For a reaction `(1)/(2)A rarr 2B`, rate of disappearance of `'A'` is related to the rate of apperance of `'B'` by the expression:

Options

A. `-(d[A])/(dt) =(1)/(2) (d[B])/(dt)`
B. `-(d[A])/(dt) =(1)/(4) (d[B])/(dt)`
C. `-(d[A])/(dt) =(d[B])/(dt)`
D. `-(d[A])/(dt) =4(d[B])/(dt)`

Explanation

To solve the problem, we need to relate the rate of disappearance of reactant A to the rate of appearance of product B for the reaction:

\[ \frac{1}{2} A \rightarrow 2B \]

### Step-by-Step Solution:

1. **Write the Rate Expressions**:
For the reaction, we can express the rate of disappearance of A and the rate of appearance of B using their stoichiometric coefficients.

The rate of disappearance of A can be expressed as:
\[ -\frac{1}{\text{stoichiometric coefficient of A}} \frac{d[A]}{dt} = -\frac{1}{\frac{1}{2}} \frac{d[A]}{dt} = -2 \frac{d[A]}{dt} \]

The rate of appearance of B can be expressed as:
\[ \frac{1}{\text{stoichiometric coefficient of B}} \frac{d[B]}{dt} = \frac{1}{2} \frac{d[B]}{dt} \]

2. **Set Up the Relationship**:
According to the stoichiometry of the reaction, we can relate these two rates:
\[ -2 \frac{d[A]}{dt} = \frac{1}{2} \frac{d[B]}{dt} \]

3. **Rearranging the Equation**:
To find the relationship between the rates, we can rearrange the equation:
\[ -2 \frac{d[A]}{dt} = \frac{1}{2} \frac{d[B]}{dt} \]
Multiply both sides by 2:
\[ -4 \frac{d[A]}{dt} = \frac{d[B]}{dt} \]

4. **Final Expression**:
We can express the rate of disappearance of A in terms of the rate of appearance of B:
\[ -\frac{d[A]}{dt} = \frac{1}{4} \frac{d[B]}{dt} \]

### Conclusion:
Thus, the relationship between the rate of disappearance of A and the rate of appearance of B is given by:
\[ -\frac{d[A]}{dt} = \frac{1}{4} \frac{d[B]}{dt} \]