Final answer:
To determine how many years it would take for a 65.0 mg sample of isotope X to decay to 29.0 mg, given a half-life of 14.2 years, use the half-life decay formula, substituting the given values and solving for the total time of decay.
Step-by-step explanation:
The question involves the concept of half-life, which is the amount of time required for half of a radioactive isotope to decay. In the case of isotope X with a half-life of 14.2 years, we need to determine how many 14.2-year intervals are needed for a 65.0 mg sample to reduce to 29.0 mg. To find the number of half-lives that have passed, we can use the equation:
N = N0 (1/2)^(t/T)
Where N is the final amount of the substance, N0 is the initial amount, t is the total time, and T is the half-life of the substance.
By rearranging the formula and solving for t, we can determine the time it takes for the sample to decay from 65.0 mg to 29.0 mg. Thus, we calculate:
t = T log2(N0/N)
Substitute the given values:
t = 14.2 years * log2(65.0 mg/29.0 mg)
After computing logarithms and multiplication, we would get the final time required for the sample to reach the specified amount.