Generally, more massive nuclides tend to be more unstable to...

Generally, more massive nuclides tend to be more unstable to alpha decay. For example, the most stable isotope of uranium, ${ }^{238} \mathrm{U},$ has an alpha decay half-life of $4.5 \times 10^{9} \mathrm{y} .$ The most stable isotope of plutonium is ${ }^{244} \mathrm{Pu}$ with an $8.0 \times 10^{7} \mathrm{y}$ half-life, and for curium we have ${ }^{248} \mathrm{Cm}$ and $3.4 \times 10^{5} \mathrm{y}$. When half of an original sample of ${ }^{238} \mathrm{U}$ has decayed, what fraction of the original sample of (a) plutonium and (b) curium is left?

Key Concepts

Exponential Decay

Exponential decay describes the process by which the quantity of a radioactive substance decreases at a rate proportional to its current amount. This mathematical model is central to understanding how unstable nuclei transform over time, using the formulation N(t) = N0 exp(-?t), where N0 is the initial quantity, ? is the decay constant, and N(t) is the remaining quantity after time t.

Half-life

The half-life is the time required for half of the radioactive nuclei in a given sample to decay. It is a critical measure that characterizes the stability of a nuclide and is often used to compare different radioactive substances. The half-life provides a straightforward way to quantify and predict the uptake or loss of material due to radioactive decay.

Decay Constant

The decay constant is a parameter that represents the probability per unit time that a single nucleus will decay. It is mathematically linked to the half-life by the relation ? = ln(2)/t½. This constant is a fundamental aspect in the exponential decay equation and helps in quantifying the rate of decay of a nuclide.

Comparative Analysis of Radioactive Stability

This concept involves comparing the decay behavior of different nuclides by examining their half-lives and decay constants. By evaluating these parameters, one can understand the relative rates at which different substances decay and predict the remaining fractions of material after a specific period, despite the nuclides having varying degrees of instability.

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