Question

For each of the questions, four choices have been provided. Select the correct alternative. For a reaction \(\mathrm{A}+\mathrm{B} \rightarrow \mathrm{C}\), the rate law is written as \(\mathrm{r}=\mathrm{k}[\mathrm{A}]^{2}[\mathrm{~B}] .\) Doubling the concentrations of both of "A" and "B" increases the rate of reaction by (a) 2 times (b) 4 times (c) 8 times (d) 16 times

Short answer

Question: When the concentrations of both reactants A and B are doubled in the chemical reaction A + B → C with the rate law r=k[A]^2[B], the rate of the reaction is ______ times the original rate. Answer: 8 times

Step-by-step solution

Write down the knowns and what we want to find

We have the rate law: r = k[A]^2[B]. We want to find how the rate (r) changes when we double the concentrations of both A and B.

Write down the initial and new concentrations

Let the initial concentrations be: [A] and [B] Let the new concentrations be: 2[A] and 2[B] (since we are doubling them)

Calculate the initial rate

The initial rate (r_initial) can be calculated using the given rate law: r_initial = k[A]^2[B]

Calculate the new rate

To find the new rate (r_new), we need to plug the new concentrations into the rate law: r_new = k(2[A])^2(2[B])

Simplify the expression for the new rate

By working out the powers and multiplications, we get: r_new = k(4[A]^2)(2[B]) r_new = 8k[A]^2[B]

Compare the new rate to the initial rate

We can find the relation between the new rate and the initial rate by dividing r_new by r_initial: rate_change = r_new / r_initial = (8k[A]^2[B]) / (k[A]^2[B])

Simplify the rate change equation

By canceling out the common factors, we have: rate_change = 8

Choose the correct alternative

As the rate change is 8, the correct choice is: (c) 8 times

Key concepts

Rate of Reaction

Understanding the rate of a chemical reaction is crucial in the field of chemistry. Simply put, the rate of reaction refers to how fast or slow reactants are transformed into products over time. The speed at which this occurs can have significant implications in industries such as pharmaceuticals, where reaction rates can impact the quality and effectiveness of a medication.

Mathematically, the rate of reaction is often expressed as the change in concentration of a reactant or product per unit time. For a generic reaction where \(A \rightarrow B\), the rate can be written as \(\text{rate} = -\frac{d[A]}{dt}\) which indicates the rate of decrease in concentration of \(A\) over time. It's important not to confuse this with the overall rate of reaction, which takes into account the stoichiometry of the reactants and products—a concept we'll delve into in chemical kinetics.

In the context of the exercise, when concentration changes are involved, the rate law equation becomes an indispensable tool to predict how fast the reaction will proceed under new conditions. By understanding how to manipulate and interpret this rate law, we can quantify the precise change in reaction rate resulting from alterations in reactant concentrations.

Concentration Effect on Reaction Rate

Concentration plays a pivotal role in determining the rate of chemical reactions. According to the collision theory, chemical reactions occur when reactant molecules collide with sufficient energy and proper orientation. The more molecules present, the more collisions can happen, which typically increases the reaction rate.

To illustrate this, consider the given rate law \(\text{r} = \text{k}[A]^2[B]\). It shows a direct relationship between the concentrations of reactants \(A\) and \(B\), and the rate \(\text{r}\). When concentrations are increased, as we've seen in the exercise, the rate of reaction is increased as well, but not in a 1:1 ratio. Instead, the rate is subject to how each reactant concentration is raised to a power, known as the reaction order.

In the exercise, by doubling both \(A\) and \(B\), we see an eightfold increase in the rate of reaction. This is because the rate of reaction is quadrupled by squaring the doubled concentration of \(A\), and then doubled again by the doubled concentration of \(B\). It’s a multiplicative effect, not additive—doubling concentrations doesn't simply double the rate of reaction unless both reactants are first-order.

Chemical Kinetics

Chemical kinetics delves into understanding the speed of reaction and the pathways by which they occur. It's a branch of physical chemistry concerned with the study of rates and mechanisms of chemical processes. The ultimate goal is to develop a detailed picture of how a reaction moves from reactants to products, which has immense practical applications ranging from designing chemical reactors to preserving food.

Key to kinetics is the concept of the reaction rate law, like the one provided in the exercise. A rate law allows chemists to determine the reaction rate from the concentrations of the reactants and the rate constant \(\text{k}\). The rate constant is a proportionality factor that is specific to each reaction and is influenced by factors such as temperature and catalysts.

The orders of reaction, integral to the rate law, provide insight into the reaction mechanism. They are typically derived from experimental data and can indicate which reactants are involved in the rate-determining step. The overall reaction rate is determined by the slowest step, known as the rate-determining step, making kinetics an essential tool for understanding complex chemical reactions. Through kinetics, we can control and optimize reactions in various fields, ensuring that we're able to efficiently produce the products we rely on.