The half -life of cobalt-60 is 5.25 years. How long after a new sample is delivered, will the activity have decreased to about one third `(1//3)` of its original value-

Options

A. 5.25 years
B. 8.3 years
C. 10.50 years
D. 15.75 years

Answer

To solve the problem of how long it will take for the activity of cobalt-60 to decrease to one third of its original value, we can follow these steps:

### Step 1: Understand the half-life and decay constant
The half-life of cobalt-60 is given as 5.25 years. The decay constant (λ) can be calculated using the formula:
\[ \lambda = \frac{\ln(2)}{t_{1/2}} \]
where \(t_{1/2}\) is the half-life.

### Step 2: Calculate the decay constant
Substituting the half-life into the formula:
\[ \lambda = \frac{\ln(2)}{5.25} \]

### Step 3: Set up the decay equation
The activity of a radioactive substance decreases according to the equation:
\[ N = N_0 e^{-\lambda t} \]
where:
- \(N_0\) is the initial activity,
- \(N\) is the final activity,
- \(t\) is the time elapsed.

### Step 4: Substitute the final activity
We want to find the time when the activity decreases to one third of its original value:
\[ N = \frac{N_0}{3} \]
Substituting this into the decay equation gives:
\[ \frac{N_0}{3} = N_0 e^{-\lambda t} \]

### Step 5: Simplify the equation
Dividing both sides by \(N_0\) (assuming \(N_0 \neq 0\)):
\[ \frac{1}{3} = e^{-\lambda t} \]

### Step 6: Take the natural logarithm of both sides
Taking the natural logarithm:
\[ \ln\left(\frac{1}{3}\right) = -\lambda t \]
This can be rewritten as:
\[ -\ln(3) = -\lambda t \]
or
\[ t = \frac{\ln(3)}{\lambda} \]

### Step 7: Substitute the decay constant back into the equation
Now substitute \(\lambda = \frac{\ln(2)}{5.25}\) into the equation:
\[ t = \frac{\ln(3)}{\frac{\ln(2)}{5.25}} = \frac{5.25 \cdot \ln(3)}{\ln(2)} \]

### Step 8: Calculate the time
Using the values of \(\ln(3) \approx 1.0986\) and \(\ln(2) \approx 0.6931\):
\[ t \approx \frac{5.25 \cdot 1.0986}{0.6931} \approx 8.3 \text{ years} \]

### Final Answer
Thus, the time after which the activity of cobalt-60 will have decreased to about one third of its original value is approximately **8.3 years**.

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