Re: How was the half-life of radioactive elements first determined?

I am going to give a general answer to this one, rather than historical 
detail. The latter can be found in reference books.

Radioactive half-lives can be determined in several different ways. It 
depends mainly on which particular half-life is under consideration. There 
is an enormous range in the actual time-scale of radioactive decay -- from 
fractions of a second through to a million times the age of the earth!

For half lives in the region of an hour or two to a year or two, the 
problem is very readily solved: you simply get your radioactive source, and 
follow the decrease in its decay rate by checking it from time to time with 
a geiger counter.

For shorter half-lives, you must again obtain your data by simply following 
the decreasing decay rate. But this involves electronic counting with much 
more sophisticated equipment. Generally speaking, isotopes with very short 
half lives were produced and investigated much later than those with medium 
or very long half-lives.

For long half-lives, the approach must be quite a different one. U-238 (the 
dominant isotope of natural uranium) has a half life of about 4 billion 
years; one could not simply get hold of a uranium sample and wait for its 
radioactive decay rate to fall to half of the original value! But from 
geiger counter readings one can infer the actual number of uranium atoms 
decomposing per minute in a sample, and from a chemical analysis one can 
measure the total amount of uranium in the sample. These two results 
together allow the half-life to be calculated.

Finally, there is a different method again for isotopes that are 
intermediates in a long decay series. Ra-226 (radium) is a good example. 
Ra-226 is formed in radioactive decay of Th-230, which originally comes 
from U-238. But Ra-226 itself decays to Rn-222 (radon) and eventually to 
Pb-208 (lead). In a series of decays like this, all of the intermediates 
eventually reach a steady state, where the rate of radium production 
exactly equals the rate of its removal; in fact, you can infer equal rates 
for every reaction down the decay series. So the rate of Ra-226 decay must 
equal the rate of U-238 decay. This leads to the interesting result that

(concentration of Ra-226)/(concentration of U-238) = (half-life of Ra-226)/
(half-life of U-238)

That is, if you know the half-life of U-238, you can determine the half-
life of Ra-226 simply by doing a chemical assay of the uranium and radium 
in your sample; there is no need to make any further measurements on the 
radioactivity! Of course that is not nearly as easy as it sounds, because 
there is only about 0.3 mg of radium per kg of uranium in uranium ore. But 
it means that when Marie Curie first isolated radium, she automatically had 
a pretty good estimate of its half-life along with her radium assay.