52^106 Te is radioactive; it αdecays to ^102 s0 Sn . " 50^102 Sn is itself radioactive and has a

step 1 In this question, we are looking at the decay of Terrillion 106 to Thin 102 via an alpha decay.

52^106 Te is radioactive; it αdecays to ^102 s0 Sn . "...

$_{52}^{106} \mathrm{Te}$ is radioactive; it $\alpha$ decays to $^{102} \mathrm{s}_{0} \mathrm{Sn} .$ " $_{50}^{102} \mathrm{Sn}$ is itself radioactive and has a half-life of 4.5 s. At $t=0,$ a sample contains 4.00 mol of $^{106}$ Te and 1.50 mol of $^{102}$ so Sn. At $t=$ $25 \mu \mathrm{s},$ the sample contains $3.00 \mathrm{mol}$ of $^{106} \mathrm{Te}$ and $2.50 \mathrm{mol}$ of $^{102} \mathrm{s} 0$ Sn. How much 102 50 Sn will there be at $t=50 \mu \mathrm{s} ?$

Key Concepts

Radioactive Decay Law

This concept describes the process by which unstable nuclei lose energy by emitting radiation over time. The decay is modeled by an exponential function where the number of remaining nuclei at any time is given by the initial number multiplied by an exponential decay factor dependent on the decay constant. This law is fundamental to understanding how radioactive materials diminish over time.

Half-Life and Decay Constant

The half-life of a radioactive substance is the time required for half of the nuclei in a sample to decay. It is mathematically connected to the decay constant (?) through the relation t1/2 = ln2/?. This relationship is essential to calculate the rate of decay and to predict how much of a radioactive isotope remains after a given period.

Sequential (Chain) Decay

In many nuclear processes, a parent isotope decays into a daughter isotope, which may itself be radioactive. Understanding sequential decay involves tracking how the decay of the parent contributes to the buildup of the daughter, while also considering any decay of the daughter isotope. This concept is critical when analyzing decay chains where intermediates can accumulate or deplete depending on their own decay rates.

Negligible Decay over Short Time Intervals

When the time interval considered is very short compared to the half-life of an isotope, the effect of its decay is often negligible. This approximation simplifies calculations by allowing one to assume that the number of atoms of a long-lived daughter remains nearly constant over the short period under study, focusing the analysis primarily on the quicker-decaying parent isotope.

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