Radioactive decay is a first-order reaction. Berkelium-250, a radioactive metallic element used in the synthesis of heavier elements, has a half-life of 3.217 hours. Determine the fraction of berkelium-250 that remains after four half-lives.

Plutonium-239 (atomic weight = 239.0522 amu, t_1/2 = 24110 y) is produced in a fast breeder reactor. Uranium-238 absorbs a neutron and then undergoes two consecutive beta decays to yield ²³⁹Pu. What is the activity of 245 mg ²³⁹Pu in Bq and Ci units?

Europium-152 has the potential to be used in external beam radiotherapy but is not yet developed for clinical use. Europium-152 has a half-life of 13.5 years. Assuming that europium also needs to be replaced when its activity drops to 75% of the original sample like in cobalt-60 machines, determine when will it be replaced if the machine is developed on January 2030.

The initial decay rate of a sample of Ac-224 is 2.50×10⁵/s. If the half-life of Ac-224 is 2.78 hours, calculate the time it will take for the decay rate to fall to 1.50×10³/s.

The following graph shows the radioactive decay of an unknown isotope. Using the information presented in the graph, determine the half-life of the unknown isotope.