Answer : The amount of the normal proportion of carbon-14 to carbon-12 is, 1.07
Explanation :
Half-life = 5730 years
First 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}{5730\text{ years}}[/tex]
[tex]k=1.21\times 10^{-4}\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]1.21\times 10^{-4}\text{ years}^{-1}[/tex]
t = time passed by the sample = 22920 years
a = let initial amount of the reactant = 100
a - x = amount left after decay process = ?
Now put all the given values in above equation, we get
[tex]22920=\frac{2.303}{1.21\times 10^{-4}}\log\frac{100}{a-x}[/tex]
[tex]a-x=6.25g[/tex]
[tex]x=100-6.25=93.8g[/tex]
Now we have to calculate the amount of the normal proportion of carbon-14 to carbon-12.
[tex]\frac{\text{carbon-14}}{\text{carbon-12}}=\frac{100}{93.8}=1.07[/tex]
Thus, the amount of the normal proportion of carbon-14 to carbon-12 is, 1.07
