In 1951, the Dead Sea Scrolls were dated using carbon-14 dating. They were found to still contain 30% of their original carbon 14. If the half life of carbon 14 is 5,730 years, how old were the scrolls?

Question

In 1951, the Dead Sea Scrolls were dated using carbon-14 dating. They were found to still contain 30% of their original carbon 14. If the half life of carbon 14 is 5,730 years,  how old were the scrolls?

Jeremiah Allen · Accepted Answer

Roughly 10,000 years old according to the data in the question. But probably it is the decay that is 30%, in which case the age would be 2900 years old.

"Half life" refers to the amount of time that passes before 1/2=50% of the carbon 14 has vanished. Pardon the clumsy pun. The rate of decay is -5730 years Let the start mass be #A_0# at the first year. The decay rate is geometric, i.e. #A_t=A_0e^(kt)# or #(A_t)/(A_0)=e^(kt)# #=> kt=ln((A_t)/(A_0))# For #t=5730# #(A_t)/(A_0)=1/2# #=>5730k=ln(1/2)# #k=ln(1/2)/5730=-0.000120968=-1.20968*10^-4# From this we get: #-1.20968*10^-4*t=ln(0.3)=-1.20397280433# #t=(-1.20397280433)/(-1.20968*10^-4) =9953# Note that the actual age of the Dead Sea Scrolls are more like 1800 to 2300 years old, so the amount of C14 given in the question is probably way too low. Should I guess, the decay has been 30%, i.e. that there is still 70% of the original C14 left. With that as the starting point we get: #t=ln(0.7)/(-1.20968*10^-4)=2949# I think you misunderstood the information provided because this figure is much more in line with the observed results.

Ruby Armstrong · Answer

undefined The age of the Dead Sea Scrolls can be estimated by determining how many half-lives of carbon-14 have passed since they were created. With a half-life of 5,730 years, the calculation is as follows:

Original carbon-14 remaining = 30%
Remaining carbon-14 = 0.30

To find the number of half-lives:

0.30 = (1/2)^n
n = log(0.30) / log(0.5)
n ≈ 1.737

Thus, approximately 1.737 half-lives have passed since the creation of the Dead Sea Scrolls.

To find the age:

Age = Number of half-lives × Half-life of carbon-14
Age ≈ 1.737 × 5,730
Age ≈ 9,970.41 years

Therefore, the Dead Sea Scrolls are approximately 9,970 years old.