4. Carbon dating is a method that has revolutionized many academic fields. It is

Question

4. Carbon dating is a method that has revolutionized many academic fields. It is based on the fact that a small fraction of the carbon in CO2 and in the bodies of all living creatures is the isotope Carbon-14 instead of Carbon-12. C-14 is produced by cosmic radiation interacting with the upper atmosphere. The radioactive isotope Carbon-14 decays with a half-life of approximately 5730 years and turns into Nitrogen-14. Let y be the mole fraction of C-14. At t = 0, an animal dies and no longer interacts with the atmosphere so all the C-14 in its body begins to decay like a first-order dynamical system.

a. Define what the half-life of a system means.

b. What is the time constant of this system?

c. Using Excel or Matlab, plot the first-order dynamical system when yi is equal to 100% (no decay) up to 100 half-lives. Plot in non-dimensional form. After 100 half-lives, how much C-14 is left in a sample?

d. Archeologists find a fossil that has 35% Carbon-14 compared to the living sample. How old is the fossil?

Explanation / Answer

A.Half Life is basically the amount of time taken by an radioactive isotope to reduce to half of it's initial value.It is a constant for first order reaction.

B.k(time constant)=ln2/(Half Life)

Upon calculating,

K=ln2/5730=0.00012 year-1

C.The amont of C-14 left after 100 half lives is

Amount=y/1.26*(10^30)

D.The equation for the amount of carbon-14 left is

A= A0(e^-kt)

Given that 35 percent of carbon-14 is left,so A=0.35*A0

So,0.35=e^-kt

We will also assume that carbon-14 quantity was A0 at the initial stage.

Now, applying log on both sides

ln(0.35)=-kt

1.0498=0.00012*t

t= 8678.50 years

Therefore the fossil is 8678.50 years old.

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