Question

For each of the questions, four choices have been provided. Select the correct alternative. If \(\mathrm{r}=\frac{-3}{2} \frac{\Delta[\mathrm{A}]}{\Delta \mathrm{t}}=\frac{-5}{2} \frac{\Delta[\mathrm{B}]}{\Delta \mathrm{t}}=\frac{+7}{3} \frac{[\Delta \mathrm{C}]}{\Delta \mathrm{t}}\), which of the following is the corresponding reaction? (a) \(2 / 3 \mathrm{~A}+2 / 5 \mathrm{~B} \rightarrow 3 / 7 \mathrm{C}\) (b) \(2 / 3 \mathrm{~A}+5 / 2 \mathrm{~B} \rightarrow 7 / 3 \mathrm{C}\) (c) \(3 / 2 \mathrm{~A}+5 / 2 \mathrm{~B} \rightarrow 7 / 3 \mathrm{C}\) (d) \(7 / 3 \mathrm{C}+5 / 2 \mathrm{~B} \rightarrow 3 / 2 \mathrm{~A}\)

Short answer

The correct reaction equation is (b) 2/3 A + 5/2 B → 7/3 C.

Step-by-step solution

Reformatting the rate equation

Given the equation: \(\mathrm{r}=\frac{-3}{2} \frac{\Delta[\mathrm{A}]}{\Delta \mathrm{t}}=\frac{-5}{2} \frac{\Delta[\mathrm{B}]}{\Delta \mathrm{t}}=\frac{+7}{3} \frac{[\Delta \mathrm{C}]}{\Delta \mathrm{t}}\). We will reformat the equation as follows: \(\frac{\Delta[\mathrm{A}]}{\Delta \mathrm{t}} = -2 \cdot \frac{1 }{3}\mathrm{r}\) \(\frac{\Delta[\mathrm{B}]}{\Delta \mathrm{t}} = -2 \cdot \frac{2 }{5}\mathrm{r}\) \(\frac{\Delta[\mathrm{C}]}{\Delta \mathrm{t}} = 3 \cdot \frac{2 }{7}\mathrm{r}\)

Comparing the rates with the reaction choices

Let's look at each choice and compare it to the reformatted rate equations. (a) \(2/3 \mathrm{A} + 2/5 \mathrm{B} \rightarrow 3/7 \mathrm{C}\) For this reaction choice, we have: \(\frac{\Delta[\mathrm{A}]}{\Delta \mathrm{t}} = -\mathrm{r}\) \(\frac{\Delta[\mathrm{B}]}{\Delta \mathrm{t}} = -\mathrm{r}\) \(\frac{\mathrm{\Delta[C]}}{\Delta \mathrm{t}} = \mathrm{r}\) Choice (a) does not match the re-formatted rate equations. (b) \(2/3 \mathrm{A} + 5/2 \mathrm{B} \rightarrow 7/3 \mathrm{C}\) For this reaction choice, we have: \(\frac{\Delta[\mathrm{A}]}{\Delta \mathrm{t}} = -2 \cdot \frac{1 }{3}\mathrm{r}\) \(\frac{\Delta[\mathrm{B}]}{\Delta \mathrm{t}} = -2 \cdot \frac{2 }{5}\mathrm{r}\) \(\frac{\Delta[\mathrm{C}]}{\Delta \mathrm{t}} = 3 \cdot \frac{2 }{7}\mathrm{r}\) Choice (b) matches the re-formatted rate equations. (c) \(3/2 \mathrm{A} + 5/2 \mathrm{B} \rightarrow 7/3 \mathrm{C}\) For this reaction choice, we have: \(\frac{\Delta[\mathrm{A}]}{\Delta \mathrm{t}} = -2 \cdot \frac{2 }{3}\mathrm{r}\) \(\frac{\Delta[\mathrm{B}]}{\Delta \mathrm{t}} = -2 \cdot \frac{2 }{5}\mathrm{r}\) \(\frac{\Delta[\mathrm{C}]}{\Delta \mathrm{t}} = 3 \cdot \frac{1 }{3}\mathrm{r}\) Choice (c) does not match the re-formatted rate equations. (d) \(7/3 \mathrm{C} + 5/2 \mathrm{B} \rightarrow 3/2 \mathrm{A}\) For this reaction choice, we have: \(\frac{\Delta[\mathrm{A}]}{\Delta \mathrm{t}} = 2 \cdot \frac{1 }{2}\mathrm{r}\) \(\frac{\Delta[\mathrm{B}]}{\Delta \mathrm{t}} = -2 \cdot \frac{2 }{5}\mathrm{r}\) \(\frac{\Delta[\mathrm{C}]}{\Delta \mathrm{t}} = -3 \cdot \frac{1 }{3}\mathrm{r}\) Choice (d) does not match the re-formatted rate equations.

Final answer

Given the analysis and comparisons above, the correct reaction equation corresponding to the given rates is: (b) \(2/3 \mathrm{A} + 5/2 \mathrm{B} \rightarrow 7/3 \mathrm{C}\)

Key concepts

Rate Equations

Rate equations are pivotal in understanding how chemical reactions progress over time.
They describe how the concentration of reactants or products change as the reaction proceeds. In the given problem, different rate expressions can be written for reactants and products, capturing these changes with respect to time.

In simpler terms, the rate of a chemical reaction is a measure of how fast the reactants are being converted into products. It is expressed by the rate equation which links the rate, often denoted by \(r\), to the concentration change of a substance over time. For instance, if you know that for a reaction \(\mathrm{A} \rightarrow \mathrm{C}\), the rate equation is linked with the decrease in concentration of \(\mathrm{A}\) and increase of \(\mathrm{C}\), it looks like:

• For reactant \(\mathrm{A}\): \(\frac{-\Delta[\mathrm{A}]}{\Delta t}\)
• For product \(\mathrm{C}\): \(\frac{+\Delta[\mathrm{C}]}{\Delta t}\)

In the context of the exercise, the negative sign indicates the concentration of \(\mathrm{A}\) or \(\mathrm{B}\) is decreasing, while for \(\mathrm{C}\) it is increasing.
This knowledge allows us to match specific equations with given reaction scenarios accurately.

Chemical Reactions

Chemical reactions are fundamental processes where substances, called reactants, are transformed into different substances known as products.
These transformations involve the breaking and forming of chemical bonds, leading to changes in the physical and chemical properties.

Chemical reactions can be represented using chemical equations. These equations provide a symbolic representation of the chemical process and include information about the reactants, products, and their respective quantities. For example, the exercise provided equations like \(2/3 \mathrm{A} + 5/2 \mathrm{B} \rightarrow 7/3 \mathrm{C}\).
This equation conveys that specific fractions or proportions of reactants \(\mathrm{A}\) and \(\mathrm{B}\) react to form a product \(\mathrm{C}\).

Understanding chemical reactions involves comprehending the stoichiometry, which is about the quantitative relationships in these transformations. It allows you to determine the amount of reactants needed for a reaction and the expected amount of products formed. This is crucial when trying to match rate equations to reaction equations as seen in the original problem.

Balancing Reactions

Balancing chemical reactions is a critical skill that ensures the reaction obeys the Law of Conservation of Mass.
This law posits that in a chemical reaction, matter is neither created nor destroyed. Thus, the number of atoms of each element must be the same on both sides of the equation.

In practice, balancing involves adjusting the coefficients of reactants and products to ensure equal numbers of each type of atom on both sides of the reaction equation.
This can seem a bit like a mathematical puzzle, fitting together numbers to maintain equality. For example, if a reaction seems to consume more of a substance than is present, that discrepancy needs to be corrected by adjusting coefficients like \(2/3\) or \(5/2\) in the given options.

Balancing reactions is also closely linked to reaction stoichiometry in determining the exact proportions needed in a reaction. This precision is crucial to achieve the reaction as it was intended in a lab setting or industrial process. It also ensures that the rate equations align properly with the given balanced chemical equations.