Question

Modeling Radioactive Decay In Exercise, complete the table for each radioactive isotope.

Answer 1

Answer :

The amount after 1000 years will be, 4.43 grams.

The amount after 10000 years will be, 1.49 grams.

Step-by-step explanation :

Half-life = 5715 years

First we have to calculate the rate constant, we use the formula :

$k=\frac{0.693}{t_{1/2}}$

$k=\frac{0.693}{5715\text{ years}}$

$k=1.21\times 10^{-4}\text{ years}^{-1}$

Now we have to calculate the amount after 1000 years.

Expression for rate law for first order kinetics is given by:

$t=\frac{2.303}{k}\log\frac{a}{a-x}$

where,

k = rate constant  = $1.21\times 10^{-4}\text{ years}^{-1}$

t = time passed by the sample  = 1000 years

a = initial amount of the reactant  = 5 g

a - x = amount left after decay process = ?

Now put all the given values in above equation, we get

$1000=\frac{2.303}{1.21\times 10^{-4}}\log\frac{5}{a-x}$

$a-x=4.43g$

Thus, the amount after 1000 years will be, 4.43 grams.

Now we have to calculate the amount after 10000 years.

Expression for rate law for first order kinetics is given by:

$t=\frac{2.303}{k}\log\frac{a}{a-x}$

where,

k = rate constant  = $1.21\times 10^{-4}\text{ years}^{-1}$

t = time passed by the sample  = 10000 years

a = initial amount of the reactant  = 5 g

a - x = amount left after decay process = ?

Now put all the given values in above equation, we get

$10000=\frac{2.303}{1.21\times 10^{-4}}\log\frac{5}{a-x}$

$a-x=1.49g$

Thus, the amount after 10000 years will be, 1.49 grams.