Question

For the reaction \(\mathrm{A}(g)+\frac{1}{2} \mathrm{~B}(g) \longrightarrow 2 \mathrm{C}(g)\) a. determine the expression for the rate of the reaction in terms of the change in concentration of each of the reactants and products. b. when \(C\) is increasing at a rate of \(0.025 \mathrm{M} / \mathrm{s},\) how fast is \(\mathrm{B}\) decreasing? How fast is A decreasing?

Short answer

The rate of the reaction is \(\text{rate} = -\frac{d[\text{A}]}{dt} = -2\frac{d[\text{B}]}{dt} = 2\frac{d[\text{C}]}{dt}\). For \(\text{C}\) increasing at 0.025 M/s, \(\text{B}\) is decreasing at 0.0125 M/s, and \(\text{A}\) is decreasing at 0.025 M/s.

Step-by-step solution

Write the Rate Expression

The rate of a chemical reaction can be expressed in terms of the rate of change of concentration of any reactant or product. For the reaction \(\text{A}(g) + \frac{1}{2} \text{B}(g) \rightarrow 2 \text{C}(g)\), the rate can be written as \(-\frac{1}{a}\frac{d[\text{A}]}{dt} = -\frac{2}{b}\frac{d[\text{B}]}{dt} = \frac{1}{c}\frac{d[\text{C}]}{dt}\), where a, b, and c are the stoichiometric coefficients of A, B, and C, respectively. Plugging in the coefficients from the reaction gives us \(-\frac{d[\text{A}]}{dt} = -2\frac{d[\text{B}]}{dt} = 2\frac{d[\text{C}]}{dt}\).

Determine the Rate of Decrease of B

Given that \([\text{C}]\) is increasing at a rate of 0.025 M/s, we can use the rate expression to find the rate at which \([\text{B}]\) is decreasing. We know that \(2\frac{d[\text{C}]}{dt} = -2\frac{d[\text{B}]}{dt}\), so if \(\frac{d[\text{C}]}{dt} = 0.025 \text{M/s}\), then \(\frac{d[\text{B}]}{dt}\) must be \(\frac{d[\text{C}]}{dt} / 2 = 0.025 \text{M/s} / 2 = 0.0125 \text{M/s}\) (since the rate of appearance of C is twice the rate of disappearance of B).

Determine the Rate of Decrease of A

Similarly, we can determine the rate at which \([\text{A}]\) is decreasing by using the stoichiometry of the reaction. Since the rate of disappearance of A is equal to the rate of appearance of C for this reaction, the rate of decrease of A will be the same as the rate of increase of C: \(-\frac{d[\text{A}]}{dt} = \frac{d[\text{C}]}{dt} = 0.025 \text{M/s}\).

Key concepts

Reaction Rate Expression

Understanding the reaction rate expression is crucial when studying the dynamics of a chemical equation. A reaction rate expression indicates how the rate of a reaction depends on the concentration of its reactants and products. In essence, it's a mathematical statement that shows how the concentration of substances changes over time.

For a balanced chemical equation, like the one given \(\mathrm{A}(g)+\frac{1}{2} \mathrm{B}(g) \longrightarrow 2\mathrm{C}(g)\), the reaction rate expression can be formed by relating the rates of change in concentration of the reactants and products. It's important to consider the stoichiometry of the reaction, which is denoted by the coefficients in the balanced equation. These coefficients help us understand the relative rates at which reactants are consumed and products are formed. For every one molecule of A consumed, half a molecule of B is used, and two molecules of C are produced.

Rate of Change of Concentration

The rate of change of concentration refers to how quickly the concentration of a substance in a reaction is increasing or decreasing over time. It is typically expressed in molarity per second (M/s).

In our exercise, the focus is to calculate the rate at which reactants are consumed and products are generated in the reaction \(\mathrm{A}(g)+\frac{1}{2} \mathrm{B}(g) \longrightarrow 2\mathrm{C}(g)\). Using the rate of change of \(\text{C}\), we can directly correlate this to the changes in concentrations of \(\text{A}\) and \(\text{B}\) using stoichiometry. This relationship allows us to solve for unknown rates by using the known rate of a single species, providing us with a comprehensive view of the reaction's dynamics.

Chemical Kinetics

Chemical kinetics is the branch of chemistry that studies the speeds, or rates, of chemical reactions and the mechanisms by which they occur. It not only involves the rate expressions but also factors that affect these rates, such as temperature, pressure, and the presence of catalysts.

In relation to the exercise provided, chemical kinetics would explore not only how fast \(\text{A}\) and \(\text{B}\) are decreasing and \(\text{C}\) is increasing but also why they are changing at these specific rates. Kinetic studies provide insights into the steps involved in the reaction mechanism, which can help us understand and control the production of desired products in various industrial and biochemical processes.

Stoichiometry

The concept of stoichiometry is applied in this exercise as a quantitative relationship between the amounts of reactants used and products formed in a chemical reaction. It is based on the conservation of mass where the total mass of the reactants equals the total mass of the products.

In the context of the reaction \(\mathrm{A}(g)+\frac{1}{2} \mathrm{B}(g) \longrightarrow 2\mathrm{C}(g)\), stoichiometry tells us that one mole of A reacts with half a mole of B to produce two moles of C. This stoichiometric relationship is crucial when we convert the rate of formation of C to the rates of consumption of A and B. Without stoichiometry, we would be unable to predict the amounts of reactants needed and products formed, making it an essential tool in chemical manufacturing and laboratory work.