The age of wine can be determined by measuring the trace...

The age of wine can be determined by measuring the trace amount of radioactive tritium, ${ }^{3} \mathrm{H}$, present in a sample. Tritium is formed from hydrogen in water vapor in the upper atmosphere by cosmic bombardment, so all naturally occurring water contains a small amount of this isotope. Once the water is in a bottle of wine, however, the formation of additional tritium from the water is negligible, so the tritium initially present gradually diminishes by a first-order radioactive decay with a half-life of 12.5 years. If a bottle of wine is found to have a tritium concentration that is $10.0 \%$ of freshly bottled wine, what is the age of the wine?

Key Concepts

Radioactive Decay

Radioactive decay is the process by which unstable atoms lose energy by emitting radiation. It occurs spontaneously and is characterized by a decay constant that determines the probability of decay per unit time, making it crucial in understanding how radioactive substances decrease over time.

First-Order Kinetics

First-order kinetics describes reactions where the rate of decay or reaction is directly proportional to the amount of the substance present. In the context of radioactive decay, this means that a constant fractional amount of the substance decays per unit time, leading to an exponential decay curve.

Half-Life

The half-life of a radioactive substance is the time required for half of the radioactive atoms in a sample to decay. It is a fundamental concept in radioactive decay, providing a measure of the rate at which the substance diminishes and is pivotal in dating techniques.

Exponential Decay

Exponential decay is the mathematical representation of a process where the quantity decreases at a rate proportional to its current value. This decay is described by an exponential function, which is essential for modeling the behavior of first-order radioactive decay.

Isotopic Dating

Isotopic dating is a technique that uses the known decay rates of radioactive isotopes to determine the age of an object or sample. By measuring the remaining concentration of a radioactive isotope and applying decay laws, it allows for the estimation of time elapsed since the formation or deposition of the sample.

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