Calculating the Age of 1.4g of Charcoal Using Beta Decay

[QuarksAbove]

Homework Statement

Suppose you have a 1.4g sample of old charcoal. It produces 0.7 beta decays per minute. How old is the charcoal.

Given:
1g of carbon current day has 6.36x10¹⁰ atoms of ¹⁴C

Homework Equations

N = N_oe^-rt

N = number of atoms in the sample (current-day)
N_o = original number of atoms (i.e. at time of death)
r = decay rate = 1.21x10^-4
t = time

The Attempt at a Solution

I know that N_o = 1.4*6.36x10¹⁰ = 8.904x10¹⁰
r = 1.21x10^-4

what I don't understand is what to do with the 0.7 decays per minute. I know I need to solve for N before I can solve for t, but I'm stuck. As soon as I solve for N, it's plug and chug.

Any help would be much appreciated!

 

[SteamKing]

It would be helpful to know the units of the decay rate r.

 

[ehild]

If you have N C14 atoms, how many of them decays in unit time? How is it related to the decay rate?

ehild

 

[QuarksAbove]

It would be helpful to know the units of the decay rate r.

r = 1.21x10^-4 /year

If you have N C14 atoms, how many of them decays in unit time? How is it related to the decay rate?

If I have N C-14 atoms, then 0.7 of those N atoms decay each minute.. 0.7 decay/minute = 367920 atom decays per year.

I'm not sure how it's related to the decay rate. Sorry, I haven't done a problem with two rates before and it's really confusing.

 

[ehild]

N is the number of the ¹⁴C atoms in the sample at present. N₀ was the number of atoms when the sample got isolated. N=N₀e^-rt, so dN/dt=-rN₀e^-rt=-rN.

rN atoms decays in a year, that is about 368000 in the 1.4 g sample. You know r. What is N then?
Originally there were 6.36x10¹⁰atoms of ¹⁴C in 1 g sample. Current-day means "fresh" sample, which can interact with the surroundings, so has supply of ¹⁴C. The charcoal is isolated, so the number of ¹⁴C atom decreases with time.

ehild