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### Topic: Carbon14/Carbon12 ratio  (Read 6219 times)

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#### Rutherford ##### Carbon14/Carbon12 ratio
« on: October 13, 2012, 01:06:29 PM »
The element carbon consists of the stable isotopes 12C (98.90% of atoms) and 13C (1.10% of atoms). In addition, carbon contains a small fraction of the radioisotope 14C (t1/2= 5730 years). What is the isotope 12C/14C ratio of carbon, which takes part in the natural CO2 cycle (1 year=365 days) if the decay rate of carbon 14 is found to be 13.6 disintegrations per minute and gram of carbon.

Because the disintegration is expressed in 1g of carbon, I can assume that I have 1g of a carbon mixture, that is 0.989g of 12C and the number of atoms is easily calculated, but how to do it for 14C?

#### Rutherford ##### Re: Carbon14/Carbon12 ratio
« Reply #1 on: October 15, 2012, 08:46:54 AM »
Any idea please? The formula they used to calculate the number of 14C atom is: N=13.6*t1/2/ln2. Could someone interpret this formula?

#### Borek ##### Re: Carbon14/Carbon12 ratio
« Reply #2 on: October 15, 2012, 10:02:26 AM »
An obvious line of attack seems to be to calculate N0-Nt from $N_t = N_0 e^{\frac t {t_{1/2}}}$.
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#### Rutherford ##### Re: Carbon14/Carbon12 ratio
« Reply #3 on: October 15, 2012, 10:29:03 AM »
I don't understand what you wrote. I know this formula of first-order reaction (with natural logarithms):
lnN=lnN0-λt
λ can be calculated using the t1/2 ([N]=1/2[N]0):
λ=ln2/t1/2
I convert all times into minutes and put λ into the first equation and then I think that I need to calculate N0, but how?
Where to put the disintegrations?

#### Borek ##### Re: Carbon14/Carbon12 ratio
« Reply #4 on: October 15, 2012, 12:14:15 PM »
I know this formula of first-order reaction (with natural logarithms):
lnN=lnN0-λt

See if you can rearrange it to get equation in terms of Nt and N0, not in terms of logs.

If you start with N0 atoms and after time t you are left with Nt atoms, how many atoms decayed?
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#### Rutherford ##### Re: Carbon14/Carbon12 ratio
« Reply #5 on: October 15, 2012, 12:29:22 PM »
Okay, this formula was given in the problem that I didn't write:
Nt=N0e-λt
N0-Nt decayed but I need N0. What now?

#### Borek ##### Re: Carbon14/Carbon12 ratio
« Reply #6 on: October 15, 2012, 12:47:46 PM »
Now you start thinking on your own. Enough spoon feeding for today.

You are a single rearrangement from the answer.
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#### Rutherford ##### Re: Carbon14/Carbon12 ratio
« Reply #7 on: October 15, 2012, 02:11:42 PM »
But I don't know how to add the disintegration period. I tried this way: calculated λ, then after 1min 13.6 atoms decay, that would be Nt, so N0=13.6/e-λt, t is 1min and λ=2.3*10-10min-1. Then N0=13.6  ?

#### Borek ##### Re: Carbon14/Carbon12 ratio
« Reply #8 on: October 15, 2012, 03:27:23 PM »
As you already stated - N0-Nt decayed. Add to that fact that $N_t = N_0 e^{\frac t {t_{1/2}}}$. There is one unknown here, all you have to do is to solve.
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#### Rutherford ##### Re: Carbon14/Carbon12 ratio
« Reply #9 on: October 16, 2012, 07:35:48 AM »
I don't understand how you got that equation.
1)N0-Nt=13.6    Nt=N0-13.6
2)Nt=N0e-λt
t is 1 so it gets out of equation 2). λ is a very very small number ≈0, so e=1. Then N0-13.6=N0 .

#### Borek ##### Re: Carbon14/Carbon12 ratio
« Reply #10 on: October 16, 2012, 08:28:40 AM »
Sorry, my mistake. It should be $N_t=N_0 2^{\frac{t}{t_{1/2}}}$. Note that they are almost identical, the only difference is the ln(2) constant by which they differ.

You are right these numbers are very close, but try to solve without any approximations.
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#### Rutherford ##### Re: Carbon14/Carbon12 ratio
« Reply #11 on: October 16, 2012, 08:56:53 AM »
If I don't approximate, then the calculator does it, and it shows 1.00000000 and again I have the same situation. By the way, their answer is 5.91*1010 atoms of 14C in 1g of mixture and the formula they used is N=13.6*t1/2/ln2 which is N=13.6/λ, but I can't think of a way to derive this.

#### Borek ##### Re: Carbon14/Carbon12 ratio
« Reply #12 on: October 16, 2012, 09:43:29 AM »
Windows calculator is able to give the correct answer. TI-89 is able to give the correct answer. You need at least 12-13 digits to get the result this way, but it is perfectly doable. Sure, some calculators won't work.

Activity is a first derivative of the number of atoms with respect to time - if you have not started to learn calculus you can't derive the formula they used, you can at best memorize it.
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#### Rutherford ##### Re: Carbon14/Carbon12 ratio
« Reply #13 on: October 16, 2012, 10:41:41 AM »
Using the math at the level I know, from reactions 1) and 2) without direct calculation I got: N0=(13.6eln2/t1/2)/(eln2/t1/2-1) here it can be seen that if I use the approximation the denominator is 0, if I don't use it, the denominator is a really small number. Using a calculator here on the net, I got the right answer. I suppose that this requires knowing integrals, so I should skip similar problems for a while. Thanks for helping me through this problem.