Final answer:
To model radioactivity level over time for a radioactive isotope like fluorine-18, an exponential decay function is used, where the amount remaining equals the initial amount times e to the power of negative decay constant times time. The decay constant (k) is computed from the half-life (t1/2) using the relationship k = 0.693 / t1/2, and the radioactivity level function is the initial dosage times e to the power of negative k times t.
Step-by-step explanation:
To model the radioactivity level of a radioactive isotope over time, we use an exponential decay function. The equation for the remaining amount of a radioactive substance, which is a function of time, is given by:
amount remaining = (amount initial) × e−(0.693 × t) / t1/2
where e is the base of the natural logarithm, t is the time elapsed in the same units as the half-life t1/2, and 0.693 is a constant that arises from the natural logarithm of 2 when solving the exponential decay function for half-life.
To find the decay constant (k), we use the relationship:
k = 0.693 / t1/2
Once we have the value of k, we can write the function that models the radioactivity level as:
radioactivity level = (initial dosage) × e−k×t
For a given half-life and time t, we can calculate the radioactivity level at any time t using this function. The half-life must be given or determined in order to perform these calculations. When working with this function, we should round the value of k to five decimal places for final calculations while not rounding intermediate values to prevent accumulation of rounding errors.