Question

The half-life for radioactive decay of C-14 is 5730 years. An archaeological artefact containing wood had only 80% of the C-14 found in a living tree. The age of the sample is?
a.) 1485 years
b.) 1845 years
c.) 530 years
d.) 4767 years

Answer 1

Hint: The age of sample can be determined by t=$\dfrac{2.303}{k}$log$\dfrac{[R]_0}{[R]}$, here substitute the values of k, [R]. The mentioned radioactive decay is a first order process, and the k represents the decay constant.

Complete step by step solution:
Firstly, let us know that we need to calculate the age of the sample. The age of sample can be found by t=$\dfrac{2.303}{k}$log$\dfrac{[R]_0}{[R]}$, here k symbolises the decay constant and t is the time i.e. age and $\dfrac{[R]_0}{[R]}$ represent the ratio of total number of samples to the decayed samples.

Now, we are given with the half-life of C-14 i.e. 5730 years, so we will calculate the decay constant that is k=$\dfrac{0.693}{t}_{1/2}$, t$_{1/2}$ is the representation of half-life.
Now substitute the value of t$_{1/2}$ then k=$\dfrac{0.693}{5730}$ years$^{-1}$.
Now, we already know t= $\dfrac{2.303}{k}$log$\dfrac{[R]_0}{[R]}$, $\dfrac{[R]_0}{[R]}$ is $\dfrac{100}{80}$, we are given 80% of decay, then t= $\dfrac{2.303}{\dfrac{0.693}{5730}}$log $\dfrac{100}{80}$= 1845 years (approximately).
Therefore, the age of the sample is 1845 years. The correct option is B.

Note: Don’t get confused between the symbols. Sometimes k, the decay constant is also represented by $\lambda$. The decay constant k and ratio of activity samples is different. The symbol k shows the dependence of half-life of the corresponding sample, whereas the ratio is how much activity is done by the sample during the decay.