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The half-life of a radioactive isotope is the time it takes for half of the initial amount of the isotope to decay. It is a crucial concept used in various scientific fields, including geology and physics.

To calculate the half-life, we can use the decay constant, which is derived from the exponential decay formula. Once we know the decay constant, the half-life can be found using the relation:

• \( T_{1/2} = \frac{\ln(2)}{\lambda} \)

Here, \( \lambda \) is the decay constant. The natural logarithm of 2, \( \ln(2) \), is approximately 0.693. By using this relationship, we can easily calculate the half-life of a radioactive substance if we have determined its decay constant from the observed decay rate.

Radioactive decay is best described by the exponential decay formula, which shows how much of an isotope remains after a certain period. This formula is given as:

• \( N = N_0 e^{-\frac{t}{\tau}} \)

In this formula, \( N \) is the remaining quantity after time \( t \), \( N_0 \) is the initial quantity, and \( \tau \) is the decay constant.

Exponential decay occurs because the rate at which a radioactive isotope decays is proportional to the amount of the isotope that remains.
As time goes on, less and less of the isotope is present, causing the rate of decay to slow down. This predictable decay pattern allows scientists to use isotopes for dating, as it provides a clear timeline for how much of a substance remains over any given period.

Dating meteorites involves determining their age by measuring the concentrations of certain radioactive isotopes within them. Radioactive isotopes are like natural clocks, ticking away as they decay, helping us understand the time that has passed since the meteorite solidified.

To date a meteorite, scientists measure the remaining percentage of a known radioactive isotope, as done in the given exercise, which showed 93.8% of a particular isotope remaining.

• This information allows scientists to apply the exponential decay formula to calculate the meteorite's age.
• Knowing the age of meteorites helps us learn about the early solar system's history and the formation of planets and other celestial bodies.

It is an essential tool for geologists and astronomers in their study of the universe.

The decay constant, symbolized as \( \tau \) or sometimes \( \lambda \), is an integral part of understanding radioactive decay. It measures the probability of decay of a radioactive atom per unit time.

This constant directly influences how quickly or slowly a radioactive isotope decays, and it can be found by rearranging the exponential decay formula:

• Take the natural logarithm of both sides of \( N = N_0 e^{-\frac{t}{\tau}} \).
• You get \( \ln(N/N_0) = -\frac{t}{\tau} \).
• Solving for \( \tau \) yields \( \tau = -\frac{t}{\ln(N/N_0)} \).

Once determined, the decay constant allows us to calculate the half-life of the isotope, offering valuable insights into the rate of decay. Understanding this constant is crucial for accurately timing the age of substances via radioactive decay.