OK, this is something I can help you out with. I am finishing up my masters in chemistry.
Here's something to remember. This is an exponential type of decay not a linear decay. Basically, every 5 days the value is multiplied by 1/2.
So lets just look at the problem and see what the value is as the half-lives go on.
We will call the original value N
0 (this is called N sub zero, N initial or N starting)
So at the first half-life we have (1/2)N
0At the second half-life we have (1/2)(1/2)N
0At the third half-life we have (1/2)(1/2)(1/2)N
0This means we have an exponential of 1/2 type of decay
So we get N = (1/2)
xN
0, where x is the number of half-lives and N is the resulting amount.
Now to prove to you this works I will show you the result of the above three (plus the starting value).
At the starting point our value for half-lives is 0. This results in (1/2)
0N
0 which results in (1)*N
0 because any number to the zeroth power = 1.
Now at the first half-life (1/2)
1N
0 = (1/2)N
0For the second half-life (1/2)
2N
0 = (1/2)(1/2)N
0For the third half-life (1/2)
3N
0 = (1/2)(1/2)(1/2)N
0Fractional Half-livesSo now that I have shown how to come to the equation N = (1/2)
xN
0. HOw do we deal with fractional half-lives. Simple, we substitue them in for x. Thats it.
The most common way to do this is to create a fraction with the half-life on the bottom and the amount of time that has passed on the top and substitute that for x.
This sounds confusing, but I think if you see it written out it will make more sense.
The common one is T/T
1/2, where T
1/2 = the half-life and T = the amount of time that has passed.
The only caveat is these need to be in the SAME units, so for example, your half-life is 5. Your fraction would be x = T/5.
So for 1 day, x = 1/5, 2 days, x = 2/5...........5 days x= 5/5 or 1 half-life.
Now if you wanted to do it in minutes, you would have to change the bottom value to be 5-days but in minutes, so 5 days * 24 hrs * 60mintes = 7200 minutes so your x = T/7200
SO our final equation is N = (1/2)
(T/T1/2N
0.
Though the fraction can be substituted for a numberical value of half-lives as well, if the teacher asks how much is left after 4.5 half-lives, you simply use 4.5 for your x value instead of the fraction.
I hope this helps.
Edited by PedroDaGR8, 16 January 2009 - 12:26 PM.