For a reaction `1//2A to 2B`, rate of disappearance of A is related to the rate of appearance of B by the expression:

Options

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

Explanation

To solve the problem regarding the relationship between the rate of disappearance of A and the rate of appearance of B for the reaction \( \frac{1}{2}A \rightarrow 2B \), we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Reaction**: The given reaction is \( \frac{1}{2}A \rightarrow 2B \). Here, A is the reactant and B is the product.

2. **Define Rate of Reaction**: The rate of a reaction can be defined in terms of the change in concentration of reactants and products over time.

3. **Rate of Disappearance of A**: The rate of disappearance of A can be expressed as:
\[ \text{Rate of disappearance of A} = -\frac{1}{\Delta t} \frac{\Delta [A]}{\Delta t} = -\frac{1}{2} \frac{\Delta [A]}{\Delta t} \]
Here, the negative sign indicates that the concentration of A is decreasing.

4. **Rate of Appearance of B**: The rate of appearance of B can be expressed as:
\[ \text{Rate of appearance of B} = \frac{1}{\Delta t} \frac{\Delta [B]}{\Delta t} = \frac{1}{2} \frac{\Delta [B]}{\Delta t} \]

5. **Relate the Rates**: According to the stoichiometry of the reaction, we can relate the rate of disappearance of A to the rate of appearance of B:
\[ -\frac{1}{2} \frac{\Delta [A]}{\Delta t} = \frac{1}{2} \frac{\Delta [B]}{\Delta t} \]

6. **Rearranging the Equation**: To express the relationship in a more straightforward manner, we can rearrange the equation:
\[ \frac{\Delta [A]}{\Delta t} = -\frac{1}{2} \frac{\Delta [B]}{\Delta t} \]
This indicates that for every 1 mole of A that disappears, 2 moles of B appear.

7. **Final Expression**: Therefore, the final relationship between the rates can be expressed as:
\[ \frac{-\Delta [A]}{2 \Delta t} = \frac{\Delta [B]}{2 \Delta t} \]
or simply:
\[ \text{Rate of disappearance of A} = \frac{1}{2} \times \text{Rate of appearance of B} \]