For `3AtoxB,(d[B])/(dt)` is found to be 2/3rd of `(d[A])/(dt)`,. Then the value of x is

Options

A. 1.5
B. 3
C. `1//2`
D. 2

Answer

To solve the problem, we need to analyze the reaction \(3A \rightarrow xB\) and the relationship between the rates of change of concentrations of reactants and products.

### Step-by-Step Solution:

1. **Write the Rate Law Expression:**
The rate of the reaction can be expressed in terms of the change in concentration of the reactants and products. For the reaction \(3A \rightarrow xB\), the rate can be written as:
\[ -\frac{1}{3} \frac{d[A]}{dt} = \frac{1}{x} \frac{d[B]}{dt} \]

2. **Use the Given Relationship:**
According to the problem, it is given that:
\[ \frac{d[B]}{dt} = \frac{2}{3} \frac{d[A]}{dt} \]
We can substitute this expression into our rate law.

3. **Substitute into the Rate Law:**
Substitute \(\frac{d[B]}{dt}\) into the rate law:
\[ -\frac{1}{3} \frac{d[A]}{dt} = \frac{1}{x} \left(\frac{2}{3} \frac{d[A]}{dt}\right) \]

4. **Simplify the Equation:**
We can simplify this equation. First, multiply both sides by \(3x\) to eliminate the fractions:
\[ -x \frac{d[A]}{dt} = 2 \frac{d[A]}{dt} \]

5. **Rearrange the Equation:**
Rearranging gives:
\[ -x \frac{d[A]}{dt} - 2 \frac{d[A]}{dt} = 0 \]
Factoring out \(\frac{d[A]}{dt}\) (assuming \(\frac{d[A]}{dt} \neq 0\)):
\[ (-x - 2) \frac{d[A]}{dt} = 0 \]

6. **Solve for x:**
Since \(\frac{d[A]}{dt} \neq 0\), we can set the expression in parentheses to zero:
\[ -x - 2 = 0 \implies x = -2 \]

7. **Final Value of x:**
However, since \(x\) must be a positive integer (as it represents the stoichiometric coefficient), we take the absolute value:
\[ x = 2 \]

### Conclusion:
The value of \(x\) is \(2\).