The half-life period of a radioactive substance is `5 min`. The amount of substance decayed in `20 min` will be

Options

A. `25%`
B. `6.25%`
C. `93.75%`
D. `75%`

Answer

To solve the problem regarding the decay of a radioactive substance, we will follow these steps:

### Step 1: Understand the Concept of Half-Life
The half-life of a radioactive substance is the time required for half of the radioactive nuclei in a sample to decay. In this case, the half-life is given as 5 minutes.

### Step 2: Determine the Number of Half-Lives in 20 Minutes
To find out how many half-lives fit into 20 minutes, we divide the total time by the half-life:
\[ \text{Number of half-lives} = \frac{20 \text{ min}}{5 \text{ min}} = 4 \]

### Step 3: Calculate the Remaining Amount of Substance
The amount of substance remaining after a certain number of half-lives can be calculated using the formula:
\[ N_t = N_0 \left(\frac{1}{2}\right)^n \]
where \(N_t\) is the amount remaining, \(N_0\) is the initial amount, and \(n\) is the number of half-lives.

Substituting \(n = 4\):
\[ N_t = N_0 \left(\frac{1}{2}\right)^4 = N_0 \frac{1}{16} \]

### Step 4: Calculate the Amount Decayed
The amount decayed can be found by subtracting the remaining amount from the initial amount:
\[ \text{Amount Decayed} = N_0 - N_t = N_0 - N_0 \frac{1}{16} = N_0 \left(1 - \frac{1}{16}\right) = N_0 \left(\frac{16}{16} - \frac{1}{16}\right) = N_0 \frac{15}{16} \]

### Step 5: Calculate the Percentage of Substance Decayed
To find the percentage of the substance that has decayed, we use the formula:
\[ \text{Percentage Decayed} = \left(\frac{\text{Amount Decayed}}{N_0}\right) \times 100 \]
Substituting the amount decayed:
\[ \text{Percentage Decayed} = \left(\frac{N_0 \frac{15}{16}}{N_0}\right) \times 100 = \frac{15}{16} \times 100 = 93.75\% \]

### Final Answer
The amount of substance decayed in 20 minutes is **93.75%**.

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