A radioactive substance contains `10^15` atoms and has an activity of `6.0xx10^11` Bq. What is its half-life?

Explanation

To find the half-life of a radioactive substance, we can use the relationship between activity (A), decay constant (λ), and the number of nuclei (N). The steps to solve the problem are as follows:

### Step 1: Understand the relationship between activity, decay constant, and number of nuclei
The activity (A) of a radioactive substance is given by the formula:
\[ A = \lambda N \]
where:
- \( A \) is the activity in Becquerels (Bq),
- \( \lambda \) is the decay constant (in s\(^{-1}\)),
- \( N \) is the number of radioactive nuclei.

### Step 2: Rearrange the formula to find the decay constant
From the formula, we can express the decay constant \( \lambda \) as:
\[ \lambda = \frac{A}{N} \]

### Step 3: Substitute the given values
We are given:
- \( A = 6.0 \times 10^{11} \) Bq,
- \( N = 10^{15} \) atoms.

Substituting these values into the equation for \( \lambda \):
\[ \lambda = \frac{6.0 \times 10^{11}}{10^{15}} \]

### Step 4: Calculate the decay constant
Perform the division:
\[ \lambda = 6.0 \times 10^{-4} \, \text{s}^{-1} \]

### Step 5: Use the decay constant to find the half-life
The half-life (\( t_{1/2} \)) is related to the decay constant by the formula:
\[ t_{1/2} = \frac{\ln(2)}{\lambda} \]
where \( \ln(2) \approx 0.693 \).

### Step 6: Substitute the decay constant into the half-life formula
Now substitute \( \lambda \) into the half-life formula:
\[ t_{1/2} = \frac{0.693}{6.0 \times 10^{-4}} \]

### Step 7: Calculate the half-life
Perform the calculation:
\[ t_{1/2} \approx \frac{0.693}{6.0 \times 10^{-4}} \approx 1155 \, \text{s} \]

### Step 8: Round the result
Rounding to two significant figures:
\[ t_{1/2} \approx 1.16 \times 10^{3} \, \text{s} \]

Thus, the half-life of the radioactive substance is approximately \( 1.16 \times 10^{3} \) seconds.

---