Carbon-14 Half-Life: Calculating Decays After 50,000 Years

[Wing2015]

Homework Statement

A sample containing carbon-14 has 16000 decays per minute. If the half life of carbon-14 is 5730 years. Aproximately how many decays per minute would be occurring after another 50 000 years.

Answer: approximately 40 decays per minute.

Homework Equations

N= No x e^-λt
T=in2/λ
A= λN ( A is the decay rate of sample I.e number of decays per second)

The Attempt at a Solution

Calculated the value of λ first from in2/5730 and got 1.2 x 10^-4.

Then tried to plug it into A= λN and tried to get the value of N by substituting the first activity of 16000.

From here I am stuck because I realized even if I find N and plug it back into the formula I'll be getting the same result. Which makes me confused about the whole question because I would have thought that the decay rate would be constant for a particular sample.

Any help would be greatly appreciated. Thanks in advance.

 

[Rellek]

Hi,

Examining your given data, you are given a time rate of change, a half-life, and you want another time rate of change after a finite amount of time.

It looks like your first plan of action should be to find the initial amount of carbon-14 present, since that is a rather integral part of your equation. I'm not sure what mathematics you have covered, but the derivative of your function N(t) with respect to t will give you an expression for the rate of change at any time t (that's what your third equation looks to be, but with time set to 0).

Hope that helps!

 

[BvU]

Hello Wing, and a belated welcome to PF

"decay rate would be constant for a particular sample" may need some more explanation: what is constant is the probability that one single carbon-14 nucleus decays in a given time. The greater that probability, the more active a sample of a radioactive material. But the activity of a sample also depends on the number of carbon-14 nuclei in the sample, and that number decreases by 1 with every decay. So we write a differential equation for the decay rate (the number of decays per unit time) $${dN\over dt} = -\lambda N$$with the solution you indicate:$$N(t) = N_0 e^{-\lambda t}$$This can also be written as $N(t) = N_0 e^{- t/\tau}$ with $\tau = 1/\lambda$, or as $$N(t) = N_0 \;2^{- t/\tau_{\scriptscriptstyle 1\over 2}}$$