Half life of a certain zero order reaction, A `rightarrow` P is 2 hour when the initial concentration of the reactant, 'A' is 4 mol `L^(-1)`. The time required for its concentration to change from 0.40 to 0.20 mol `L^(-1)` is

Options

A. 0.75 min
B. 2 min
C. 12 min
D. 8 min

Answer

To solve the problem step by step, we will use the information provided about the zero-order reaction and the half-life formula.

### Step 1: Understand the given data
We know that:
- The half-life (T_half) of the zero-order reaction is 2 hours.
- The initial concentration (A_0) of the reactant A is 4 mol/L.

### Step 2: Write the half-life formula for a zero-order reaction
For a zero-order reaction, the half-life is given by the formula:
\[ T_{1/2} = \frac{[A_0]}{2k} \]
Where:
- \( T_{1/2} \) is the half-life,
- \( [A_0] \) is the initial concentration,
- \( k \) is the rate constant.

### Step 3: Calculate the rate constant (k)
We can rearrange the half-life formula to solve for \( k \):
\[ k = \frac{[A_0]}{2T_{1/2}} \]

Substituting the values:
\[ k = \frac{4 \, \text{mol/L}}{2 \times 2 \, \text{hours}} \]
\[ k = \frac{4}{4} = 1 \, \text{mol/L/hour} \]

### Step 4: Use the rate constant to find the time required for concentration change
We need to find the time required for the concentration to change from 0.40 mol/L to 0.20 mol/L.

Using the zero-order reaction formula:
\[ [A_t] = [A_0] - kt \]
Where:
- \( [A_t] \) is the concentration at time t,
- \( [A_0] \) is the initial concentration,
- \( k \) is the rate constant,
- \( t \) is the time.

Here, we can set:
- \( [A_t] = 0.20 \, \text{mol/L} \)
- \( [A_0] = 0.40 \, \text{mol/L} \)

Substituting these values into the equation:
\[ 0.20 = 0.40 - (1)(t) \]

### Step 5: Solve for t
Rearranging the equation gives:
\[ t = 0.40 - 0.20 \]
\[ t = 0.20 \, \text{hours} \]

### Step 6: Convert time to minutes
To convert hours into minutes, we multiply by 60:
\[ t = 0.20 \, \text{hours} \times 60 \, \text{minutes/hour} = 12 \, \text{minutes} \]

### Final Answer
The time required for the concentration to change from 0.40 to 0.20 mol/L is **12 minutes**.

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