Percentage of a radioactive element decayed after 20 s when half-life is 4 s

Options

A. 92.25
B. 96.87
C. 50
D. 75

Explanation

To solve the problem of finding the percentage of a radioactive element that has decayed after 20 seconds, given that the half-life is 4 seconds, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Half-Life**:
The half-life (t₁/₂) of the radioactive element is given as 4 seconds.

2. **Calculate the Decay Constant (λ)**:
The decay constant (λ) can be calculated using the formula:
\[ \lambda = \frac{0.693}{t_{1/2}} \]
Substituting the half-life:
\[ \lambda = \frac{0.693}{4 \, \text{s}} = 0.17325 \, \text{s}^{-1} \]

3. **Use the First-Order Decay Formula**:
The amount of substance remaining after time t can be calculated using the first-order decay equation:
\[ N = N_0 e^{-\lambda t} \]
Here, \(N_0\) is the initial amount, and \(N\) is the remaining amount after time t. We want the ratio \( \frac{N}{N_0} \):
\[ \frac{N}{N_0} = e^{-\lambda t} \]

4. **Substitute the Values**:
For \(t = 20 \, \text{s}\):
\[ \frac{N}{N_0} = e^{-0.17325 \times 20} \]
Calculate the exponent:
\[ -0.17325 \times 20 = -3.465 \]
Now calculate \(e^{-3.465}\):
\[ \frac{N}{N_0} = e^{-3.465} \approx 0.0314 \]

5. **Calculate the Amount Decayed**:
The amount decayed can be found using:
\[ \text{Amount Decayed} = 1 - \frac{N}{N_0} = 1 - 0.0314 = 0.9686 \]

6. **Convert to Percentage**:
To find the percentage of the element that has decayed:
\[ \text{Percentage Decayed} = 0.9686 \times 100 \approx 96.86\% \]

### Final Answer:
The percentage of the radioactive element that has decayed after 20 seconds is approximately **96.86%**.

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