Answer :
The parent and daughter concentrations (in percentages) is, 60 % and 40 % respectively.
The age of rock is [tex]3.32\times 10^9\text{ years}[/tex]
Explanation :
First we have to calculate the parent and daughter concentrations (in percentages).
[tex]\text{Parent concentrations}=\frac{1.8g}{3g}\times 100=60\%[/tex]
and,
[tex]\text{Daughter concentrations}=\frac{(3-1.8)g}{3g}\times 100=40\%[/tex]
As we know that, the half-life of uranium-238 = [tex]4.5\times 10^9[/tex] years
Now we have to calculate the rate constant, we use the formula :
[tex]k=\frac{0.693}{t_{1/2}}[/tex]
[tex]k=\frac{0.693}{4.5\times 10^9\text{ years}}[/tex]
[tex]k=1.54\times 10^{-10}\text{ years}^{-1}[/tex]
Now we have to calculate the time passed.
Expression for rate law for first order kinetics is given by:
[tex]t=\frac{2.303}{k}\log\frac{a}{a-x}[/tex]
where,
k = rate constant = [tex]1.54\times 10^{-10}\text{ years}^{-1}[/tex]
t = time passed by the sample = ?
a = initial amount of the reactant = 3 g
a - x = amount left after decay process = 1.8 g
Now put all the given values in above equation, we get
[tex]t=\frac{2.303}{1.54\times 10^{-10}}\log\frac{3}{1.8}[/tex]
[tex]t=3.32\times 10^9\text{ years}[/tex]
Therefore, the age of rock is [tex]3.32\times 10^9\text{ years}[/tex]
