One gram of `.^(226)Ra` has an activity of nearly 1 Ci. The half life of `.^(226)Ra` is,

Options

A. 1582 yrs
B. 12.5 hrs
C. 140 days
D. `4.5 xx 10^(9)` yrs

Explanation

To find the half-life of \(^{226}\text{Ra}\) given that 1 gram has an activity of nearly 1 Ci, we can follow these steps:

### Step 1: Understand the relationship between activity, decay constant, and number of atoms
The activity \(A\) of a radioactive substance is given by the formula:
\[ A = \lambda N \]
where:
- \(A\) is the activity in disintegrations per second (Becquerels, Bq),
- \(\lambda\) is the decay constant,
- \(N\) is the number of radioactive atoms present.

### Step 2: Convert the activity from Ci to Bq
1 Ci (Curie) is equivalent to \(3.7 \times 10^{10}\) Bq. Therefore, the activity of 1 gram of \(^{226}\text{Ra}\) in Bq is:
\[ A = 3.7 \times 10^{10} \text{ Bq} \]

### Step 3: Calculate the number of atoms \(N\)
To find \(N\), we need to know the molar mass of \(^{226}\text{Ra}\), which is approximately 226 g/mol. The number of moles in 1 gram of \(^{226}\text{Ra}\) is:
\[ \text{Number of moles} = \frac{1 \text{ g}}{226 \text{ g/mol}} = \frac{1}{226} \text{ mol} \]
Using Avogadro's number (\(6.022 \times 10^{23}\) atoms/mol), the number of atoms \(N\) is:
\[ N = \left(\frac{1}{226}\right) \times (6.022 \times 10^{23}) \approx 2.67 \times 10^{21} \text{ atoms} \]

### Step 4: Relate decay constant \(\lambda\) to half-life
The decay constant \(\lambda\) is related to the half-life (\(t_{1/2}\)) by the equation:
\[ \lambda = \frac{0.693}{t_{1/2}} \]

### Step 5: Substitute \(N\) and \(\lambda\) into the activity equation
Substituting \(N\) and \(\lambda\) into the activity equation:
\[ 3.7 \times 10^{10} = \left(\frac{0.693}{t_{1/2}}\right) \times (2.67 \times 10^{21}) \]

### Step 6: Solve for half-life \(t_{1/2}\)
Rearranging the equation to solve for \(t_{1/2}\):
\[ t_{1/2} = \frac{0.693 \times (2.67 \times 10^{21})}{3.7 \times 10^{10}} \]
Calculating this gives:
\[ t_{1/2} \approx \frac{1.85 \times 10^{11}}{3.7 \times 10^{10}} \approx 5.0 \times 10^{0} \text{ seconds} \approx 1582 \text{ years} \]

### Final Answer
The half-life of \(^{226}\text{Ra}\) is approximately **1582 years**.

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