A radioactive element is disintegrating having half-life 6.93 s. The fractional change in number of nuclei of the radioactive element during 10 s is

Options

A. (a) 0.37
B. (b) 0.63
C. (c) 0.25
D. (d) 0.50

Explanation

To solve the problem of finding the fractional change in the number of nuclei of a radioactive element during 10 seconds, we can follow these steps:

### Step 1: Understand the given data
We are given:
- Half-life (T₁/₂) = 6.93 seconds
- Time (t) = 10 seconds

### Step 2: Calculate the decay constant (λ)
The decay constant (λ) can be calculated using the formula:
\[ \lambda = \frac{\ln(2)}{T_{1/2}} \]
Substituting the half-life:
\[ \lambda = \frac{\ln(2)}{6.93 \, \text{s}} \approx \frac{0.693}{6.93} \approx 0.1 \, \text{s}^{-1} \]

### Step 3: Calculate the number of nuclei remaining after time t
The number of nuclei remaining (N) after time t can be calculated using the formula:
\[ N = N_0 e^{-\lambda t} \]
Where:
- \(N_0\) = initial number of nuclei
- \(t\) = time elapsed

Substituting the values:
\[ N = N_0 e^{-0.1 \times 10} = N_0 e^{-1} \]
Using the value of \(e^{-1}\):
\[ N \approx N_0 \times 0.3679 \]

### Step 4: Calculate the fractional change in the number of nuclei
The fractional change in the number of nuclei is given by:
\[ \text{Fractional change} = \frac{N_0 - N}{N_0} = \frac{N_0 - N_0 e^{-1}}{N_0} = 1 - e^{-1} \]
Calculating this:
\[ \text{Fractional change} = 1 - 0.3679 \approx 0.6321 \]

### Final Answer
The fractional change in the number of nuclei during 10 seconds is approximately **0.6321**.

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