Question

Express the rate of this reaction in terms of the change in concentration of each of the reactants and products: $$ 2 \mathrm{~A}(g) \rightarrow \mathrm{B}(g)+\mathrm{C}(g) $$ When \([\mathrm{C}]\) is increasing at \(2 \mathrm{~mol} / \mathrm{L} \cdot \mathrm{s},\) how fast is \([\mathrm{A}]\) decreasing?

Short answer

The concentration of [A] is decreasing at 4 mol/L·s

Step-by-step solution

Identify the given information

The reaction is given by: \[2 \mathrm{~A}(g) \rightarrow \mathrm{B}(g) + \mathrm{C}(g) \] The rate of increase in concentration of \([\mathrm{C}]\) is given as \(2 \mathrm{~mol} / \mathrm{L} \cdot \mathrm{s}\).

Write the rate expression for reactants and products

For the reaction \[2 \mathrm{~A}(g) \rightarrow \mathrm{B}(g) + \mathrm{C}(g)\], the rate of the reaction can be written in terms of the concentration changes of reactants and products. The rate \(r\) is: \[ -\frac{1}{2} \frac{d[A]}{dt} = \frac{d[B]}{dt} = \frac{d[C]}{dt} \]

Relate the given rate to the concentration change

Given that \( \frac{d[C]}{dt} = 2 \mathrm{~mol} / \mathrm{L} \cdot \mathrm{s} \), use the rate expression to find the rate of change in concentration of \([ \mathrm{A} ]\). Substituting the given value: \[ \frac{d[C]}{dt} = 2 \mathrm{~mol} / \mathrm{L} \cdot \mathrm{s} \] From the rate expression: \[ -\frac{1}{2} \frac{d[A]}{dt} = 2 \mathrm{~mol} / \mathrm{L} \cdot \mathrm{s} \]

Solve for the rate of decrease in concentration of A

To find the value of \(\frac{d[A]}{dt}\), equate to the expression: Rearrange and solve for \(\frac{d[A]}{dt}\): \[ \frac{d[A]}{dt} = -2 \times 2 \mathrm{~mol} / \mathrm{L} \cdot \mathrm{s} \] \[ \frac{d[A]}{dt} = -4 \mathrm{~mol} / \mathrm{L} \cdot \mathrm{s} \]

Key concepts

reaction rate

In chemistry, the reaction rate is a measure of how quickly a chemical reaction occurs. It is typically expressed as the change in concentration of a reactant or product per unit time. For example, in the reaction \[2 \mathrm{~A}(g) \rightarrow \mathrm{B}(g)+\mathrm{C}(g)\], the reaction rate can be observed by how fast the concentration of substances A, B, and C change over time.
The reaction rate is important because it helps us understand the speed at which reactions proceed, allowing for better control and optimization of chemical processes.
To express the rate of this reaction, we need to consider the decrease in concentration of reactant A and the increase in concentration of products B and C. These changes are linked together by the stoichiometry of the reaction, which tells us the proportions in which reactants and products participate in the reaction.

concentration change

Concentration change refers to the variation in the amount of reactants or products in a given volume over time. In our exercise, we are looking at the change in concentrations of A, B, and C.
For the reaction \[2 \mathrm{~A}(g) \rightarrow \mathrm{B}(g)+\mathrm{C}(g)\], the changes in concentration are related to one another. As the reaction proceeds:

• The concentration of A decreases because A is being consumed.
• The concentrations of B and C increase because they are being produced.

Given that the concentration of C is increasing at \(2 \mathrm{~mol} / \mathrm{L} \cdot \mathrm{s}\), we use this information to find the concentration change of A. This direct relationship between the reactants and products allows us to calculate the rate at which the concentration of any participant in the reaction changes.

chemical kinetics

Chemical kinetics is the study of reaction rates and the factors that influence them. It involves understanding the steps through which reactants convert into products.
The rate at which a reaction occurs depends on various factors such as:

• The concentration of reactants
• The temperature of the system
• The presence of a catalyst
• The surface area of reactants

In our case, chemical kinetics helps us determine how the increase in concentration of product C affects the decrease in concentration of reactant A. This relationship is essential for predicting and controlling the behavior of chemical reactions in both laboratory and industrial settings.

rate expression

The rate expression is a mathematical way of linking the rate of a reaction to the concentration changes of its reactants and products.
For the reaction \[2 \mathrm{~A}(g) \rightarrow \mathrm{B}(g)+\mathrm{C}(g)\], the general rate expression can be written as:
\[ -\frac{1}{2} \frac{d[A]}{dt} = \frac{d[B]}{dt} = \frac{d[C]}{dt} \]
This expression shows the relationship between the rates of decrease of A and the rates of increase of B and C. The negative sign indicates that the concentration of A is decreasing, while the concentrations of B and C are increasing.
Given that \( \frac{d[C]}{dt} = 2 \mathrm{~mol} / \mathrm{L} \cdot \mathrm{s}\), we can substitute this value into the rate expression to find the rate of decrease of A:
\[ -\frac{1}{2} \frac{d[A]}{dt} = 2 \mathrm{~mol} / \mathrm{L} \cdot \mathrm{s} \]
Solving for \( \frac{d[A]}{dt} \) gives us:
\[ \frac{d[A]}{dt} = -4 \mathrm{~mol} / \mathrm{L} \cdot \mathrm{s} \] This tells us that the concentration of A decreases at a rate of \(4 \mathrm{~mol} / \mathrm{L} \cdots\), consistent with the stoichiometry of the reaction.