The correct answer is (4) \(\frac{1}{16}\).
The concept of half-life is used to describe the time it takes for half of a substance to decay or transform. The formula for calculating the remaining fraction after a certain number of half-lives is given by:
\[ \text{Remaining fraction} = \left( \frac{1}{2} \right)^{\text{Number of half-lives}} \]
In this case, the isotope \( _{19}K^{42} \) has a half-life of 12 hours. To find the fraction remaining after 48 hours, we need to determine how many half-lives have passed in that time period. We can do this by dividing the total time (48 hours) by the half-life (12 hours):
\[ \text{Number of half-lives} = \frac{\text{Total time}}{\text{Half-life}} = \frac{48 \, \text{hours}}{12 \, \text{hours/half-life}} = 4 \, \text{half-lives} \]
Now, we can use the formula to find the remaining fraction:
\[ \text{Remaining fraction} = \left( \frac{1}{2} \right)^4 \]
\[ \text{Remaining fraction} = \frac{1}{2^4} \]
\[ \text{Remaining fraction} = \frac{1}{16} \]
So, after 48 hours, only \(\frac{1}{16}\) of the initial concentration of \( _{19}K^{42} \) remains. This means that over the course of four half-lives, the original amount has undergone multiple cycles of halving, leading to a significant reduction in quantity. |