For 1000k atoms I will have a 500k after 1 half life, and 250k after 2 half lives. But when I try to do the same with fractions something doesn’t add up. Because after “quarter life” i will have 750k atoms, but then I’ll have 562500 atoms, because of 750k multiplied by 0.75. So quarter life should be equal to 0.5 of half life, but half life doesn’t work that way. I am confused.
In: Physics
Your calculations are right. Two quarter lives are not equal to a half life.
A quarter life is the time it takes to lose one quarter. But after the first quarter is gone, the next quarter life is the time it takes to lose one quarter of what’s left, which is smaller than the first quarter you lost.
You are right that a quarter life is not half of a half life. There are actually 2.409 quarter lives in a half life. The decay is not linear like you expect but exponential.
The problem here is that half-life is a non-linear decay, which means you can’t just add things up.
Imagine it like this: After a quarter-life, you have three quarters left. But if you now lose another quarter, in the next quarter-span, it is a quarter of those three quarters you started the second period with. A quarter of three quarters however is 3 / 16, which is slightly less than the quarter you lost in the first period.
The actual decay you get in a given period is always one quarter less than the decay you got in the last period.
To put it more simply: The decay you get is always depending on the amount at the start of each decay period. So it becomes less and less. For calculating the length of decay periods, this is somewhat factored in, so a half-life has a longer decay period than just double the quarter-life.
Case in point: The full-life period is not just double the half-life period. It is in fact infinitely long!
You calculate the half-life like this: T_half = 0.693 / K, where K is some speed constant of the stuff you are looking at.
Quarter life would be T_quarter = 0.287 / K.
Ignoring K, as that does not change for a given process, you can see that the relation of quarter life to half life is always 0.287 / 0.693 = 0.415, so the quarter life is always 41.5% of the half life value.
It doesn’t work that way for the same reason two half-lives are not a “full life”.
A “quarter life” would be about 0.415 half lives. An exact calculation needs the exponential function, but we can check things in small steps to verify: After some unknown time t, 0.1% will have decayed so 99.9% = 0.999 are left. After 2 t, 0.999^2 are left and so on. 0.999^288 = 0.7496 so 288 time steps are around the time a quarter has decayed. 0.999^693 = 0.4999 so that’s around the time half of it has decayed. 288/693 = 0.416
A quarter life would be less than 1/2 the time of half life
So if after 1 year your 1,000k atoms are now 500k atoms, the first quarter life would have happened before the 6 month point and the second quarter life that took you down to 562,500 atoms would have happened before the 1 year half life point
I’m not going to do the maths for it but for argument sake say it was 5 months for the quarter life then you would have had two quarter life points taking you down to 562,500 atoms by 10 months. Then by 12 months you would still hit the half life point
Follows fractions of atoms remaining after 1-3 half lives passed
– 1HL: 1/2
– 2HL: 1/2*1/2 =1/4
– 3HL: 1/4*1/2=1/8
Follows fractions of atoms remaining after 1-3 quarter lives passed
– 1QL: 3/4
– 2QL: 3/4*3/4 = 9/16
– 3QL: 9/16*3/4= 27/64
Just like HL does not mean you remove a half of the original after each HL (because then you would loose all atoms after two HL) QL wouldn’t mean you loose a quarter of the original, after each QL, instead in both cases you loose a defined fraction of **the remainder**
This is the difference between “linear decay” and “exponential decay”.
With linear decay, things work sort of like you’d expect. So you’ll lose a certain number of atoms per second. So if we have 500k atoms after 8 seconds, we’re losing 62,500 atoms per second. After 4 seconds we will indeed have 750k atoms. But this only really happens if we’re doing something like pouring atoms away. In this situation, after 16 seconds we’ll have nothing left at all.
Half life is exponential decay. We can’t model this in the same way. Instead after half the time we have √½ times as many atoms.
So if the half life is 8 second, initially we have 1000k atoms, and after 4 seconds we have √½ = 0.7071 times as many, or 707.1k atoms. After another 4 seconds we multiply again by 0.7071 and have 500k atoms as you expect.
The actual calculation for atoms remaining after a time is not really all that easily explained to a 5 year old, but I’ll put it here anyway, for time t,
Number of atoms remaining = Starting number of atoms *x* (0.5)^(t/half_life)
The whole “2 quarter lives isn’t 1 half life” thing has been well explained, but I wanted to point out that other fractional-lives are used a lot. In many applications (one big one being electrical engineering) you’ll run into something similar to half-life called the ‘time constant’ τ which a measure of the time it takes for ~63.2% (1 – 1/e) of the thing to happen. This value is chosen because it can make some kinds of math a lot easier to do.
Half-life has just become the standard measure of exponential decay. Sure you can use quarter life or whatever, and it can be mathematically recalculated, but we’ve chosen half-life as a standard — nothing particularly special about it.
Half life is a characteristic measure of a non-linear process. It turns out that specifying three facts about a process:
1. It is exponential
2. The quantity at one time
3. The quantity at another time
fully fixes that behavior. By convention the time interval is given such that the ratio of quantities is 1:2, but it’s just convention. There’s nothing special about that particular fraction. If the half life of material was 100 years we could say the 1/3rd life was a different number and it would contain identical information.
Converting between different fractions isn’t trivial, as you’ve discovered. The math isn’t necessarily difficult, just not generally you can do without some thought.
I believe the conceptual problem is one of language. You read “half life” and consider “life” to be some quantity of time to which “half life” is 0.5 multiplied by that quantity. I.e. if half life is 50 years then quarter life must be 25, tenth life must be 5, etc.
You’re thinking “half life” is “half of a life” but it really means “life of a half” or to put it in unmistakable language “time of a halving.”
As an aside, the term “quarter life” I took to mean the time interval in which the material was reduced **to** one quarter, not **by** one quarter. I don’t know which interpretation is more common. Obviously the term half life doesn’t suffer from this confusion because both interpretations are numerically the same.
The time to reduce all of the material is infinity. We know that the time to reduce the material by half is not half of infinity. By the same thinking the time it takes to reduce by one quarter is not half of a “time of halving”.
In mathematical practice when you know the “time of halving” and want to know the “time of ___” you need to find the equation of behavior and solve it for this different fraction.
The general equation for exponential decay is:
X = A * e^(-kT)
where X is how much at time T, A is how much at time zero, and k is the decay constant.
Half life (H) is a particular time T=H where X/A is 0.5. With this information the value of k can be discovered and the full character of the decay process known at all times.
The rearrangment of the equation for k looks like:
k = -ln(X/A)/H
If the equation seems scary, don’t worry, the point is that k is a number. Just remember the value of k depends of the units of time used in T. If you use the same k and T=10 years or T=10 seconds you would get the same decay when we know they shouldn’t be. It’s another aside but it’s a classic test question mistake like half life is 10 years how much is left after 20 days and people answer one fourth.
So you want to know the time where X/A=0.75? Well go back to the equation:
X/A = 0.75 = e^-(kT)
Now we want to know what T is. Rearrange:
T = -ln(0.75)/k
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