The half-life of a radioactive substance against `alpha-`decay is `1.2 xx 10^7 s`. What is the decay rate for `4 xx 10^15` atoms of the substance ?

Options

A. `2.3xx10^8` atom/s
B. `3.2xx10^8` atom/s
C. `2.3xx10^11` atom/s
D. `3.2xx10^11` atom/s

Answer

To solve the problem, we need to calculate the decay rate of a radioactive substance given its half-life and the number of atoms. Here’s a step-by-step solution:

### Step 1: Understand the relationship between half-life and decay constant
The half-life (\( t_{1/2} \)) of a radioactive substance is related to the decay constant (\( \lambda \)) by the formula:
\[ t_{1/2} = \frac{\ln(2)}{\lambda} \]
where \( \ln(2) \approx 0.693 \).

### Step 2: Rearrange the formula to find the decay constant
We can rearrange the formula to solve for \( \lambda \):
\[ \lambda = \frac{\ln(2)}{t_{1/2}} \]

### Step 3: Substitute the given half-life into the equation
Given that the half-life \( t_{1/2} = 1.2 \times 10^7 \) seconds, we substitute this value into the equation:
\[ \lambda = \frac{0.693}{1.2 \times 10^7} \]

### Step 4: Calculate the decay constant
Now, we perform the calculation:
\[ \lambda \approx \frac{0.693}{1.2 \times 10^7} \approx 5.775 \times 10^{-8} \text{ s}^{-1} \]

### Step 5: Use the decay rate formula
The decay rate (\( \frac{dn}{dt} \)) is given by the formula:
\[ \frac{dn}{dt} = -\lambda n \]
where \( n \) is the number of atoms.

### Step 6: Substitute the values of \( \lambda \) and \( n \)
Given \( n = 4 \times 10^{15} \) atoms, we substitute \( \lambda \) and \( n \) into the decay rate formula:
\[ \frac{dn}{dt} = -\left(5.775 \times 10^{-8}\right)(4 \times 10^{15}) \]

### Step 7: Calculate the decay rate
Now, we perform the multiplication:
\[ \frac{dn}{dt} \approx -2.31 \times 10^{8} \text{ atoms/s} \]

### Step 8: Final result
The decay rate is approximately:
\[ \frac{dn}{dt} \approx -2.3 \times 10^{8} \text{ atoms/s} \]
The negative sign indicates a decrease in the number of atoms over time.

### Summary
The decay rate for \( 4 \times 10^{15} \) atoms of the substance is approximately \( 2.3 \times 10^{8} \) atoms per second.

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