One gram of `Ra^(226)` has an activity of nearly `1 Ci`the half life of `Ra^(226)` is
a. 1500 years b. 300 years
c. 1582 years d. 200 years

Options

A. a. 1500 years
B. b. 300 years
C. c. 1582 years
D. d. 200 years

Explanation

To find the half-life of Radium-226 (Ra-226) 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 half-life
The activity (A) of a radioactive substance is given by the formula:
\[ A = \lambda \cdot N \]
where \( \lambda \) is the decay constant and \( N \) is the number of radioactive atoms present.

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

### Step 2: Calculate the number of atoms (N) in 1 gram of Ra-226
To find \( N \), we use the formula:
\[ N = \frac{\text{mass}}{\text{atomic weight}} \times N_A \]
where:
- mass = 1 gram
- atomic weight of Ra-226 = 226 g/mol
- \( N_A \) (Avogadro's number) = \( 6.022 \times 10^{23} \) atoms/mol

Calculating \( N \):
\[ N = \frac{1 \text{ g}}{226 \text{ g/mol}} \times 6.022 \times 10^{23} \text{ atoms/mol} \]
\[ N \approx 2.67 \times 10^{24} \text{ atoms} \]

### Step 3: Use the activity value
Given that the activity \( A \) is approximately 1 Ci, we convert this to disintegrations per second:
\[ 1 \text{ Ci} = 3.7 \times 10^{10} \text{ disintegrations/second} \]

### Step 4: Substitute values into the activity formula
Now we can substitute the values into the activity formula:
\[ 3.7 \times 10^{10} = \lambda \cdot (2.67 \times 10^{24}) \]

### Step 5: Solve for the decay constant (\( \lambda \))
Rearranging gives:
\[ \lambda = \frac{3.7 \times 10^{10}}{2.67 \times 10^{24}} \]
\[ \lambda \approx 1.38 \times 10^{-14} \text{ s}^{-1} \]

### Step 6: Calculate the half-life (\( t_{1/2} \))
Using the relationship between \( \lambda \) and \( t_{1/2} \):
\[ t_{1/2} = \frac{0.693}{\lambda} \]
Substituting for \( \lambda \):
\[ t_{1/2} = \frac{0.693}{1.38 \times 10^{-14}} \]
\[ t_{1/2} \approx 5.02 \times 10^{13} \text{ seconds} \]

### Step 7: Convert seconds to years
To convert seconds to years:
1. Divide by 60 to convert to minutes.
2. Divide by 60 to convert to hours.
3. Divide by 24 to convert to days.
4. Divide by 365 to convert to years.

Calculating:
\[ t_{1/2} \text{ in years} = \frac{5.02 \times 10^{13}}{60 \times 60 \times 24 \times 365} \approx 1582 \text{ years} \]

### Conclusion
The half-life of Ra-226 is approximately **1582 years**. Therefore, the correct answer is option **c. 1582 years**.

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