The common form of uranium (^238 U) has a half-life of 4.5 billion years. It decays through a se

The common form of uranium (^238 U) has a half-life of 4.5...

The common form of uranium $\left(^{238} \mathrm{U}\right)$ has a half-life of 4.5 billion years. It decays through a sequence of unstable isotopes to lead $\left(^{206} \mathrm{Pb}\right),$ releasing a number of helium nuclei in the process. (Radioactive decay is the source of most of the helium in Earth's atmosphere today.) Find the ratio of the number of $^{238} \mathrm{U}$ to $^{206} \mathrm{Pb}$ atoms you would expect to find for rocks that last melted $1,2,3,$ and 4 billion years ago.

Key Concepts

Radioactive Decay

Radioactive decay is the spontaneous process by which an unstable atomic nucleus loses energy and transforms into a different nucleus or a different energy state by emitting radiation such as alpha particles, beta particles, or gamma rays. This process follows well-defined probabilistic laws and is a fundamental mechanism by which isotopes change over time.

Half-Life

Half-life is the characteristic time required for half of the nuclei in a sample of a radioactive isotope to decay. It is a critical concept in understanding the rate of radioactive decay and is used to determine the age of rocks and minerals by comparing the proportion of parent and daughter isotopes present.

Exponential Decay Law

The exponential decay law describes the mathematical relationship that governs the rate at which a radioactive substance decays. It is expressed by the equation N(t) = N0 e^(-?t), where N0 is the initial quantity of the substance, N(t) is the quantity that remains after time t, and ? is the decay constant. This law underpins calculations that determine the remaining fraction of a radioactive parent isotope and the accumulated amount of its decay product.

Decay Constant

The decay constant (?) is a parameter that quantifies the probability per unit time that a given atomic nucleus will decay. It is directly related to the half-life by the relation ? = ln(2)/t1/2 and is essential for converting between the half-life of a radioactive isotope and the rate at which its quantity decreases over time.

Radiometric Dating

Radiometric dating is a method used to determine the age of rocks and other materials by comparing the relative abundances of a parent radioactive isotope and its daughter products. It relies on the predictable decay process described by the exponential decay law, allowing scientists to back-calculate the time elapsed since the rock or mineral solidified.

Transcript

00:01 This is a moderately challenging question that requires you to think about a different way in which you can carry out a first -order integrated rate -lock calculation for the decay of uranium -238 and uranium -235.

00:17 You are given the current relative abundances at 99 .28 % for uranium 238 and a much smaller amount at 0 .72 % for uranium 235.

00:28 It says that the respective half -lifes are 4 .5 times 10 to the 9 for 238 and 7 .1 times 10 to the 8 for 235.

00:40 So we see that uranium 235 has a shorter half -life, so it decays more quickly.

00:47 Therefore, we must have had 4 .5 billion years ago when the solar system the earth was created, we must have had more uranium 235 with respect to uranium 238 than we do now, because the uranium 235 has decayed more quickly than the uranium 238.

01:06 To carry out these calculations, we need to convert our half -lifes into rate constants or decay constants.

01:13 We'll do that by using the half -life equation, where k is equal to natural log of 2 divided by the half -life.

01:21 Knowing these two values, we should be able to now calculate, in two separate calculations, the amount of each of these nuclides 4 .5 billion years ago, assuming right now we have a 100 gram sample and we had back then a 100 gram sample.

01:43 So this is the key to answering this question.

01:46 If we assume that we have 100 gram sample, then right now at time t, there are 99 .8, 0 .28 grams of the 100 gram sample would be uranium 238...

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