Activity of radioactive element decreased to one third of original activity `R_0` in `9` years. After further `9` years, its activity will be

Options

A. `I_0`
B. `2/3 I_0`
C. `I_(0)//9`
D. `I_0//3`

Answer

To solve the problem step by step, we will analyze the decay of the radioactive element's activity over time.

### Step 1: Understand the given information
The initial activity of the radioactive element is denoted as \( R_0 \). After 9 years, the activity decreases to one third of its original activity, which can be expressed as:
\[ R = \frac{R_0}{3} \]

### Step 2: Use the radioactive decay formula
The activity of a radioactive substance can be expressed using the formula:
\[ R = R_0 e^{-\lambda t} \]
where:
- \( R \) is the remaining activity after time \( t \),
- \( R_0 \) is the initial activity,
- \( \lambda \) is the decay constant,
- \( t \) is the time elapsed.

### Step 3: Substitute the known values for the first 9 years
For the first 9 years, we can substitute \( R \) and \( t \) into the decay formula:
\[ \frac{R_0}{3} = R_0 e^{-9\lambda} \]

### Step 4: Simplify the equation
Dividing both sides by \( R_0 \) (assuming \( R_0 \neq 0 \)):
\[ \frac{1}{3} = e^{-9\lambda} \]

### Step 5: Take the natural logarithm of both sides
Taking the natural logarithm on both sides gives:
\[ \ln\left(\frac{1}{3}\right) = -9\lambda \]

### Step 6: Solve for the decay constant \( \lambda \)
Rearranging the equation:
\[ \lambda = -\frac{1}{9} \ln\left(\frac{1}{3}\right) \]

### Step 7: Calculate the activity after a total of 18 years
Now, we need to find the activity after a total of 18 years. We will use the same decay formula:
\[ R = R_0 e^{-18\lambda} \]

### Step 8: Substitute \( \lambda \) into the equation
We can express \( e^{-18\lambda} \) in terms of \( e^{-9\lambda} \):
\[ e^{-18\lambda} = (e^{-9\lambda})^2 \]
From Step 5, we know that \( e^{-9\lambda} = \frac{1}{3} \):
\[ e^{-18\lambda} = \left(\frac{1}{3}\right)^2 = \frac{1}{9} \]

### Step 9: Substitute back into the activity equation
Now substituting back into the activity equation:
\[ R = R_0 \cdot \frac{1}{9} \]

### Step 10: Final result
Thus, the activity after a total of 18 years will be:
\[ R = \frac{R_0}{9} \]

### Conclusion
The activity of the radioactive element after a total of 18 years will be \( \frac{R_0}{9} \).

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