Half-life of a substance is `10` years. In what time, it becomes `(1)/(4) th` part of the initial amount ?

Options

A. 15 yr
B. 20 yr
C. 25 yr
D. 30 yr

Explanation

To solve the problem of finding the time it takes for a substance to become one-fourth of its initial amount given its half-life, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Half-Life**:
The half-life (t_half) of a substance is the time required for half of the substance to decay. In this case, t_half = 10 years.

2. **Initial Amount**:
Let's denote the initial amount of the substance as \( N_0 \).

3. **Final Amount**:
We want to find the time when the remaining amount \( N_t \) is \( \frac{1}{4} N_0 \).

4. **Using the Exponential Decay Formula**:
The amount remaining after time \( t \) can be expressed as:
\[ N_t = N_0 e^{-\lambda t} \]
where \( \lambda \) is the decay constant.

5. **Setting Up the Equation**:
We set \( N_t = \frac{1}{4} N_0 \):
\[ \frac{1}{4} N_0 = N_0 e^{-\lambda t} \]

6. **Canceling \( N_0 \)**:
Dividing both sides by \( N_0 \) (assuming \( N_0 \neq 0 \)):
\[ \frac{1}{4} = e^{-\lambda t} \]

7. **Taking Natural Logarithm**:
Taking the natural logarithm of both sides gives:
\[ \ln\left(\frac{1}{4}\right) = -\lambda t \]

8. **Expressing \( \frac{1}{4} \)**:
We can rewrite \( \frac{1}{4} \) as \( 2^{-2} \):
\[ \ln(2^{-2}) = -\lambda t \]
This simplifies to:
\[ -2 \ln(2) = -\lambda t \]
or
\[ 2 \ln(2) = \lambda t \]

9. **Finding \( \lambda \)**:
We know that the half-life is related to the decay constant \( \lambda \) by:
\[ t_{half} = \frac{\ln(2)}{\lambda} \]
Rearranging gives:
\[ \lambda = \frac{\ln(2)}{t_{half}} = \frac{\ln(2)}{10 \text{ years}} \]

10. **Substituting \( \lambda \) Back**:
Substituting \( \lambda \) into the equation \( 2 \ln(2) = \lambda t \):
\[ 2 \ln(2) = \left(\frac{\ln(2)}{10}\right) t \]

11. **Solving for \( t \)**:
Multiplying both sides by 10:
\[ 20 \ln(2) = \ln(2) t \]
Dividing both sides by \( \ln(2) \) (assuming \( \ln(2) \neq 0 \)):
\[ t = 20 \text{ years} \]

### Final Answer:
The time it takes for the substance to become one-fourth of its initial amount is **20 years**.