A researcher wishes to extract

140
La from
140
Ba. She obtains a 1000 MBq virgin
Ba-140 sample and must wait until the maximum activity of La-140 is available.

a) Write down the decay relationship (include half-lives), and
include the final stable product. How long must the researcher
wait until there is maximum
140
La available?

b)What activity of
140
La does she collect, is she extracts the
140
La
completely from the
140
Ba, when the
140
La is at maximum
activity?

The reactions involved are

140Ba -> 140La + e-,
half life 12.8 days

and

140La -> 140Ce(stable) + e-,
half life 40.2 hours

Set up and solve the differential equation for [140La] and solve for the time is reaches its maximum value, which would be when d/dt[La] = 0

The population of 140Ba is

[140Ba^]*(1/2)^(t/12.8)

This will allow you to write an equation for the rate La is created. Set that equal to the rate La is destroyed by decay. You should be able to solve for [La] at that time. It is probably easier to do with decay coefficients rather than half lives.

a) The decay relationship for this process can be represented as follows:

Ba-140 (initial) -> La-140 (final) + X (unstable intermediate product)

The half-life of Ba-140 is not specified in the question, but we can assume it is much longer than that of La-140. The half-life of La-140 is 1.67 days.

To determine how long the researcher must wait until there is maximum La-140 available, we need to consider the fact that the activity of a radioactive sample is proportional to the number of radioactive atoms present. The activity of a sample decreases over time as the number of radioactive atoms decays.

Given that the Ba-140 sample starts with a radioactivity of 1000 MBq, we know that at some point it will completely decay into La-140. To determine the maximum activity of La-140, we need to wait until all of the Ba-140 has decayed.

The time required for a radioactive sample to completely decay is approximately 10 times the half-life of the radionuclide. In this case, since the half-life of La-140 is 1.67 days, we would need to wait approximately 16.7 days (10 times the half-life) until there is maximum La-140 available.

b) If the researcher successfully extracts all of the La-140 from the Ba-140 sample, then the activity of the collected La-140 would be equal to the initial activity of the Ba-140 sample, which is 1000 MBq.