Yes, the curve is exponential, and it's not hard to play tricks with the math. One could calculate 1/4 lives, 1/3 lives, or 1/277 lives for decaying elements. All will result in different exponential curves.
Sadly for you this is a totally false statement. Very simply, the "1/4 life" would be the time it takes for a sample to decay to the point where only 1/4 of the original amount is left.
No it's not. From
a dictionary1. Physics The time required for half the nuclei in a sample of a specific isotopic species to undergo radioactive decay.
Therefore, a quarter life is the time it takes for 1/4 of the material to decay away.
This would take the same time, and end up at the same concentration, as waiting for 2 half-lives of any isotope. This is pretty simple math. After the first 1/4 life, 3/4 remain. After the next 1/4 life, 9/16 remain. 9/16 ain't 1/2.
1/4 x 1 = 1/4 First ql
1/4 x 3/4 = 3/16 Second ql
1/4 + 3/16 = 7/16 Total
-7/16 + 1 = 9/16 Result
So 2 quarterlives do not give the same result as one halflife. Other fractions will do this too. Care to see four 1/8 lives?
That only works on linear functions. Since this is an exponential decay, a quarter-life is not equal to half of a half-life. Quarter-life and half-life are measures of the time it takes for a certain amount of the substance to decay away (1/4 of the material in the first case and 1/2 of the material in the second case). Those radioactive decays have been
observed to occur following this function:
N(t) = N(0)e^-?t
Where N(t) = The amount of material at a particular time
N(0) = The initial amount of material.
t = The time expended
? = the decay rate, which is differs between materials
So, for a 1/2 life: N(t)/N(0) = 1/2 = e^-?t(half)
You then solve for t(half) and get: t(half) = (-ln(1/2))/? = .693/?
for a 1/4 life: N(t)/N(0) = 3/4 = e^-?t(quarter)
You then solve for t and get:
t(quarter) = (-ln(3/4))/? = .288/?
So t(quarter)/t(half) = (.288/?)/(.693/?)
Solve for t(quarter):
t(quarter)= ((.288/?)/(.693/?))* t(half) =(.288/.693)*t(half)
t(quarter) = .415 * t(half)
.415 is less than .5 so two quarter-lives is less time than one half-life. Therefore two quarter-lives can leave 9/16 of the material while one half-life can only leave 1/2 of the material.
I hope this helps
