The decay constant , for a given radioactive sample, is 0.3465 `"day"^(-1)` .What percentage of this sample will get decayed in a period of 4 days ?

Options

A. `100%`
B. `50%`
C. `75%`
D. `10%`

Explanation

To solve the problem of determining what percentage of a radioactive sample decays over a period of 4 days given a decay constant of 0.3465 day^(-1), we can follow these steps:

### Step 1: Understand the relationship between decay constant and half-life
The half-life (T_1/2) of a radioactive sample is related to the decay constant (λ) by the formula:
\[ T_{1/2} = \frac{\ln(2)}{\lambda} \]
where \(\ln(2) \approx 0.693\).

### Step 2: Calculate the half-life
Substituting the given decay constant into the formula:
\[ T_{1/2} = \frac{0.693}{0.3465} \approx 2 \text{ days} \]

### Step 3: Determine the number of half-lives in 4 days
Since the half-life is 2 days, we can find how many half-lives fit into 4 days:
\[ \text{Number of half-lives} = \frac{4 \text{ days}}{2 \text{ days}} = 2 \]

### Step 4: Calculate the remaining amount of the sample
If we start with an initial amount \(N_0\), after one half-life, the remaining amount is:
\[ N_1 = \frac{N_0}{2} \]
After the second half-life, the remaining amount will be:
\[ N_2 = \frac{N_1}{2} = \frac{N_0}{4} \]

### Step 5: Calculate the decayed amount
The amount that has decayed can be calculated as:
\[ \text{Decayed amount} = N_0 - N_2 = N_0 - \frac{N_0}{4} = \frac{3N_0}{4} \]

### Step 6: Calculate the percentage of the decayed sample
To find the percentage of the sample that has decayed, we use the formula:
\[ \text{Percentage decayed} = \left(\frac{\text{Decayed amount}}{N_0}\right) \times 100\% \]
Substituting the decayed amount:
\[ \text{Percentage decayed} = \left(\frac{\frac{3N_0}{4}}{N_0}\right) \times 100\% = \frac{3}{4} \times 100\% = 75\% \]

### Final Answer
Thus, the percentage of the sample that will get decayed in a period of 4 days is **75%**.

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