note that not all reactions will show such a straightforward relationship, as the order of each reactant in the rate law must be determined empirically for each reaction.