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

Step-by-step explanation:

a.

Initial mass of the isotope = x

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

Formula used :

$N=N_o\times e^{-\lambda t}\\\\\lambda =\frac{0.693}{t_{\frac{1}{2}}}$

where,

$N_o$ = initial mass of isotope  = x

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

$t_{\frac{1}{2}}$ = half life of the isotope  = 4.5 billion years

$\lambda$ = rate constant

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

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

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 :

$N=N_o\times e^{-\lambda t}\\\\\lambda =\frac{0.693}{t_{\frac{1}{2}}}$

where,

$N_o$ = initial mass of isotope  = x

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

$t_{\frac{1}{2}}$ = half life of the isotope  = 4.5 billion years

$\lambda$ = rate constant

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

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

t = 2.8 billion years

The age a rock is 2.8 billion years.