Question

For reaction: \(4 \mathrm{~A}+\mathrm{B} \rightarrow 2 \mathrm{C}+2 \mathrm{D}\), the only incorrect statement is (a) The rate of disappearance of \(\mathrm{B}\) is one-fourth the rate of disappearance of \(\mathrm{A}\) (b) The rate of appearance of \(\mathrm{C}\) is half the rate of disappearance of \(\mathrm{B}\) (c) The rate of formation of \(\mathrm{D}\) is half the rate of consumption of \(\mathrm{A}\) (d) The rates of formation of \(\mathrm{C}\) and \(\mathrm{D}\) are equal

Short answer

The incorrect statement is (b), because the rate of appearance of C is equal to the rate of disappearance of B, not half.

Step-by-step solution

- Write the Rate Expression for each Substance

Based on the stoichiometry of the balanced equation, the rate of consumption or formation of each substance can be related to one another. This relationship is determined by their coefficients. For substance A with coefficient 4, B with coefficient 1, and C and D with coefficient 2 each.

- Analyze Each Given Statement

(a) Since B has a coefficient of 1 and A has a coefficient of 4, the rate of disappearance of B should be one-fourth the rate of disappearance of A if their rates are based on coefficients. (b) The appearance rate of C should be the same as the disappearance rate of B, because the stoichiometric coefficients of C and B are the same (both have coefficients of 2 and 1 respectively). (c) The formation rate of D should be the same as the rate of consumption of A, by their stoichiometric coefficients (both have coefficients of 2 and 4 respectively). (d) The formation rates of C and D are indeed equal because both have the same coefficient of 2 in the balanced equation.

- Identify the Incorrect Statement

After comparing the rates described in the given statements with the stoichiometric coefficients: (a) is correct, (b) is incorrect because the rate of appearance of C is equal to, not half of, the rate of disappearance of B, (c) is correct, (d) is correct. Thus, the incorrect statement is (b).

Key concepts

Rate of Reaction Stoichiometry

In the realm of chemistry, the rate of reaction stoichiometry plays a pivotal role in understanding how different reactants contribute to the rate of a chemical reaction. Stoichiometry, to put it simply, is the calculation of reactants and products in chemical reactions. It’s based on the balanced chemical equation, which shows the relative amounts of reactants and products involved in the reaction. For instance, the chemical equation \(4 \mathrm{A}+\mathrm{B} \rightarrow 2 \mathrm{C}+2\mathrm{D}\) indicates that 4 molecules of A react with 1 molecule of B to produce 2 molecules each of C and D.

In terms of reaction rates, stoichiometry helps to establish the relationship between the rates at which reactants disappear and products appear. The stoichiometric coefficients, the numbers in front of the chemical species, determine the proportional rates. For example, if the rate of disappearance of A is given, stoichiometry dictates that the rate of disappearance of B is one-fourth of that of A, due to their stoichiometric coefficients (4 for A and 1 for B). Similarly, the appearance rates of C and D are directly related to these stoichiometric ratios. This concept creates a framework that allows chemists to predict and measure the rates at which substances are consumed or formed over time.

Chemical Kinetics

Digging deeper, chemical kinetics is the study of the speed or rate at which chemical reactions occur, and what affects these rates. The rate of a chemical reaction can be affected by various factors including temperature, pressure, concentration of reactants, surface area, and the presence of catalysts.

Understanding kinetics is crucial because it not only tells us how fast a reaction proceeds but also provides clues to the reaction's mechanism—i.e., the step-by-step pathway from reactants to products. For instance, in the reaction \(4 \mathrm{A} + \mathrm{B} \rightarrow 2 \mathrm{C} + 2\mathrm{D}\), a kinetic study might reveal that A and B interact in a very fast initial step, which then dictates the overall rate of the reaction. Moreover, kinetics principles are applied to optimize reactions in industrial processes, ensuring that products are created as efficiently as possible.

Rate of Disappearance

Focusing on the rate of disappearance, this term describes how quickly a reactant is consumed in a chemical reaction. It is measured in terms of the concentration decrease per unit of time, typically moles per liter per second (M/s). Using the stoichiometry of the balanced chemical equation, we can link the rates of disappearance for all reactants.

For instance, in our equation \(4 \mathrm{A} + \mathrm{B} \rightarrow 2 \mathrm{C} + 2\mathrm{D}\), the rate of disappearance of B is related to A by the stoichiometric ratio of their coefficients. When A disappears at a certain rate, B disappears four times slower because it takes four molecules of A to react with one molecule of B. This rate is essential for calculating how long it will take for a reactant to be completely consumed in a given reaction.

Rate of Appearance

Similarly, rate of appearance looks at how quickly a product is formed. It’s the flip side of disappearance and measures the increase in concentration of a product in moles per liter per second (M/s). This rate should sync with the stoichiometry of the reaction, ensuring the conservation of mass and atoms.

In our case, the rate of appearance of product C is directly connected to the disappearance of B. In a balanced reaction, no atoms disappear; they just change partners. Thus, when B is consumed, C is formed; for every one B molecule that reacts, one C molecule appears. If the rate of disappearance of B is known, it equals the rate of appearance of C, not half as wrongly suggested in the incorrect statement from our original exercise. This concept is fundamental for determining how much product can be expected from a given amount of reactant, a key factor in the design of chemical reactors and the scale-up of chemical processes.