Half life of a zero order reaction is 250sec. `t_(75%), t_(100%)` of the reaction respectively in sec. are

Options

A. 500, 375
B. 375, 500
C. 300, 575
D. 575, 300

Explanation

To solve the problem of finding \( t_{75\%} \) and \( t_{100\%} \) for a zero-order reaction with a half-life of 250 seconds, we can follow these steps:

### Step 1: Understand the Zero-Order Reaction
In a zero-order reaction, the rate of reaction is constant and does not depend on the concentration of the reactants. The general formula for a zero-order reaction is:
\[ [A] = [A_0] - kt \]
where:
- \([A]\) is the concentration at time \( t \),
- \([A_0]\) is the initial concentration,
- \( k \) is the rate constant,
- \( t \) is the time.

### Step 2: Calculate the Rate Constant \( k \)
The half-life (\( t_{1/2} \)) for a zero-order reaction is given by the formula:
\[ t_{1/2} = \frac{[A_0]}{2k} \]
Given that \( t_{1/2} = 250 \) seconds, we can rearrange this to find \( k \):
\[ k = \frac{[A_0]}{2 \cdot t_{1/2}} = \frac{[A_0]}{2 \cdot 250} \]
Assuming \( [A_0] = 100 \) (for simplicity), we have:
\[ k = \frac{100}{500} = \frac{1}{5} \text{ M/s} \]

### Step 3: Calculate \( t_{75\%} \)
To find \( t_{75\%} \), we need to determine how much concentration is left when 75% of the reactant has been consumed. If 75% is consumed, then 25% remains:
\[ [A] = [A_0] - 0.75[A_0] = 0.25[A_0] \]
Substituting into the equation:
\[ 0.25 \cdot 100 = 100 - k \cdot t_{75\%} \]
This simplifies to:
\[ 25 = 100 - \left(\frac{1}{5}\right) t_{75\%} \]
Rearranging gives:
\[ \left(\frac{1}{5}\right) t_{75\%} = 100 - 25 = 75 \]
Thus:
\[ t_{75\%} = 75 \cdot 5 = 375 \text{ seconds} \]

### Step 4: Calculate \( t_{100\%} \)
For \( t_{100\%} \), the reaction is complete, meaning all of the reactant has been consumed:
\[ [A] = 0 \]
Using the equation:
\[ 0 = 100 - k \cdot t_{100\%} \]
This simplifies to:
\[ k \cdot t_{100\%} = 100 \]
Substituting \( k = \frac{1}{5} \):
\[ \left(\frac{1}{5}\right) t_{100\%} = 100 \]
Thus:
\[ t_{100\%} = 100 \cdot 5 = 500 \text{ seconds} \]

### Final Answers
- \( t_{75\%} = 375 \) seconds
- \( t_{100\%} = 500 \) seconds