Question

For the reaction \(A+2 B \longrightarrow 2 C\), the rate of reaction is \(1.76 \times 10^{-5} \mathrm{M} \mathrm{s}^{-1}\) at the time when \([\mathrm{A}]=0.3580 \mathrm{M}.\) (a) What is the rate of formation of \(\mathrm{C}\) ? (b) What will \([\mathrm{A}]\) be 1.00 min later? (c) Assume the rate remains at \(1.76 \times 10^{-5} \mathrm{M} \mathrm{s}^{-1}\) How long would it take for \([\mathrm{A}]\) to change from 0.3580 to \(0.3500 \mathrm{M} ?\)

Short answer

The rate of formation of C is \(3.52 \times 10^{-5} \mathrm{M} \mathrm{s}^{-1}\). [A] is 0.357 M after 1.00 minute. It would take approximately 7.58 minutes for [A] to change from 0.3580 M to 0.3500 M.

Step-by-step solution

Calculation of the rate of formation of C

The stoichiometry of the reaction shows that 2 moles of C are formed for every mole of A that reacts. Thus, the rate of formation of C is twice the rate of reaction. This is calculated by multiplying the given rate of reaction (\(1.76 \times 10^{-5} \mathrm{M} \mathrm{s}^{-1}\)) by 2.

Calculation of [A] after 1.00 minute

The rate of reaction is the rate of decrease of [A], therefore the change in [A] after 1.00 minute can be calculated by multiplying the rate by the time in seconds (1.00 min = 60 s). This value should then be subtracted from the initial concentration [A] to get [A] after 1.00 minute.

Find the time for [A] to change from 0.3580 M to 0.3500 M

This time can be calculated by dividing the change in concentration (0.008 M) by the rate of the reaction. The result gives the time in seconds, which then should be converted into minutes by dividing by 60.

Key concepts

Rate of Reaction

The rate of reaction is a fundamental concept in chemical kinetics. It tells us how quickly a reactant is consumed or a product is formed in a chemical reaction. In this exercise, the rate of reaction is given as \(1.76 \times 10^{-5} \text{ M s}^{-1}\). This means that every second, the concentration of reactant A decreases by this amount.
Understanding the rate of reaction helps us predict how fast a chemical process will occur. The given rate refers to the speed at which the reactants turn into products at a specific concentration and time. In our case, it specifically measures how fast A is consumed in the reaction \(A+2B \rightarrow 2C\).
This value will remain constant under stable conditions, unless concentration, temperature, or other reaction conditions change. It's important to comprehend this as it lays the foundation for calculating other reaction parameters, such as the rate of formation of products and changes in concentration over time.

Stoichiometry

Stoichiometry is the part of chemistry that deals with the quantities and proportional relationships of reactants and products in reactions. In the provided reaction \(A+2B \rightarrow 2C\), stoichiometry tells us that one mole of A and two moles of B react to form two moles of C.
This relationship is crucial for understanding how the rate of reaction affects the rate of formation of products. Because two moles of C are formed for every mole of A reacted, it is directly connected to how fast C will appear in the system.
In our exercise, this means that if the rate of reaction (decrease in [A]) is \(1.76 \times 10^{-5} \text{ M s}^{-1}\), the rate of formation of C will be twice that amount, due to the 1:2 ratio between A and C in the reaction equation. This reflects how stoichiometry helps us understand the balance and conversion between reactants and products.

Concentration Change

Concentration change refers to how the amount of a substance in a solution varies over time. It plays a key role in analyzing chemical reactions, as it's directly linked to the rate of reaction.
In this exercise, we monitor the concentration of A to determine how it decreases with time. Initially, the concentration of A is given as 0.3580 M, and we are tasked with finding out what [A] will be after 1.00 minute.

• First, we calculate the change in concentration using the rate of reaction.
• The rate is multiplied by the time interval (60 seconds in this case) to get the total concentration change.
• This value is then subtracted from the initial concentration to find [A] at the new time.

This helps us predict how the reaction progresses and aids in planning or adjusting processes in industrial and laboratory settings.

Rate of Formation

The rate of formation is a specific way to describe how quickly a product is generated in a reaction. In our example, this relates to the substance C being formed. Based on the stoichiometry \(A+2B \rightarrow 2C\), we know that for every mole of A that reacts, 2 moles of C are produced.
The calculation involves taking the rate of reaction (consumption of A) and scaling it according to the stoichiometric coefficients. Since we produce 2 moles of C for every mole of A consumed, we multiply the rate of reaction by 2. Thus:\[ \text{Rate of formation of } C = 2 \times 1.76 \times 10^{-5} \text{ M s}^{-1} \] Understanding the rate of formation is vital in fields like chemical engineering where the efficiency and yield of products can directly impact the viability of a process.