139k views
0 votes
The half-life of
^(14)C (Carbon-14) is 5730 years. That is, it takes this many years for half of a sample of
^(14)C to decay. If the decay of
^(14)C is modeled by x' = -rx, where x is the amount of
^(14)C, find the decay constant r. (Answer: r = 0.000121 yr⁻¹). In an artifact the percentage of the original
^(14)C remaining at the present day was measured to be 20 %. How old is the artifact?

1 Answer

0 votes

Answer:

a)
(1)/(2) x_o = x_o e^(-r(5730)

We can cancel
x_o and we got:


(1)/(2)= e^(-r(5730))

We apply natural log and we got:


ln((1)/(2)) = -5730 r

And
r =(ln((1)/(2)))/(-5730)=0.000121

b)
0.2 x_o = x_o e^(-0.000121 t)

We can cancel
x_o in both sides and we got:


0.2 = e^(-0.000121 t)

Now we can apply natural log on both sides and we got:


ln(0.2) = -0.00121 t

And if we solve for t we got:


t =(ln(0.2))/(-0.000121)=13301.139 yars

So in order to have 20% of the original amount
x_o the total time is 13301.14 years approximately.

Explanation:

Part a

For this case we have the following model given by the differential equation:


x' = (dx)/(dt)= -rt

The solution for this model is given by:


X(t) = x_o e^(-rt)

Using the half life we know that for 5730 years we need to have 1/2 of the initial amount
x_o so if we replace we have this:


(1)/(2) x_o = x_o e^(-r(5730))

We can cancel
x_o and we got:


(1)/(2)= e^(-r(5730))

We apply natural log and we got:


ln((1)/(2)) = -5730 r

And
r =(ln((1)/(2)))/(-5730)=0.000121

Where
x_o represent the initial amount. For this case we know the value for the rate of decay
r = 0.000121 (1)/(year)

Part b

And the half like is 5730 years. And we want to find the time in years in order to have 20% of the original amount. So we can write the following expression:


0.2 x_o = x_o e^(-0.000121 t)

We can cancel
x_o in both sides and we got:


0.2 = e^(-0.000121 t)

Now we can apply natural log on both sides and we got:


ln(0.2) = -0.00121 t

And if we solve for t we got:


t =(ln(0.2))/(-0.000121)=13301.139 yars

So in order to have 20% of the original amount
x_o the total time is 13301.14 years approximately.

User Claudiu Matei
by
7.2k points
Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.