At a certain instant a piece of radioactive material contains `10^(12)` atoms. The half life of material is 30 days. Calculate the no. of disintegrations in one second

Answer

To solve the problem step by step, we will follow these calculations:

### Step 1: Identify the given values
- Number of atoms (N₀) = \(10^{12}\)
- Half-life (T₁/₂) = 30 days

### Step 2: Convert half-life from days to seconds
Since we need to calculate the disintegrations per second, we must convert the half-life from days to seconds.

\[ \text{T}_{1/2} = 30 \text{ days} \times 24 \text{ hours/day} \times 60 \text{ minutes/hour} \times 60 \text{ seconds/minute} \]

Calculating this gives:

\[ \text{T}_{1/2} = 30 \times 24 \times 60 \times 60 = 2,592,000 \text{ seconds} \]

### Step 3: Calculate the decay constant (λ)
The decay constant (λ) can be calculated using the formula:

\[ \lambda = \frac{\ln(2)}{T_{1/2}} \]

Where \(\ln(2) \approx 0.693\). Substituting the values:

\[ \lambda = \frac{0.693}{2,592,000} \]

Calculating this gives:

\[ \lambda \approx 2.676 \times 10^{-7} \text{ s}^{-1} \]

### Step 4: Calculate the rate of disintegration (R)
The rate of disintegration (R) can be calculated using the formula:

\[ R = \lambda \cdot N \]

Where N is the number of atoms. Substituting the values:

\[ R = (2.676 \times 10^{-7} \text{ s}^{-1}) \cdot (10^{12}) \]

Calculating this gives:

\[ R \approx 2.676 \times 10^{5} \text{ disintegrations/second} \]

### Final Answer
The number of disintegrations in one second is approximately \(2.676 \times 10^{5}\) disintegrations/second.

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