Half life

[DagS132]

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Problem: Consider radioactive nuclide X. If 88% of a pure sample decays in 24 hours, what is the half-life?

The way DAT Destroyer answers it is by the following table:

Time Amount

0 X

? X/2

? X/4

24 1/8

(not sure how well this table will look in this thread but the values under time is 0, ?, ?, and 24. The values under amount is X, X/2, X/4, 1/8.

They get 1/8 from the 12% thats left of the decay but i dont know why they are using X/2 and X/4 or why the time is cut up into 4ths. I thought i just find out what the total time of the decay is by doing 24/.88 to get about 27 hours in total and then do half of 27 to get half life. Is a decay not a constant rate?

 

[SafetyDog]

You have ~12.5% of pure sample left after 24 hours.

That is equivalent to 3 half lives (100 -> 50 -> 25 -> 12.5).

So you have ~3 half lives over a period period of 24 hours which would be a half life of 8 hours each time the sample is cut in half. Since a little more decayed, it should be a little bit more than 8 hours. Maybe like 8 hours and 5 minutes.

At X/2, that means 50% is remaining and at X/2, that means there is 25% remaining. The question marks represent the amount of time that has passed.

 

[Toadesque]

This is how I would do it

Since 88% decayed then 12% would be remaining after 24 hours

100% -> 50% -> 25% -> 12.5%

It had to be halved 3 times in order to reach 12.5% (which is close to 12%)

So 24 hours/3 = 8 half-lives

 

[aznriptide859]

In this case, half life is constant.

The question is kind of worded badly; What DATD means to say is 88% (or, more specifically, 87.5%) of the sample has decayed. That means 12.5% of the sample is remaining.

If only 12.5% of the sample is remaining, that means it went through 3 half-lives (100% -> 50% -> 25% -> 12.5%). If three half-lives went past in the span of 24 hours, that means the half-life is 24/3 = 8 hours.

I tend to find progressing through the half-lives as a flow chart helps with these kinds of problems. After each half-life, note the time elapsed and remaining sample. If one of these matches your situation, then apply it to your answer. I will say that the real DAT is definitely not this vague regarding numbers, and will give you exact decimal values when necessary.

But, ultimately: DAT Destroyer rounding fail.

EDIT: Damn, got ninja'ed after so many edits lol.

 

[DagS132]

hmmmmmm, i guess i dont seem to be understanding half life. Just to help me out a bit can you guys help me with two different scenarios then?

What if it had decayed 62% in those 24 hours. What would be the half life be then?
Also what if it was 88% decayed in 36 hours?

 

[aznriptide859]

hmmmmmm, i guess i dont seem to be understanding half life. Just to help me out a bit can you guys help me with two different scenarios then?

What if it had decayed 62% in those 24 hours. What would be the half life be then?
Also what if it was 88% decayed in 36 hours?

Let's do an example.

Say a substance has 100 grams, and a half life of 50 days.

Therefore:

After 50 days, the substance has undergone 1 half-life, lost 50 grams, and has 50 grams remaining.
After 100 days, the substance has undergone 2 half-lives, lost 75 (50 + 25) grams, and has 25 grams remaining.
After 150 days, the substance has undergone 3 half-lives, lost 87.5 (50 + 25 + 12.5) grams, and has 12.5 grams remaining.
After 200 days, the substance has undergone 4 half lives, lost 93.75 (50 + 25 + 12.5 + 6.25) grams, and has 6.25 grams remaining.

You can see half-life is basically a progression of (1/2)^n, where n is the number of half-lives. After each half-life "time period", a substance will lose half it's amount (hence why it's called "half-life"). This progresses continuously, but for each subsequent half-life, you lose the previous amount by half. So 25 is half of 50, 12.5 is half of 25, and so on.

As for your questions:

1. If your substance decayed 62% in 24 hours, that means you only have 38% left of your substance. Because 50% > 38% > 25%, that means it's half-life is under over 24 hours but under 48 hours. There's some complex equation you can numerically find the specific, but usually (USUALLY) the DAT won't ask you something this complex. Half-life questions will almost always involve some multiple of the original half-life and/or original substance amount.

2. If 88% (or 87.5%) decayed in 36 hours, it again means you have 12.5% left (meaning it's undergone 3 half-lives - see above). If 36 hours have passed and 3 half-lives were undertaken, then the half-life is 36/3 = 12 hours.

 

[DagS132]

thanks aznriptide859! i understand now