Question

You perform a series of experiments for the reaction \(\mathrm{A} \longrightarrow \mathrm{B}+\mathrm{C}\) and find that the rate law has the form rate \(=k[\mathrm{~A}]^{x}\). Determine the value of \(x\) in each of the following cases: (a) There is no rate change when [A] is tripled. (b) The rate increases by a factor of 9 when [A] is tripled. (c) When [A] is doubled, the rate increases by a factor of 8 . [Section 14.3]

Short answer

For each case: (a) Since the rate does not change when [A] is tripled, x = 0. (b) The rate increases by a factor of 9 when [A] is tripled, x = 2. (c) When [A] is doubled, the rate increases by a factor of 8, x = 3.

Step-by-step solution

Write down the initial rate law

We are given the rate law: rate = k[A]^x

Triple the concentration of A

Let's triple the concentration of A and denote the new concentration as 3[A]: new_rate = k[(3[A])]^x

Compare the new rate to the initial rate

According to the problem statement, the rate doesn't change when [A] is tripled. So, new_rate = initial rate: k[(3[A])]^x = k[A]^x

Solve for x

Divide both sides by \(k[A]^x\), then \(3^x = 1\). The only value for x that satisfies this equation is \(x=0\). #Case B: The rate increases by a factor of 9 when [A] is tripled.#

Write down the initial rate law

We already have the rate law: rate = k[A]^x

Triple the concentration of A

As in Case A, let's triple the concentration of A: new_rate = k[(3[A])]^x

Compare the new rate to the initial rate

According to the problem statement, the rate increases by a factor of 9 when [A] is tripled: new_rate = 9 * k[A]^x

Solve for x

Setting the equations equal: k[(3[A])]^x = 9 * k[A]^x Divide both sides by \(k[A]^x\), then \(3^x = 9\). The value for x that satisfies this equation is \(x=2\). #Case C: When [A] is doubled, the rate increases by a factor of 8 .#

Write down the initial rate law

We again start with the rate law: rate = k[A]^x

Double the concentration of A

Let's now double the concentration of A: new_rate = k[(2[A])]^x

Compare the new rate to the initial rate

According to the problem statement, the rate increases by a factor of 8 when [A] is doubled: new_rate = 8 * k[A]^x

Solve for x

Setting the equations equal: k[(2[A])]^x = 8 * k[A]^x Divide both sides by \(k[A]^x\), then \(2^x = 8\). The value for x that satisfies this equation is \(x=3\). So, the values of x for each case are: (a) x=0, (b) x=2, and (c) x=3.

Key concepts

Chemical Kinetics

Chemical kinetics is the study of the rates of chemical processes and the factors affecting them. It aims to understand the speeds at which reactions occur and how various conditions, such as temperature, pressure, and concentration of reactants, influence these rates. This field is crucial as it helps chemists to optimize conditions for desired reactions, to understand reaction mechanisms, and to design chemical processes efficiently.

In the context of the exercise provided, when the concentration of reactant A changes, chemical kinetics helps us predict how the rate of the reaction will be affected. For instance, understanding that tripling the concentration of A leads to no rate change in case (a) implies that the concentration of A does not influence the reaction speed—an insight that is valuable in forming reaction mechanisms and theories.

Reaction Rates

Reaction rates refer to the speed at which reactants are converted into products in a chemical reaction. It is generally expressed as the concentration of a reactant consumed or the concentration of a product formed per unit time. Factors like reactant concentrations, temperature, and the presence of a catalyst can greatly affect the rate of a reaction.

In the exercise, by manipulating the concentration of reactant A and observing the change in reaction rate, we obtain empirical data that quantifies how the reaction rate responds to changes in concentration. Noticing that the rate does not vary with the concentration in (a), increases nine-fold in (b), or eight-fold in (c) when the concentration of A is altered, gives us critical information about how sensitive the reaction is to changes in concentration of A.

Rate Law Expression

The rate law expression is a mathematical equation that describes the relationship between the rate of a chemical reaction and the concentrations of its reactants. It takes the form: rate = k[A]^x, where 'rate' is the rate of the reaction, 'k' is the rate constant, '[A]' signifies the concentration of reactant A, and 'x' is the order of the reaction with respect to A, a decisive factor that must be determined experimentally.

In each part of the exercise, experimental data is used to determine the order 'x'. For example, if the rate is unaffected by a change in [A], then the reaction is zero-order with respect to A, which gives us (a) x=0. If the rate increases by a factor equal to the change in concentration raised to a power, this power is the reaction order 'x.' Thus, in (b) when the rate increases by a factor of 9 (which is 3^2) when [A] is tripled, 'x' is determined to be 2. Similarly, a rate increase by a factor of 8 (which is 2^3) when [A] is doubled, yields (c) x=3, indicating a third-order reaction with regard to A.