Physics HELP!!

There's this question on half-life that baffles me😔

Carbon-14 is radioactive and has a half-life of 5,700 years.

The number of radioactive carbon-14 atoms in a very old piece of wood is found to have decreased from 1,000,000 to 125,000.

Determine the age of the piece of wood.

Reply 1

chlorinek

7

so keep dividing 1,000,000 by 2 until you get to 125,000 i think, so that would be 3, therefore 3 half lives. So if one half life is 5,700 i think you would just multiply that by 3 to get 17100 years. might be wrong.

Reply 2

tastish

OP

9

Original post by chlorinek

so keep dividing 1,000,000 by 2 until you get to 125,000 i think, so that would be 3, therefore 3 half lives. So if one half life is 5,700 i think you would just multiply that by 3 to get 17100 years. might be wrong.

thank you!! :smile:

Reply 3

DarkShadow101

12

Original post by chlorinek

so keep dividing 1,000,000 by 2 until you get to 125,000 i think, so that would be 3, therefore 3 half lives. So if one half life is 5,700 i think you would just multiply that by 3 to get 17100 years. might be wrong.

would it not be 8 half lives? 1000000/8 = 125000 then you would do 8 times 5700? 45600? Not sure tho

Reply 4

tastish

OP

9

Original post by DarkShadow101

would it not be 8 half lives? 1000000/8 = 125000 then you would do 8 times 5700? 45600? Not sure tho

It definitely not that, since that's what I did and got marked wrong.

Reply 5

chlorinek

7

Original post by tastish

thank you!! :smile:

its okay! i hope its right lol

Reply 6

Sinnoh

22

Original post by DarkShadow101

would it not be 8 half lives? 1000000/8 = 125000 then you would do 8 times 5700? 45600? Not sure tho

No, that would suggest that after 9 half-lives the radioactivity is completely gone, which cannot be the case.
The half-life is defined as the time taken for the activity of a radioactive substance to halve. This is constant no matter how much of it was there to begin with. Radioactivity is an exponential decay because the number of decays occurring per second is proportional to how many nucleons there are of it. So after 1 half-life, it's down to 500,000. Then another half-life passes and it's down to 250,000. Then another half-life passes and it's down to 125,000. So with n half-lives that pass, the number of radioactive nuclei is multiplied by (0.5)ⁿ

Just as an extra bit:
You could generalise this as

\text{number of remaining nucleons }N(t) = N_0 (\frac12)^n

where n is the number of half-lives that have passed and N₀ is how many there were to begin with. If you don't have an integer number of half-lives then you'd have to fiddle with logarithms but this only matters in A-level physics.

Physics HELP!!

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