Question

Consider the general reaction $$ \mathrm{aA}+\mathrm{bB} \longrightarrow \mathrm{cC} $$ and the following average rate data over some time period \(\Delta t :\) $$ \begin{aligned}-\frac{\Delta \mathrm{A}}{\Delta t} &=0.0080 \mathrm{mol} / \mathrm{L} \cdot \mathrm{s} \\\\-& \frac{\Delta \mathrm{B}}{\Delta t}=0.0120 \mathrm{mol} / \mathrm{L} \cdot \mathrm{s} \\ \frac{\Delta \mathrm{C}}{\Delta t} &=0.0160 \mathrm{mol} / \mathrm{L} \cdot \mathrm{s} \end{aligned} $$ Determine a set of possible coefficients to balance this general reaction.

Short answer

The balanced general reaction with the given average rate data is: \(2\mathrm{A} + 3\mathrm{B} \longrightarrow 4\mathrm{C}\)

Step-by-step solution

Write the general reaction with given coefficients

We are given a general reaction with reactants A and B and product C, represented as follows: \[ \mathrm{aA} + \mathrm{bB} \longrightarrow \mathrm{cC} \] Our goal is to find the coefficients a, b, and c that will balance this reaction.

Analyze the rate data given

We are given the following average rate data: - \(-\frac{\Delta \mathrm{A}}{\Delta t} = 0.0080 \, \mathrm{mol} \, \mathrm{L}^{-1} \mathrm{s}^{-1}\) - \(-\frac{\Delta \mathrm{B}}{\Delta t} = 0.0120 \, \mathrm{mol} \, \mathrm{L}^{-1} \mathrm{s}^{-1}\) - \(\frac{\Delta \mathrm{C}}{\Delta t} = 0.0160 \, \mathrm{mol} \, \mathrm{L}^{-1} \mathrm{s}^{-1}\)

Determine the relationship between the coefficients

To find the relationship between the coefficients, let's divide the reactions rates: By dividing the average rate of A by the average rate of B, we get: \[ \frac{-\frac{\Delta \mathrm{A}}{\Delta t}}{-\frac{\Delta \mathrm{B}}{\Delta t}} = \frac{a}{b} \] Similarly, divide the average rate of C by either the average rate of A or B: \[ \frac{\frac{\Delta \mathrm{C}}{\Delta t}}{-\frac{\Delta \mathrm{A}}{\Delta t}} = \frac{c}{a} \] Now, let's solve for the coefficients.

Solve for the coefficients

We can now substitute the rate data into the equations: 1. \(\frac{-\frac{0.0080}{1}}{-\frac{0.0120}{1}} = \frac{a}{b}\) 2. \(\frac{\frac{0.0160}{1}}{-\frac{0.0080}{1}} = \frac{c}{a}\) By solving the equations: 1. \(\frac{0.0080}{0.0120} = \frac{a}{b}\) 2. \(\frac{0.0160}{0.0080} = \frac{c}{a}\) Simplify the fractions: 1. \(\frac{2}{3} = \frac{a}{b}\) 2. \(2 = \frac{c}{a}\)

Find the set of coefficients

From the simplified equations, we can find one possible set of coefficients that will balance the reaction: 1. \(a = 2\) and \(b = 3\) 2. \(c = 2 \times a = 2 \times 2 = 4\) So, one possible set of coefficients that will balance the reaction is: \[ 2\mathrm{A} + 3\mathrm{B} \longrightarrow 4\mathrm{C} \] This is one possible balanced general reaction with the given average rate data.

Key concepts

Reaction Stoichiometry

In chemistry, stoichiometry is the calculation of reactants and products in chemical reactions. It provides a quantitative relationship between the substances as they participate in various reactions. In our wroked exercise, we had a general reaction represented as \(\mathrm{aA} + \mathrm{bB} \longrightarrow \mathrm{cC}\),\(\) and we needed to determine the coefficients \( a, b, \) and \( c \). The stoichiometric coefficients tell us the ratio in which reactants react and products are formed.Understanding stoichiometry is crucial because it allows chemists to predict how much of each substance is needed and produced. For example, using stoichiometry, if you know the amount of one reactant, you can calculate how much of another reactant you will need and how much product you will get from a reaction.

Average Reaction Rate

The average reaction rate in a chemical reaction is a measure of how quickly the concentration of a reactant or product changes over a given time period. Specifically, reaction rates can be expressed in terms of the change in concentration of a reactant or product per unit time.In the given exercise, the rate data was provided:

• For \(A\), \(-\frac{\Delta \mathrm{A}}{\Delta t} = 0.0080 \, \mathrm{mol} \, \mathrm{L}^{-1} \mathrm{s}^{-1}\)
• For \(B\), \(-\frac{\Delta \mathrm{B}}{\Delta t} = 0.0120 \, \mathrm{mol} \, \mathrm{L}^{-1} \mathrm{s}^{-1}\)
• For \(C\), \(\frac{\Delta \mathrm{C}}{\Delta t} = 0.0160 \, \mathrm{mol} \, \mathrm{L}^{-1} \mathrm{s}^{-1}\)

These rates indicate how fast reactants are consumed and products are formed.The average rate is important because it helps in understanding the dynamics of a reaction. It tells us whether a reactant is being used up quickly or slowly, and how fast a product is being formed, which is crucial in industrial applications to ensure processes are efficient and safe.

Rate Law

Rate law is an expression that links the rate of a reaction to the concentration of its reactants. It shows the relationship between the concentrations of the reactants and the rate, usually expressed as:\[ Rate = k [A]^m [B]^n \]Where \(k\) is the rate constant, and \(m\) and \(n\) are the reaction orders with respect to each reactant.Within the scope of the given exercise, we utilized the reaction rates to explore the stoichiometric coefficients. While rate laws give us insight into the kinetics of a reaction, they are different from the stoichiometry as they are derived from experimental data rather than from the balanced equation.Understanding rate laws is crucial as they provide insights on how alterations in concentration affect reaction rates, thereby allowing chemists to manipulate conditions to control how fast or slow a reaction proceeds.

Balancing Chemical Equations

Balancing chemical equations is a fundamental skill in chemistry. It involves making sure that the number of each type of atom is the same on both sides of the equation.In our exercise, after analyzing the average rate data, we determined the coefficients that balanced the equation. Starting from the general reaction \(\mathrm{aA} + \mathrm{bB} \longrightarrow \mathrm{cC}\),\(\) we found that the balanced equation was \(2\mathrm{A} + 3\mathrm{B} \longrightarrow 4\mathrm{C}\).Balancing equations is critical because it reflects the law of conservation of mass, which states that mass in an isolated system is neither created nor destroyed by chemical reactions. A balanced equation also provides the foundation needed for further calculations involving reaction stoichiometry, as it allows us to understand the exact amounts of each reactant and product in a reaction. The balanced equation is not just a requirement for stoichiometry but also sets the stage for determining other key chemical concepts such as limiting reactants and theoretical yields.