Final answer:
The half-life of Californium-253 (Cf-253) is approximately 5.29 days.
Step-by-step explanation:
The half-life of an isotope is the time it takes for half of a sample to decay. In this case, we can use the information given to calculate the half-life of Californium-253 (Cf-253). We know that after 89 days, 46.875 grams of Cf-253 remain. We can set up an equation using the half-life formula:
Remaining mass = Initial mass * (1/2)^(t/half-life)
Substituting the given values:
46.875 g = 1.5 kg * (1/2)^(89/half-life)
We can solve this equation to find the value of the half-life:
46.875 g / 1500 g = (1/2)^(89/half-life)
0.03125 = (1/2)^(89/half-life)
To simplify, we can take the logarithm of both sides:
log(0.03125) = (89/half-life) * log(1/2)
Using logarithm properties:
log(0.03125) = -5 = (89/half-life) * -0.301
Solving for half-life:
half-life = 89 * (-0.301) / -5
half-life ≈ 5.29
Therefore, the half-life of Californium-253 is approximately 5.29 days.