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## Homework Statement

Analyzing a rock sample, it is found that it contains 1.58 mg of

^{238}U and 0.342 mg of

^{206}Pb, which is the stable final product of the disintegration of

^{238}U. Assume that all

^{206}Pb found comes from the disintegration of

^{238}U originally contained in sample. How old is this rock?

## Homework Equations

Radioactive decay:

[tex]N = N_0 e^{- \lambda t},[/tex] where [itex]\lambda[/itex] stands for the decay constant, and [itex]N_0[/itex] and [itex]N(t)[/itex] stands for the initial and the number of particles in some instant of time. To convert the mass of the Uranium and Plumbum of the sample, it will be necessary to use the number of Avogadro [itex]N_A = 6.02 \times 10^23[/itex], and their respective molar mass, 238.02891u and 207.2 u.

## The Attempt at a Solution

My attempt consisted in convert the mass of Uranium and Plubum to [itex]N_U[/itex] and [itex]N_{Pb}[/itex] using their molar mass, which resulted in [itex]4.00\times 10^{21}[/itex] and [itex] 9.94\times 10^{20} [/itex] respectively.

In my first try, I used that [itex]N_0 = N_U + N_Pb[/itex] in the Radioactive decay, and got [itex] 9.99\times 10^8 [/itex] years.

The author claims that this time is [itex] 1.45\times 10^9 [/itex] years.

In my second try, I observed the

^{238}U series here, https://en.wikipedia.org/wiki/Decay_chain#Uranium_series, and verified that there are 8 alpha decays before one Plubum atom appears. With this in mind I used that [itex] N_0 = N_U + 32*N_{Pb} [/itex] (I considered that an alpha decay takes 4u of mass from the sample) and got [itex] 9.86\times 10^9 [/itex] years.

Please, I really need some help, I just can't figure out any better assumptions to solve this. I must be using some very incorrect reasoning.