Radium has atomic weight 226 and half life of 1600 years. The number of disintegrations produced per second from one gram is

Options

A. a. `4.8 xx 10^(10)`
B. b. `3.7 xx 10^(10)`
C. c. `9.2 xx 10^(6)`
D. d. `3.7 xx 10^(8)`

Explanation

To find the number of disintegrations produced per second from one gram of radium, we will follow these steps:

### Step 1: Determine the decay constant (λ)
The decay constant (λ) can be calculated using the half-life (t½) of the radioactive substance. The formula to calculate λ is:

\[ \lambda = \frac{0.693}{t_{1/2}} \]

Given that the half-life of radium is 1600 years, we first need to convert this into seconds.

### Step 2: Convert half-life from years to seconds
1 year = 365 days (approximately)
1 day = 24 hours
1 hour = 3600 seconds

Thus,

\[ t_{1/2} = 1600 \text{ years} \times 365 \text{ days/year} \times 24 \text{ hours/day} \times 3600 \text{ seconds/hour} \]

Calculating this gives:

\[ t_{1/2} = 1600 \times 365 \times 24 \times 3600 \approx 5.04 \times 10^{10} \text{ seconds} \]

### Step 3: Calculate λ
Now substituting the half-life in seconds into the decay constant formula:

\[ \lambda = \frac{0.693}{5.04 \times 10^{10}} \approx 1.375 \times 10^{-11} \text{ s}^{-1} \]

### Step 4: Calculate the number of atoms (N) in 1 gram of radium
The number of atoms (N) in one gram can be calculated using Avogadro's number (6.022 x 10²³) and the atomic weight of radium (226 g/mol):

\[ N = \frac{6.022 \times 10^{23}}{226} \approx 2.66 \times 10^{21} \text{ atoms} \]

### Step 5: Calculate the disintegration rate (R)
The disintegration rate (R) can be calculated using the formula:

\[ R = \lambda \times N \]

Substituting the values we have:

\[ R = (1.375 \times 10^{-11} \text{ s}^{-1}) \times (2.66 \times 10^{21} \text{ atoms}) \approx 3.66 \times 10^{10} \text{ disintegrations/second} \]

### Step 6: Final answer
Rounding off, we find that the number of disintegrations produced per second from one gram of radium is approximately:

\[ \boxed{3.7 \times 10^{10} \text{ disintegrations/second}} \]