The number of atoms of a radioactive substance of half-life T is `N_(0)` at t = 0. The time necessary to decay from `N_(0)//2` atoms to `N_(0)//10` atoms will be

Options

A. `(5)/(2)T`
B. T 1n 5
C. `T 1n ((5)/(2))`
D. `T (1n5)/(1n2)`

Answer

To solve the problem of determining the time necessary for a radioactive substance to decay from \( \frac{N_0}{2} \) atoms to \( \frac{N_0}{10} \) atoms, we can follow these steps:

### Step-by-Step Solution:

1. **Understand the decay law**: The number of atoms remaining at time \( t \) is given by the equation:
\[ N_t = N_0 e^{-\lambda t} \]
where \( N_t \) is the number of atoms at time \( t \), \( N_0 \) is the initial number of atoms, and \( \lambda \) is the decay constant.

2. **Determine the time \( t_1 \) for decay to \( \frac{N_0}{2} \)**:
- Set \( N_t = \frac{N_0}{2} \):
\[ \frac{N_0}{2} = N_0 e^{-\lambda t_1} \]
- Dividing both sides by \( N_0 \):
\[ \frac{1}{2} = e^{-\lambda t_1} \]
- Taking the natural logarithm of both sides:
\[ \ln\left(\frac{1}{2}\right) = -\lambda t_1 \]
- Therefore, we have:
\[ t_1 = \frac{\ln 2}{\lambda} \]

3. **Determine the time \( t_2 \) for decay to \( \frac{N_0}{10} \)**:
- Set \( N_t = \frac{N_0}{10} \):
\[ \frac{N_0}{10} = N_0 e^{-\lambda t_2} \]
- Dividing both sides by \( N_0 \):
\[ \frac{1}{10} = e^{-\lambda t_2} \]
- Taking the natural logarithm of both sides:
\[ \ln\left(\frac{1}{10}\right) = -\lambda t_2 \]
- Therefore, we have:
\[ t_2 = \frac{\ln 10}{\lambda} \]

4. **Calculate the time difference \( t_2 - t_1 \)**:
- Now we find the time necessary to decay from \( \frac{N_0}{2} \) to \( \frac{N_0}{10} \):
\[ t_2 - t_1 = \frac{\ln 10}{\lambda} - \frac{\ln 2}{\lambda} \]
- Factoring out \( \frac{1}{\lambda} \):
\[ t_2 - t_1 = \frac{1}{\lambda} (\ln 10 - \ln 2) \]
- Using the property of logarithms \( \ln a - \ln b = \ln\left(\frac{a}{b}\right) \):
\[ t_2 - t_1 = \frac{1}{\lambda} \ln\left(\frac{10}{2}\right) = \frac{1}{\lambda} \ln 5 \]

5. **Relate \( \lambda \) to half-life \( T \)**:
- The half-life \( T \) is given by:
\[ T = \frac{\ln 2}{\lambda} \implies \lambda = \frac{\ln 2}{T} \]
- Substituting \( \lambda \) into the equation for \( t_2 - t_1 \):
\[ t_2 - t_1 = \frac{T}{\ln 2} \ln 5 \]

### Final Answer:
Thus, the time necessary to decay from \( \frac{N_0}{2} \) atoms to \( \frac{N_0}{10} \) atoms is:
\[ t_2 - t_1 = T \frac{\ln 5}{\ln 2} \]