The half-life of `K^40` is `1.30xx10^9` y. A sample if `1.00g` of pure `KCI` gives `"160 dps"`. The percentage fraction of `K^40` present in the sample is:

Answer

To find the percentage fraction of \( K^{40} \) present in a sample of \( KCl \), we can follow these steps:

### Step 1: Calculate the decay constant \( \lambda \)
The decay constant \( \lambda \) is given by the formula:
\[ \lambda = \frac{0.693}{T_{1/2}} \]
where \( T_{1/2} \) is the half-life. Given that the half-life of \( K^{40} \) is \( 1.30 \times 10^9 \) years, we need to convert this into seconds for our calculations.

1 year = \( 365 \times 24 \times 3600 \) seconds.

Calculating \( T_{1/2} \) in seconds:
\[ T_{1/2} = 1.30 \times 10^9 \times 365 \times 24 \times 3600 \approx 4.1 \times 10^{16} \text{ seconds} \]

Now substituting this value into the decay constant formula:
\[ \lambda = \frac{0.693}{4.1 \times 10^{16}} \approx 1.69 \times 10^{-17} \text{ s}^{-1} \]

### Step 2: Relate the decay rate to the number of atoms \( N \)
The decay rate \( R \) is given as \( 160 \text{ dps} \) (decays per second). The relationship between the decay rate, decay constant, and the number of radioactive atoms is given by:
\[ R = \lambda N \]
Rearranging this to find \( N \):
\[ N = \frac{R}{\lambda} = \frac{160}{1.69 \times 10^{-17}} \approx 9.465 \times 10^{18} \text{ atoms} \]

### Step 3: Calculate the number of moles of \( KCl \)
The molar mass of \( KCl \) is approximately \( 74.5 \text{ g/mol} \). The number of moles in \( 1 \text{ g} \) of \( KCl \) is:
\[ \text{Number of moles} = \frac{1 \text{ g}}{74.5 \text{ g/mol}} \approx 0.0134 \text{ moles} \]

### Step 4: Calculate the total number of atoms in the sample
Using Avogadro's number \( N_A = 6.022 \times 10^{23} \text{ atoms/mol} \), the total number of atoms in \( 0.0134 \) moles of \( KCl \) is:
\[ \text{Total atoms} = 0.0134 \times 6.022 \times 10^{23} \approx 8.06 \times 10^{21} \text{ atoms} \]

### Step 5: Calculate the fraction of \( K^{40} \)
The fraction of \( K^{40} \) in the sample is given by:
\[ \text{Fraction of } K^{40} = \frac{N}{\text{Total atoms}} = \frac{9.465 \times 10^{18}}{8.06 \times 10^{21}} \approx 0.00117 \]

### Step 6: Convert to percentage
To find the percentage fraction:
\[ \text{Percentage fraction} = 0.00117 \times 100 \approx 0.117\% \]

### Final Answer
The percentage fraction of \( K^{40} \) present in the sample is approximately \( 0.12\% \).

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