The half-life of a radioactive substance is 100 years, Calculate in how many years the activity will decay to 1/10th of its initial value.

Options

A. 332.3 years
B. 232.3 years
C. 432.3 years
D. 532.3 years

Answer

To solve the problem of how many years it will take for the activity of a radioactive substance to decay to 1/10th of its initial value, we can follow these steps:

### Step 1: Understand the relationship between half-life and decay constant
The half-life (T₁/₂) of a radioactive substance is the time required for half of the radioactive nuclei in a sample to decay. The decay constant (λ) is related to the half-life by the formula:
\[ \lambda = \frac{0.693}{T_{1/2}} \]
Given that the half-life is 100 years, we can calculate λ.

### Step 2: Calculate the decay constant (λ)
Using the half-life:
\[ \lambda = \frac{0.693}{100 \text{ years}} = 0.00693 \text{ years}^{-1} \]

### Step 3: Set up the equation for decay
The activity (A) of a radioactive substance at any time t can be expressed as:
\[ A(t) = A_0 e^{-\lambda t} \]
where \(A_0\) is the initial activity.

### Step 4: Set the activity to 1/10th of its initial value
We want to find the time t when the activity is 1/10th of its initial value:
\[ \frac{A_0}{10} = A_0 e^{-\lambda t} \]
Dividing both sides by \(A_0\) gives:
\[ \frac{1}{10} = e^{-\lambda t} \]

### Step 5: Take the natural logarithm of both sides
Taking the natural logarithm of both sides:
\[ \ln\left(\frac{1}{10}\right) = -\lambda t \]
This can be simplified to:
\[ -\ln(10) = -\lambda t \quad \Rightarrow \quad t = \frac{\ln(10)}{\lambda} \]

### Step 6: Substitute the value of λ
Now substituting the value of λ:
\[ t = \frac{\ln(10)}{0.00693} \]
Using the value of \(\ln(10) \approx 2.303\):
\[ t = \frac{2.303}{0.00693} \approx 332.2 \text{ years} \]

### Final Answer
Thus, the time it will take for the activity to decay to 1/10th of its initial value is approximately **332 years**.

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