# Shorter half-life and therefore very radioactive -- why?

• B
Summary:
In reading through The Physics of Energy, the textbook describes the decay chain of U-238:
"The longest half-life of any descendent in the chain is less 1 million years. Many half-lives are much shorter, making those nuclides very radioactive."

In reading through The Physics of Energy, the textbook describes the decay chain of U-238:
"The longest half-life of any descendent in the chain is less 1 million years. Many half-lives are much shorter, making those nuclides very radioactive."

Relative to the time available for particle emissions from the long-lived parent radionuclide (U-238), the short-lived descendants have much less time to perform all the necessary particle emissions. And therefore, the short-lived radionuclides will have much higher radioactivity, as they will be emitting particles more frequently.
(Am I correct?)

However, quantitatively, I'm stuck.

The amount of radioactivity (Bq) must be related to the number of disintegrations per gram per second.
But is there an equation relating these quantities?

dRic2
Gold Member
The activity (##A##), which is the number of disintegration per unit time, is given by
$$A = \lambda N$$
where ##\lambda## is the decay-constant and ##N## is the number of particles in the sample. If you assume that the number of particles ##N## in the sample does not change significantly during the period of time in which you measure the radioactivity, then you see that the higher the ##\lambda##, the higher the activity (number of disintegrations). It also turns out that the decay-constant ##\lambda## and the half-life ##\tau_{1/2}## are related by:
$$\tau_{1/2} = \frac {\ln 2} {\lambda}$$.
To summarize, small half-life -> big decay-constant ->big number of disintegrations per seconds = high activity.

• artis, Astronuc, Keith_McClary and 2 others
Thank you.

• Keith_McClary, dRic2 and berkeman
DrDu