How can activity increase with radioactive decay?

You are correct that radioactive
materials, by definition, do decay. However, some radionuclides decay to
species that are themselves radioactive. Indeed, a parent radionuclide may
decay through numerous radioactive species, often referred to as radioactive
progeny, that comprise a long chain that finally terminates in a stable end
product. Depleted uranium is mostly made up of a long-lived radioisotope of
uranium, ²³⁸U. The ²³⁸U has a half-life of about 4.5
billion years. As it decays by alpha particle emission it yields a second
radionuclide, ²³⁴Th, which has a 24-day half-life and decays by
emitting beta radiation to produce a radionuclide referred to as ^234mPa,
which has a short half-life of a bit more than one minute, and decays by beta
emission to the ²³⁴U, a radionuclide with a half-life of 2.47 x 10⁵
years.

When depleted uranium (DU) is newly
prepared, the progeny through ^234mPa grow into the DU fairly quickly,
and within four or five months the activity (decay rate) of each of these
progeny is equal to the decay rate of the ²³⁸U. Because the ²³⁴U
has quite a long half-life, it acts rather like a time stop in the chain so that
apparent total activity of the DU does not appear to change noticeably after four
or five months have elapsed since preparation. So if you assess the DU 5 or
10 years after it has been prepared, the radioactive constituents do not appear
significantly different from what they were at five months. If one is concerned
about much longer intervals of time, however—e.g., many thousands of years—the
picture changes as sufficient time passes that significant decays of ²³⁴U
have occurred, and these decays lead to other shorter-lived progeny in the long
chain. There are a total of 10 major plus three minor radionuclides in the
chain beyond ²³⁴U. These include some well-known nuclides such as
radium, ²²⁶Ra, and radon, ²²²Rn, in addition to isotopes
of thorium, polonium, lead, and bismuth. It would require about 1.25 million
years following initial preparation of DU for all of the radionuclides in the
chain to reach a state of equilibrium in which each of the progeny has the same
activity as the original ²³⁸U. Over this 1.25 million years,
however, the original activity of the ²³⁸U would have decreased by
only 0.0192 percent. Thus while the ²³⁸U decreased by a tiny amount,
the total activity that includes all 13 of the major progeny would have
increased by 1,300 percent.

I hope this clarifies some of the
statements to which you have referred.

George Chabot, PhD, CHP