Q.The half life period for a radioactive decay of C-14 is 5730 years. An archaeological artifact containing wood had only 60% of the C-14 found in a living tree. Estimate the age of the sample. OR In general, it is observed that the rate of a chemical reaction becomes double with every 10° rise in temperature. If this generalisation holds for a reaction in the temperature range of 298K to 308K, what would be the value of activation energy for this reaction? (R = 8.314 J K⁻¹ mol⁻¹)

Radioactive Decay Dating

Imagine you have a campfire that's been burning all night. When you wake up, you see the pile of ash and a few remaining embers. If you know how fast wood burns, you could look at the ratio of ash to unburnt wood and figure out roughly when the fire was lit. That's the core idea behind radioactive decay dating — except instead of wood turning to ash, we have unstable atoms turning into stable ones, and the "burn rate" is incredibly precise and constant.

The Intuition

Some atomic nuclei are unstable. They spontaneously transform into a different nucleus by emitting radiation — this is radioactive decay. The original unstable atom is called the parent, and the atom it becomes is the daughter.

Here's the key: every radioactive substance has a fixed half-life — the time it takes for exactly half of any sample to decay. If you start with 1000 parent atoms, after one half-life you'll have 500 parents and 500 daughters. After two half-lives, 250 parents and 750 daughters. After three, 125 parents and 875 daughters.

This is not a guess. It's a statistical certainty for large numbers of atoms. The half-life of carbon-14 is 5,730 years. The half-life of uranium-238 is 4.5 billion years. These numbers never change — not by heat, pressure, or chemical reactions.

So if you measure how much parent is left and how much daughter has formed, you can calculate how many half-lives have passed. Multiply by the half-life, and you get the age.

The Precise Statement

Radioactive decay follows first-order kinetics. The rate of decay at any instant is proportional to the number of parent atoms present:

where $N$ is the number of parent atoms, $t$ is time, and $\lambda$ is the decay constant — a unique number for each radioactive isotope.

Solving this differential equation gives the exponential decay law:

where $N_0$ is the number of parent atoms at time $t = 0$.

The half-life $t_{1/2}$ is related to $\lambda$ by:

This formula assumes no daughter atoms were present initially and none have been lost or added — a critical assumption we'll come back to.

How It's Actually Done

You can't count individual atoms in a rock. Instead, scientists measure the ratio of parent to daughter isotopes using a mass spectrometer. They also need to know the initial amount of daughter — often zero, but sometimes they use a trick called an isochron plot to figure it out.

The Three Big Assumptions

Every dating method rests on three assumptions. If any fails, the date is wrong.

1. The initial condition is known. You must know how much daughter was present at $t=0$. For carbon-14, this means assuming the atmospheric ratio of $^{14}\text{C}$ to $^{12}\text{C}$ has been constant. (It hasn't been perfectly constant — that's why calibration curves exist.)

2. The system has been closed. No parent or daughter atoms have entered or left the sample since formation. If groundwater leached out uranium, or if heat drove off argon, the calculated age will be wrong.

3. The decay rate has been constant. This is on extremely solid ground — experiments and observations of supernova light curves confirm that half-lives haven't changed over billions of years.

Common Methods at a Glance