Question

"You are dating Moon rocks based on their proportions of uranium-238 (half-life of about 4.5 billion years) and its ultimate decay product, lead. Find the age for each of the following."a. A rock for which you determine that 59 % of the original uranium-238 remains, while the other 41 % has decayed into lead.b. A rock for which you determine that 65 % of the original uranium-238 remains, while the other 35 % has decayed into lead.

Answer 1

Explanation:

a.

Initial mass of the isotope = x

Time taken by the sample to decay its mass by 41% = t

Formula used :

`N=N_o* e^(-\lambda t)\\\\\lambda =\frac{0.693}{t_{(1)/(2)}}`

where,

`N_o` = initial mass of isotope = x

N = mass of the parent isotope left after the time, (t) = 59% of x = 0.59x

`t_{(1)/(2)}` = half life of the isotope = 4.5 billion years

`\lambda` = rate constant

Now put all the given values in this formula, we get

`0.59x=x* e^{-(\frac{0.693}{\text{4.5 billion years}})* t}`

t = 3.4 billion years

The age a rock is 3.4 billion years.

b.

Initial mass of the isotope = x

Time taken by the sample to decay its mass by 35%= t

Formula used :

`N=N_o* e^(-\lambda t)\\\\\lambda =\frac{0.693}{t_{(1)/(2)}}`

where,

`N_o` = initial mass of isotope = x

N = mass of the parent isotope left after the time, (t) = 65% of x = 0.65x

`t_{(1)/(2)}` = half life of the isotope = 4.5 billion years

`\lambda` = rate constant

Now put all the given values in this formula, we get

`0.65x=x* e^{-(\frac{0.693}{\text{4.5 billion years}})* t}`

t = 2.8 billion years

The age a rock is 2.8 billion years.