## buttermequeasy 3 years ago Radium has a half-life of 1,620 years. In how many years will a 1 kg sample of radium decay and reduce to 0.125 kg of radium? 1,620 years 3,240 years 4,860 years 6,480 years

1. kma230

What do you have so far?

2. buttermequeasy

Well, I don't really understand this topic. How do you find the answer?

3. kma230

Ah okay. So half-life's just the amount of time it takes for half your initial amount to decay, right? If P is your principle amount, one half life leaves you with P/2, two half lives leave you with P/4, three with P/8, and so on. We express this mathematically as such: \[\Large A(t) = P*(1/2)^{t/H} \] where A(t) is the amount left after time t, P is your principle amount, and H is the length of time for your half life (in this case, 1,620 years). Though this may seem a bit foreign, it's really just the same as what we said above. Let's say one half life, or 1620 years have passed. t/H = 1, therefore, your amount = P/2. Exactly what we said above. In this question, we're actually given A(t) and we have to solve for t. Why don't you give it a try and see what you can get?

4. buttermequeasy

I'm sorry but I don't understand this at all.

5. kma230

Are you comfortable with the concept of the half-life? i.e., are you good before we jump to the part with the equation?

6. buttermequeasy

The problem is, I don't know what half-life is. I was presented this question and there wasn't much explanation.

7. kma230

Ah I see...okay let me rewind then. Certain atoms have unstable nuclei. There's a complex interplay of forces keeping the nuclei stable, and certain combinations of protons and neutrons don't last (relatively speaking) very long! Say you have a collection of Uranium-238, a radioactive element. It's radioactive because we know that its nucleus is unstable- each atom, will eventually blow apart. The interesting thing is, not every atom blows apart at the same time. Rather, the pattern of "atoms blowing apart" or decay, follows a neat little pattern. If you have 200 atoms of Uranium-238, after a certain amount of time t, you'll only have 100 atoms left. When another interval of time t passes, you'll have 50 atoms. This time t is called the half life of the element. The pattern, visually, is shown below (exponential decay; I'm sure you've encountered it in math class).

8. kma230

The basic premise is - after some constant amount of time, half of what's left of the element will have decayed.

9. buttermequeasy

I see

10. kma230

Yep! And quantifying the above relationship and pattern simply gives you the equation I typed up there. To simplify things a little bit, the term t/H is really just the number of half lives that have elapsed. If three half lives have elapsed, you're taking half..of half..of half of what was originally there, right?

11. buttermequeasy

Would it be the second choice?

12. kma230

Hmm would you mind sharing your steps?

13. buttermequeasy

Well, I multiplied 1620 by 2 and came up with the second choice..

14. kma230

Why 2, though?

15. buttermequeasy

I thought you were supposed to multiply it by 2?

16. buttermequeasy

Do you multiply it by 1/2?

17. kma230

That was unfortunate. My browser restarted in the midst of my typing up a several paragraph long explanation. Just letting you know I'm still here, retyping everything :P

18. buttermequeasy

I'm sorry that I'm making you work double. Thanks for helping me.

19. kma230

Alright so! Let's go back to the equation - it's your handy tool for defeating monstrous questions like these. \[ \Large A(t)=P∗(1/2)^{t/H} \] in regular ol' words, this is simply: \[\large Amount\ left\ after\ decay = Starting\ amount * (1/2) ^{Number\ of\ half\ lives} \] A(t), or the left side of the equals sign, is simply the amount that results after decay has occurred. In other words, it's your ending or 'desired' amount. In this case, it's 0.125 kg. Let's plug in some more values...we know the principle amount (the amount you start with ) is 1kg, and your half life is 1620 years. \[ \Large 0.125=1*(1/2)^{t/1620} \] Simplify a bit, and rewrite that decimal into a fraction...makes it easier! \[ \Large 1/8 = (1/2)^{t/1620} \] Okay so let's think what we need to do to solve this...1/2 to the third power yields 1/8, right? So we can set 3 equal to t/1620. Solving for t yields 4860 years. Here's the above step broken down. \[ \Large (1/2)^3 = (1/2)^{t/1620} \] Therefore, \[ \Large 3 = t /1620 \] Let's review what we did. We are looking for the amount of time it takes for 1 kg of an element to decay to 1/8th kg. We can do this by looking at the amount of half lives it takes. 1* (1/2) * (1/2) * (1/2) = 1/8th. It takes three half lives for this decay to occur. The TOTAL time it takes then, is 3*1620 years. It's a tough concept to get. If you want, I can make up some practice problems for you?

20. UnkleRhaukus

after 0 half lives 1 kg after 1 half life 0.5 kg after 2 half lives 0.25 kg after 3 half lives 0.125 kg so that is three half lives , or 3 times 1,620 years = 4860 years

21. arcticf0x

Excellent Explanation @kma230! Thank you so much for bringing back some old memories.