The half-life of radium is `1620 years` and its atomic weight is `226`. The number of atoms that will decay from its `1 g` sample per second will be .

Options

A. `3.6 xx 10^(10)`
B. `3.6 xx 10^(12)`
C. `3.1 xx 10^(15)`
D. `31.1 xx 10^(15)`

Explanation

To solve the problem of how many atoms of radium will decay from a 1 g sample per second, we will follow these steps:

### Step 1: Calculate the number of atoms in 1 g of radium.
To find the number of atoms in 1 gram of radium, we use the formula:

\[ N_0 = \frac{m}{M} \times N_A \]

Where:
- \( m \) = mass of the sample = 1 g
- \( M \) = molar mass of radium = 226 g/mol
- \( N_A \) = Avogadro's number = \( 6.022 \times 10^{23} \) atoms/mol

Substituting the values:

\[ N_0 = \frac{1 \, \text{g}}{226 \, \text{g/mol}} \times 6.022 \times 10^{23} \, \text{atoms/mol} \]

Calculating this gives:

\[ N_0 \approx 2.66 \times 10^{21} \, \text{atoms} \]

### Step 2: Calculate the decay constant (\( \lambda \)).
The decay constant can be calculated using the half-life formula:

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

Where:
- \( T_{1/2} \) = half-life of radium = 1620 years

First, we need to convert the half-life from years to seconds:

\[ T_{1/2} = 1620 \, \text{years} \times 365 \, \text{days/year} \times 24 \, \text{hours/day} \times 3600 \, \text{seconds/hour} \]

Calculating this gives:

\[ T_{1/2} \approx 5.126 \times 10^{10} \, \text{seconds} \]

Now, substituting this into the decay constant formula:

\[ \lambda = \frac{0.693}{5.126 \times 10^{10} \, \text{s}} \approx 1.35 \times 10^{-11} \, \text{s}^{-1} \]

### Step 3: Calculate the activity (\( R \)).
The activity, or the number of decays per second, can be calculated using the formula:

\[ R = \lambda N_0 \]

Substituting the values we have:

\[ R = (1.35 \times 10^{-11} \, \text{s}^{-1}) \times (2.66 \times 10^{21} \, \text{atoms}) \]

Calculating this gives:

\[ R \approx 3.61 \times 10^{10} \, \text{decays/second} \]

### Final Answer:
The number of atoms that will decay from a 1 g sample of radium per second is approximately \( 3.61 \times 10^{10} \) decays/second.

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