Final answer:
To calculate how long it takes for 90% of the sample of (_90^234)Th to decay, we use the half-life of 24 days and the formula for exponential decay. Approximately 78.26 days are required for 90% of the isotope to decay to (_91^234)Pa.
Step-by-step explanation:
To calculate how long it takes for 90% of a sample of isotope (_90^234)Th to decay to (_91^234)Pa, we need to understand the concept of half-life. The half-life of (_90^234)Th is 24 days, meaning that every 24 days, half of the remaining sample will decay. To decay 90%, we need to reach a point where only 10% is left un-decayed.
To find the appropriate number of half-lives that correspond to 90% decay, we can use the formula:
N(t) = N0 * (1/2)^(t/T)
Where N(t) is the remaining amount, N0 is the initial amount, t is the time in days, and T is the half-life in days.
For 90% decay, N(t) = 0.10 * N0:
0.10 = (1/2)^(t/24)
We need to solve for t, the time in days:
t/24 = log(0.10) / log(0.5)
t = 24 * log(0.10) / log(0.5)
Calculating the time yields approximately 78.26 days for 90% of the sample to decay.