Question

A reaction in which \(A, B,\) and \(C\) react to form products is zero order in A, one-half order in B, and second order in C. a. Write a rate law for the reaction. b. What is the overall order of the reaction? c. By what factor does the reaction rate change if [A] is doubled (and the other reactant concentrations are held constant)? d. By what factor does the reaction rate change if [B] is doubled (and the other reactant concentrations are held constant)? e. By what factor does the reaction rate change if [C] is doubled (and the other reactant concentrations are held constant)? f. By what factor does the reaction rate change if the concentrations of all three reactants are doubled?

Short answer

a) Rate = k[A]^0[B]^{1/2}[C]^2. b) The overall order is 2.5. c) The rate does not change when [A] is doubled. d) The rate increases by a factor of \(\sqrt{2}\) when [B] is doubled. e) The rate quadruples when [C] is doubled. f) The rate increases by a factor of \(4\sqrt{2}\) when all reactant concentrations are doubled.

Step-by-step solution

Writing the Rate Law

Since the reaction is zero order in A, one-half order in B, and second order in C, we combine the orders with respect to each reactant. The rate law can be written as \( Rate = k[A]^0[B]^{1/2}[C]^2 \).Here, \(k\) is the rate constant, \([A]\) is the concentration of A, \([B]\) is the concentration of B, and \([C]\) is the concentration of C.

Determining the Overall Order

The overall order of the reaction is the sum of the orders of each reactant. Add the exponents of the concentrations in the rate law: \(0 + \frac{1}{2} + 2 = 2.5\). The overall order of the reaction is 2.5.

Reaction Rate Change with Doubling [A]

Since the reaction is zero order with respect to A, changing [A] will have no effect on the rate. Doubling [A] means multiplying by 2, so the rate change factor due to [A] is \(2^0 = 1\).

Reaction Rate Change with Doubling [B]

The reaction is one-half order with respect to B, so the rate of the reaction is proportional to \([B]^{1/2}\). If [B] is doubled, the rate change factor is \(2^{1/2} = \sqrt{2} \approx 1.41\).

Reaction Rate Change with Doubling [C]

Since the reaction is second order with respect to C, the rate is proportional to \([C]^2\). Doubling [C] will lead to a rate change factor of \(2^2 = 4\).

Reaction Rate Change with Doubling All Concentrations

When doubling all reactant concentrations, we need to multiply the change factors for each reactant: \(2^0\) for [A], \(2^{1/2}\) for [B], and \(2^2\) for [C]. The total rate change factor is \(2^0 \cdot 2^{1/2} \cdot 2^2 = 1 \cdot \sqrt{2} \cdot 4 = 4\sqrt{2}\).

Key concepts

Chemical Kinetics

Chemical kinetics is the branch of chemistry that studies the speed or rate at which chemical reactions occur and the factors affecting these rates. It is essential for understanding reaction mechanisms and the behavior of reactions over time. The rate of a chemical reaction is the change in concentration of a reactant or product per unit time. Factors that affect reaction rates include concentration of reactants, temperature, surface area, catalysts, and the presence of light.

For example, in an exercise where the rate law for a reaction involving reactants A, B, and C is determined, chemical kinetics provides the framework for analyzing how the concentrations of these reactants affect the reaction rate. It's important to remember that while some reactants may influence the rate significantly, others may not affect it at all. Understanding the principles of kinetics can aid in predicting and controlling reaction behavior, which is crucial both in scientific research and industrial processes.

Reaction Order

The reaction order with respect to a particular reactant is a numerical value that represents how the rate of a reaction is affected by the concentration of that reactant. It can be zero, a whole number, or a fraction. The individual orders for each reactant are determined experimentally and can be summed to find the overall order of the reaction.

In the context of our exercise, reactant A has a reaction order of zero, meaning that changes in its concentration do not affect the reaction rate. Reactant B has a one-half order, signifying that the rate is affected by the square root of its concentration. Reactant C has a second-order reaction, indicating the rate is proportional to the square of its concentration. Establishing the reaction order is not only crucial for the rate law expression but also helps in predicting the effect of concentration changes on the reaction rate, as shown in the step-by-step solutions.

Rate Constant

The rate constant, symbolized by the letter 'k', is a proportionality factor in the rate law equation of a chemical reaction. It is a measure of the reaction speed that is unique for every reaction at a given temperature. The rate constant connects the reaction rate to the concentrations of reactants raised to their respective orders in the rate law.

In the solution of our exercise, the rate law is given as \( Rate = k[A]^0[B]^{1/2}[C]^2 \). Here, the rate constant 'k' would remain constant as long as the temperature and other environmental conditions do not change. It's crucial to understand that the rate constant is not just a number—it embodies the intrinsic rate of the reaction and encapsulates factors such as the frequency and the energetic success of molecular collisions. Knowing the value of the rate constant allows us to calculate the reaction rates for any combination of reactant concentrations and is particularly helpful in chemical engineering and reaction design.