Finding Original Carbon nuclei from given sample.

[rcubed]

Homework Statement

A small animal bone fragment found in an archaelogical site has a carbon mass of 155g. When the animal was alive, the ratio of radioactive ¹⁴₆C to the stable ¹²₆C was 1.31×10^-12. What was the number of ¹⁴₆C nuclei found in the sample when the animal was alive?

Homework Equations

None given, but I would assume:
N=N₀e^-λt

The Attempt at a Solution

Not too sure where to start so I got the decay constant, λ by using half life of Carbon14, 5730 Years
0.5=e^-λ(5730)
λ=1.21×10^-4

Then I solved the for the number of years since the animal was alive by plugging everything back into the original equation, assuming N/N₀ = 1.31×10^-12

t=-ln(1.31×10-12)/-1.21×10^-4 = 226180 Years

Up until here I don't think I did anything wrong, but here is where I am unsure of what to do.

I tried using the same formula to solve for N₀, but this time using the given 155g.

155=N₀e^{-(1.2110-4)(226180)}
N₀=1.18×10¹⁴g

I don't think in doing the right thing here. Can anyone give me some guidance?

Thanks!

 

[collinsmark]

Hello rcubed,

Welcome to Physics Forums!

Homework Statement

A small animal bone fragment found in an archaelogical site has a carbon mass of 155g. When the animal was alive, the ratio of radioactive ¹⁴₆C to the stable ¹²₆C was 1.31×10^-12. What was the number of ¹⁴₆C nuclei found in the sample when the animal was alive?

Homework Equations

None given, but I would assume:
N=N₀e^-?t

The Attempt at a Solution

Not too sure where to start so I got the decay constant, ? by using half life of Carbon14, 5730 Years
0.5=e^-?(5730)
?=1.21×10^-4

Okay, you've found that
$$\frac{N}{N_0} = e^{-1.21 \times 10^4 \ t}$$

Although I don't think that helps for this problem. (Maybe it does later in a different part of the problem not listed in the above statement).

Then I solved the for the number of years since the animal was alive by plugging everything back into the original equation, assuming N/N₀ = 1.31×10^-12

t=-ln(1.31×10-12)/-1.21×10^-4 = 226180 Years

No, wait. You're using the 1.31×10^-12 out of context. It is *not* the ratio of the final amount of ¹⁴C to the original amount of ¹⁴C.

As a matter of fact, you don't even know what the final amount of ¹⁴C is. That information is not given in the problem statement. And since the age of the animal is not given either, you can't even calculate it (at least not without additional information).

Can anyone give me some guidance?

Thanks!

All you're trying to find is the original amount of ¹⁴C (when the animal was alive). For this particular problem, don't worry about how much of the sample is ¹⁴C today, or even how old the sample is. There's not enough information given anyway.

So here is what we know. There is 155 g sample of carbon. When the animal was alive, the fraction of carbon (in terms of the ratio of the number of nuclei) that was ¹⁴C was 1.31×10^-12 (Technically that number is the ratio of ¹⁴C to ¹²C, but it's also approximately the same ratio as the number of ¹⁴C to total). So when the animal was alive, how much of that sample was ¹⁴C?

[Hint: You might want to start by determining how many carbon nuclei (primarily ¹²C) are in 155 g of carbon. Then, since you know what fraction of that that was ¹⁴C, determine the number of ¹⁴C nuclei].

-------------
Edit:

By the way, I am presently interpreting the 1.31×10^-12 ratio as the ratio of number of nuclei. If instead it is a ratio of masses or weights, then my advice needs to be modified to take that into account (i.e. ¹⁴C is heavier than ¹²C, which needs consideration).

 

[rcubed]

^Thanks for the reply!

So its just simply finding the number of ¹²C Molecules first, then using the ratio to find the ¹⁴C?
155g/12gmol^-1*NA = 7.78e24

7.78e24*1.32e-12 = 1.026e13 ?

Is that it?

 

[collinsmark]

^Thanks for the reply!

So its just simply finding the number of ¹²C Molecules first, then using the ratio to find the ¹⁴C?
155g/12gmol^-1*NA = 7.78e24

7.78e24*1.32e-12 = 1.026e13 ?

Is that it?

It looks like the right idea to me. But I think the original problem statement said that the ratio was 1.31 x 10^-12 (you used 1.32 x 10^-12).

 

[asteriatic]

Based on the information given, it seems like you have correctly calculated the decay constant (λ) and the number of years since the animal was alive (t). However, your approach to solving for the number of 146C nuclei (N0) may not be accurate. The formula N=N0e-λt is typically used to calculate the number of nuclei remaining after a certain amount of time has passed, not the initial number of nuclei. In this case, we are trying to find the initial number of 146C nuclei, so we need to use a different formula.

One approach could be to use the ratio of 146C to 126C to calculate the initial number of 146C nuclei. Since the ratio is given as 1.31×10-12, we can set up the following equation:

1.31×10-12 = N0/N0e-λt
1.31×10-12 = 1/e-λt
e-λt = 1/1.31×10-12
-λt = ln(1/1.31×10-12)
t = -ln(1/1.31×10-12)/λ

Now, we can plug in the values for λ and t that we calculated earlier to solve for N0:

t = (-ln(1/1.31×10-12))/-1.21×10-4 = 226180 years
N0 = 1/e-(-1.21×10-4)(226180) = 1.31×10-12 * e1.21×10-4 * 226180 = 1.31×10-12 * 1.34 = 1.75×10-12

Therefore, the initial number of 146C nuclei in the sample when the animal was alive would be 1.75×10-12. I hope this helps guide you in the right direction.