(sorry, my keyboard is starting to get old, so there might be some missing e's and I don't have time to edit)
First, I have a couple of links I wanted you to check out. Rutherford's curve that I posted is around 3.5 days, which matched the wiki table for the radium (historical name thorium X by Rutherford), it is not Th232. This is where you'l find Rutherford's expriments http://web.lemoyne.e...ta/ruthsod.html You need to look at the Thorium series table provided at wiki, and you will find the historical names, as well as the modern names. Thorium x is radium, and the thorium emanation is radon gas. You'll see that it was not thorium 232, unless the throium x (radium) was coming from the daughters of Th 232. The daughters after radium in the thorium series are short half lives, on the order of hours, minutes and seconds. And I understand what a half life curve is and it's nature. That it is an exponentially slowing decrease into the next daughter product, either by beta or alpha decay. Each half life series has quantum decreases in atomic weight of 4.
Here's a few links that challenge the dogma of immutabl half lives.
I think there may be a mixup on which curve is being discussed. The decreasing activity is the curve for the radium only sample. The activity is decreasing because there is nothing creating more radium and the daughter products decay so quickly they can't accumulate.
The increasing sample contained only thorium initially. The reason it is increasing is because th232 decays slowly (low activity), while it's daughter products decay faster (higher activity) but are able to stick around long enough to increase in amount and therefore increase the activity of the sample.
The half life of thorium is long enough that only a very small amount of Ra-224 will be produced in 3 days. However, only half of that produced ra-224 will decay in that same 3 day span. This means that the ra-224 will accumulate until the amount of ra-224 lost every 3 days is equal to that produced every 3 days by the thorium sample.
The equation is (atoms leftover)=(# starting atoms)*.5 ^ (time/half life)
Thorium 232 decays slowly enough that over the course of a few months the amount of decays every 3 days is essentially a constant X. For simplicity in the below example, I'm treating each set of decays as occurring in 3 day intervals rather than continuously. It illustrates the principle if not the exact numbers. You can use the half life equation to get real numbers if you'd prefer.
Let X be the number of radium atoms produced by thorium at 3 day intervals. The total radium in the sample will be X plus whatever was leftover from the previous 3 day interval. Half of the total radium will decay leaving half for the next 3 day interval.
After day 3 we produced X, X/2 decayed. we have X/2 leftover radium atoms in the sample.
After day 6 we produced X, X + X/2 = 3X/2, after half of that decays we have 3X/4 leftover.
After day 9 we produced X, X + 3X/4 = 7X/4. after half of that decays we have 7X/8 leftover.
Notice that the leftover radium is getting closer and closer to X and the amount of decays (activity) is increasing as well.
This will continue until we have X produced with X leftover. X + X = 2X. after half of that decays, we have X leftover.
This will show up at an increase in activity that eventually reaches a maximum. After the maximum is reached there will be a very slow decrease in activity with the rate of decrease being related to the half life of the thorium.