Radioactive decay The mass of radioactive material in a sample has decreased by 30% since the decay began. Assuming a half-life of 1500 years, how long ago did the decay begin?

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Identify the initial mass of the radioactive material as \(M_0\) and the remaining mass after decay as \(M\). Since the mass has decreased by 30%, the remaining mass is \(M = 0.7 M_0\).

Recall the formula for radioactive decay based on half-life: \(M = M_0 \left(\frac{1}{2}\right)^{\frac{t}{T}}\), where \(t\) is the time elapsed and \(T\) is the half-life.

Substitute the known values into the decay formula: \(0.7 M_0 = M_0 \left(\frac{1}{2}\right)^{\frac{t}{1500}}\).

Divide both sides by \(M_0\) to simplify: \(0.7 = \left(\frac{1}{2}\right)^{\frac{t}{1500}}\).

Take the natural logarithm of both sides to solve for \(t\): \(\ln(0.7) = \frac{t}{1500} \ln\left(\frac{1}{2}\right)\). Then isolate \(t\) by multiplying both sides by 1500.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Radioactive Decay and Exponential Decay

Radioactive decay is a process where unstable nuclei lose mass over time, following an exponential decay model. The amount of substance decreases at a rate proportional to its current mass, which can be described by the formula N(t) = N_0 * e^(-kt), where k is the decay constant.

Half-Life

The half-life of a radioactive substance is the time required for half of the material to decay. It relates to the decay constant k by the formula t_(1/2) = ln(2)/k. Knowing the half-life allows us to determine the decay constant and model the decay process mathematically.

Solving Exponential Equations

To find the time elapsed during decay, we solve exponential equations involving the decay formula. This typically requires isolating the variable t using logarithms, especially when given a percentage decrease and the half-life, enabling calculation of the elapsed time since decay began.

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