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

Options

A. 1.5
B. 3
C. 2
D. 5

Explanation

To solve the problem, we need to analyze the given reaction and the relationship between the rates of change of the concentrations of reactants and products.

### Step-by-Step Solution:

1. **Write the Reaction**:
The reaction is given as:
\[ 3A \rightarrow xB \]

2. **Understand Rate of Change**:
The rate of change of concentration for reactants and products can be expressed as:
\[ -\frac{d[A]}{dt} = \frac{1}{3} \frac{d[B]}{dt} \]
Here, the negative sign indicates that the concentration of A is decreasing.

3. **Given Relationship**:
We are given that:
\[ \frac{d[B]}{dt} = \frac{2}{3} \frac{d[A]}{dt} \]

4. **Substitute into Rate Equation**:
From the rate equation, we can express \(\frac{d[B]}{dt}\) in terms of \(\frac{d[A]}{dt}\):
\[ \frac{1}{3} \frac{d[B]}{dt} = -\frac{d[A]}{dt} \]
Substituting the given relationship into this equation:
\[ \frac{1}{3} \left(\frac{2}{3} \frac{d[A]}{dt}\right) = -\frac{d[A]}{dt} \]

5. **Simplify**:
This simplifies to:
\[ \frac{2}{9} \frac{d[A]}{dt} = -\frac{d[A]}{dt} \]

6. **Equate the Coefficients**:
To find \(x\), we also know that:
\[ \frac{1}{x} \frac{d[B]}{dt} = \frac{1}{3} \frac{d[A]}{dt} \]
Substituting \(\frac{d[B]}{dt} = \frac{2}{3} \frac{d[A]}{dt}\):
\[ \frac{1}{x} \left(\frac{2}{3} \frac{d[A]}{dt}\right) = \frac{1}{3} \frac{d[A]}{dt} \]

7. **Cancel \(\frac{d[A]}{dt}\)**:
Assuming \(\frac{d[A]}{dt} \neq 0\), we can cancel \(\frac{d[A]}{dt}\) from both sides:
\[ \frac{2}{3x} = \frac{1}{3} \]

8. **Solve for \(x\)**:
Cross-multiplying gives:
\[ 2 = x \]
Thus, the value of \(x\) is:
\[ x = 2 \]

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

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