A certain particle has a half life of 60 seconds. The fraction of the particles that will decay at the end of 10 seconds is

Options

A. `2^(1//6)`
B. `(1-2^(1//6))`
C. `(2^6-1)`
D. `(1-2^(-1//6))`

Answer

To solve the problem of determining the fraction of particles that will decay at the end of 10 seconds given a half-life of 60 seconds, we can follow these steps:

### Step 1: Understand the half-life and decay constant
The half-life (T_1/2) of the particle is given as 60 seconds. The decay constant (λ) can be calculated using the formula:
\[ \lambda = \frac{\ln(2)}{T_{1/2}} \]
Substituting the half-life into the formula:
\[ \lambda = \frac{\ln(2)}{60 \text{ seconds}} \]

### Step 2: Use the radioactive decay formula
The number of particles remaining (N) after time (t) can be calculated using the formula:
\[ N = N_0 e^{-\lambda t} \]
Where \(N_0\) is the initial number of particles and \(t\) is the time elapsed. In this case, we want to find N after 10 seconds, so we substitute \(t = 10\) seconds:
\[ N = N_0 e^{-\lambda \cdot 10} \]

### Step 3: Substitute the decay constant into the equation
Substituting the value of λ from Step 1 into the equation:
\[ N = N_0 e^{-\left(\frac{\ln(2)}{60}\right) \cdot 10} \]
This simplifies to:
\[ N = N_0 e^{-\frac{10 \ln(2)}{60}} = N_0 e^{-\frac{\ln(2)}{6}} = N_0 \left(2^{-\frac{1}{6}}\right) \]

### Step 4: Calculate the number of particles that have decayed
The number of particles that have decayed (D) is given by:
\[ D = N_0 - N \]
Substituting the expression for N:
\[ D = N_0 - N_0 \left(2^{-\frac{1}{6}}\right) = N_0 \left(1 - 2^{-\frac{1}{6}}\right) \]

### Step 5: Calculate the fraction of particles that have decayed
The fraction of particles that have decayed (f) is given by:
\[ f = \frac{D}{N_0} = 1 - 2^{-\frac{1}{6}} \]

### Final Result
Thus, the fraction of particles that will decay at the end of 10 seconds is:
\[ f = 1 - 2^{-\frac{1}{6}} \]