QJEEM24Chef77: JEE Main 2024 Chemistry - Class 12 JEEM24Chef77 Solution: The ratio of in a piece of wood is part that of atmosphere. If…

The given problem involves calculating the age of a wood sample using the carbon-14 dating method. Carbon-14 ( $^{14}C$ ) is a radioactive isotope of carbon that decays over time, and its ratio compared to carbon-12 ( $^{12}C$ ) can be used to date organic materials. The ratio of $\frac{^{14}C}{^{12}C}$ in the wood is $\frac{1}{8}$ th of that in the atmosphere, suggesting the $^{14}C$ has decayed. The half-life of $^{14}C$ is given as 5730 years, which is the time for half the $^{14}C$ to decay.

To solve for the age of the wood, we use the formula for radioactive decay:

where $N$ is the remaining amount of $^{14}C$ , $N_0$ is the original amount of $^{14}C$ , $\lambda$ is the decay constant, and $t$ is the time (age of the wood).

Since the ratio $\frac{N}{N_0} = \frac{1}{8}$ , we infer that $N = \frac{N_0}{8}$ . Substituting into the decay equation and solving for $t$ , first we find the decay constant $\lambda$ :

Substitute $\lambda$ and rearrange to solve for $t$ :

This calculation shows that the wood sample is 17190 years old.