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) \(7 / 3 \mathrm{C}+5 / 2 \mathrm{~B} \rightarrow 3 / 2 \mathrm{~A}\)

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: \(\frac{\Delta[\mathrm{A}]}{\Delta \mathrm{t}} = -2 \cdot \frac{1 }{3}\mathrm{r}\) \(\frac{\Delta[\mathrm{B}]}{\Delta \mathrm{t}} = -2 \cdot \frac{2 }{5}\mathrm{r}\) \(\frac{\Delta[\mathrm{C}]}{\Delta \mathrm{t}} = 3 \cdot \frac{2 }{7}\mathrm{r}\)

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.

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