The rate of disintegration was observed to be `10^(17)` disintegrations per sec when its half life period is `1445` years. The original number of particles are.

Options

A. `8.9xx10^(27)`
B. `6.6xx10^(27)`
C. `1.4xx10^(16)`
D. `1.2xx10^(17)`

Answer

To find the original number of particles given the rate of disintegration and the half-life period, we can follow these steps:

### Step 1: Convert Half-Life from Years to Seconds
The half-life \( t_{1/2} \) is given as 1445 years. We need to convert this into seconds.

\[ t_{1/2} = 1445 \text{ years} \times 365 \text{ days/year} \times 24 \text{ hours/day} \times 60 \text{ minutes/hour} \times 60 \text{ seconds/minute} \]

Calculating this gives:

\[ t_{1/2} = 1445 \times 365 \times 24 \times 60 \times 60 \approx 4.55 \times 10^{10} \text{ seconds} \]

### Step 2: Calculate the Decay Constant \( \lambda \)
The decay constant \( \lambda \) can be calculated using the formula:

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

Where \( \ln(2) \approx 0.693 \). Substituting the value of \( t_{1/2} \):

\[ \lambda = \frac{0.693}{4.55 \times 10^{10}} \approx 1.52 \times 10^{-11} \text{ s}^{-1} \]

### Step 3: Use the Rate of Disintegration to Find Original Number of Particles
The rate of disintegration \( \frac{dn}{dt} \) is given as \( 10^{17} \) disintegrations per second. The relationship between the rate of disintegration, decay constant, and the number of particles \( N \) is given by:

\[ \frac{dn}{dt} = \lambda N \]

Rearranging this to solve for \( N \):

\[ N = \frac{\frac{dn}{dt}}{\lambda} \]

Substituting the known values:

\[ N = \frac{10^{17}}{1.52 \times 10^{-11}} \approx 6.58 \times 10^{27} \]

### Step 4: Final Result
Thus, the original number of particles is approximately:

\[ N \approx 6.58 \times 10^{27} \]

### Summary
The original number of particles is \( 6.58 \times 10^{27} \).

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