What should be the mass ratio ^208 Pb /^232 Th in a meteorite that is approximately 2.7 ×10^9 ye

step 1 Okay, for the given problem, we are provided with the age of metrival, which is 2 .7, multiplied

What should be the mass ratio ^208 Pb /^232 Th in a...

What should be the mass ratio $^{208} \mathrm{Pb} /^{232} \mathrm{Th}$ in a meteorite that is approximately $2.7 \times 10^{9}$ years old? The half-life of $^{232} \mathrm{Th}$ is $1.39 \times 10^{10}$ years. [Hint: One $^{208} \mathrm{Pb}$ atom is the final decay product of one $^{232}$ Th atom.

Key Concepts

Radioactive Decay Law

The radioactive decay law explains how the quantity of a radioactive substance decreases over time in an exponential manner. It is commonly expressed in the form N(t) = N0 * (1/2)^(t/T1/2), where N0 is the initial quantity, t is the elapsed time, and T1/2 is the half-life. This law is pivotal for calculating the remaining fraction of a parent nuclide after a certain period, as well as for determining the quantity of decay products formed.

Half-Life

Half-life is the time required for half of a given amount of a radioactive substance to decay. It is an intrinsic property of the nuclide and directly impacts the rate at which the substance transforms into its daughter product. Half-life serves as a key parameter in decay equations and is essential for predicting the proportions of isotopes present after a given time period.

Parent-Daughter Relationship in Radioactive Decay

In radioactive decay, each decaying parent atom transforms into a daughter atom, establishing a one-to-one correspondence in many decay schemes. This relationship allows for the calculation of the amount of daughter isotope produced from the decay of the parent. When linked with the decay law, it becomes possible to estimate the proportion of daughter to parent atoms in a sample over time.

Mass Ratio Calculation in Radioactive Systems

Mass ratio calculations in radioactive systems involve converting the number of atoms of parent and daughter isotopes into masses by considering their individual atomic masses. This approach is important when determining the isotopic composition of a sample, as it requires accounting for both the decay branch (where each decayed parent atom produces one daughter atom) and the differing atomic weights, thereby allowing scientists to infer important chronological or compositional information.

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