The reaction `2X to B` is a zeroth order reaction .If the initial concention of X is 0.2 M, the half-life is 6 h. When the initial concetration of X is 0.5 M, the time required to reach its final concetration of 0.2 M will be :

Explanation

To solve the problem, we need to determine the time required for the concentration of X to decrease from 0.5 M to 0.2 M in a zeroth-order reaction. Here are the steps to derive the solution:

### Step 1: Understand the zeroth-order reaction
In a zeroth-order reaction, the rate of reaction is constant and does not depend on the concentration of the reactant. The rate can be expressed as:
\[ \text{Rate} = k \]
where \( k \) is the rate constant.

### Step 2: Use the half-life formula for zeroth-order reactions
The half-life (\( t_{1/2} \)) for a zeroth-order reaction is given by:
\[ t_{1/2} = \frac{[A]_0}{2k} \]
where \([A]_0\) is the initial concentration of the reactant.

Given that the half-life is 6 hours when the initial concentration of X is 0.2 M, we can use this information to find the rate constant \( k \).

### Step 3: Calculate the rate constant \( k \)
Substituting the known values into the half-life formula:
\[ 6 = \frac{0.2}{2k} \]
Rearranging gives:
\[ 2k = \frac{0.2}{6} \]
\[ k = \frac{0.2}{12} = \frac{1}{60} \, \text{M/h} \]

### Step 4: Determine the change in concentration
Next, we need to find the time required for the concentration of X to decrease from 0.5 M to 0.2 M. The change in concentration (\( \Delta [X] \)) is:
\[ \Delta [X] = [X]_0 - [X]_f = 0.5 \, \text{M} - 0.2 \, \text{M} = 0.3 \, \text{M} \]

### Step 5: Use the zeroth-order kinetics equation
For a zeroth-order reaction, the change in concentration over time can be expressed as:
\[ \Delta [X] = k \cdot t \]
Substituting the values we have:
\[ 0.3 = \left(\frac{1}{60}\right) \cdot t \]

### Step 6: Solve for time \( t \)
Rearranging the equation to solve for \( t \):
\[ t = 0.3 \cdot 60 \]
\[ t = 18 \, \text{hours} \]

### Conclusion
The time required to reach the final concentration of 0.2 M from an initial concentration of 0.5 M is **18 hours**.

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