A certain radioactive element has half-life of `4` days. The fraction of material that decays in `2 `days is

Options

A. `1//2`
B. `1//(sqrt2)`
C. `(sqrt2)`
D. `(sqrt2)-1//(sqrt2)`

Explanation

To solve the problem of finding the fraction of a radioactive element that decays in 2 days given its half-life of 4 days, we can follow these steps:

### Step 1: Understand the half-life
The half-life (t_half) of a radioactive substance is the time required for half of the radioactive atoms in a sample to decay. In this case, the half-life is given as 4 days.

### Step 2: Calculate the decay constant (λ)
The decay constant (λ) can be calculated using the formula:
\[ \lambda = \frac{\ln(2)}{t_{half}} \]
Substituting the half-life:
\[ \lambda = \frac{\ln(2)}{4 \text{ days}} \]

### Step 3: Use the decay law
The number of undecayed nuclei (N) at time t can be expressed as:
\[ N = N_0 e^{-\lambda t} \]
where \(N_0\) is the initial number of nuclei, and \(t\) is the time elapsed (2 days in this case).

### Step 4: Substitute the values
Substituting the values into the equation:
\[ N = N_0 e^{-\left(\frac{\ln(2)}{4}\right) \cdot 2} \]
This simplifies to:
\[ N = N_0 e^{-\frac{\ln(2)}{2}} = N_0 e^{\ln\left(\frac{1}{\sqrt{2}}\right)} = N_0 \cdot \frac{1}{\sqrt{2}} \]

### Step 5: Calculate the fraction of material that decays
The fraction of material that decays can be calculated as:
\[ \text{Fraction decayed} = \frac{N_0 - N}{N_0} = \frac{N_0 - \frac{N_0}{\sqrt{2}}}{N_0} \]
This simplifies to:
\[ \text{Fraction decayed} = 1 - \frac{1}{\sqrt{2}} = \frac{\sqrt{2} - 1}{\sqrt{2}} \]

### Final Answer
Thus, the fraction of material that decays in 2 days is:
\[ \frac{\sqrt{2} - 1}{\sqrt{2}} \]

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