If `r=(-3)/(2)(Delta[A])/(Deltat)=(-5)/(2)(Delta[B])/(Deltat)=(+7)/(3)([DeltaC])/(Deltat)`, which of the following is the corresponding reaction ?

Options

A. 2/3 A + 2/5 B `to` 3/7 C
B. 2/3 A +5/2 B `to` 7/3 C
C. 3/2A + 5/2B `to` 7/3C
D. 7/3C + 5/2 B `to` 3/2A

Answer

To determine the corresponding reaction from the given rate expressions, we need to analyze the relationships between the changes in concentrations of reactants and products. Let's break down the steps to find the reaction.

### Step-by-Step Solution:

1. **Understand the Rate Expression**:
The rate of reaction is given by:
\[ r = -\frac{3}{2} \frac{\Delta[A]}{\Delta t} = -\frac{5}{2} \frac{\Delta[B]}{\Delta t} = +\frac{7}{3} \frac{\Delta[C]}{\Delta t} \]
Here, \(A\) and \(B\) are reactants, and \(C\) is the product.

2. **Set Up the Rate Equation**:
For a general reaction of the form:
\[ aA + bB \rightarrow cC \]
The rate can be expressed as:
\[ r = -\frac{1}{a} \frac{\Delta[A]}{\Delta t} = -\frac{1}{b} \frac{\Delta[B]}{\Delta t} = \frac{1}{c} \frac{\Delta[C]}{\Delta t} \]

3. **Relate the Given Rates to Stoichiometric Coefficients**:
From the given rate expressions, we can equate the coefficients:
- From \(-\frac{3}{2} \frac{\Delta[A]}{\Delta t}\), we have \(a = \frac{2}{3}\).
- From \(-\frac{5}{2} \frac{\Delta[B]}{\Delta t}\), we have \(b = \frac{2}{5}\).
- From \(+\frac{7}{3} \frac{\Delta[C]}{\Delta t}\), we have \(c = \frac{3}{7}\).

4. **Formulate the Reaction**:
The reaction can be written as:
\[ \frac{2}{3}A + \frac{2}{5}B \rightarrow \frac{3}{7}C \]

5. **Clear the Fractions**:
To express the reaction with whole numbers, we can multiply through by the least common multiple (LCM) of the denominators (3, 5, and 7), which is 105:
\[ 105 \left(\frac{2}{3}A + \frac{2}{5}B \rightarrow \frac{3}{7}C\right) \]
This gives:
\[ 70A + 42B \rightarrow 45C \]

6. **Final Reaction**:
The corresponding reaction is:
\[ 70A + 42B \rightarrow 45C \]