To solve this problem, let us first calculate for the rate constant k using the half life formula:
t1/2 = ln 2 / k
where t1/2 = half life period = 24,000 years, therefore k is:
k = ln 2 / 24,000
k = 2.89 x 10^-5 / yr
Now we use the rate equation:
A = Ao e^(-k t)
where,
A = mass of Plutonium-239 after number of years
Ao = initial mass of Plutonium-239
t = number of years
A. t = 12,000 years, find A
A = 100g e^(- 2.89 x 10^-5 * 12,000)
A = 70.7 g
B. t = 24,000 years, find A
A = 100g e^(- 2.89 x 10^-5 * 24,000)
A = 50 g
C. t = 96,000 years, find A
A = 100g e^(- 2.89 x 10^-5 * 96,000)
A = 6.24 g