Question

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

Amount Amount
after after
Half-life Initial 1000 10,000
Isotope (In years) quantity years years
226Ra 1599 8 grams

Answer 1

The amount after 1000 years will be, 5.19 grams.

The amount after 10000 years will be, 0.105 grams.

Step-by-step explanation :

Half-life = 1599 years

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

`k=(0.693)/(t_(1/2))`

`k=\frac{0.693}{1599\text{ years}}`

`k=4.33* 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=(2.303)/(k)\log(a)/(a-x)`

where,

k = rate constant =
`4.33* 10^(-4)\text{ years}^(-1)`

t = time passed by the sample = 1000 years

a = initial amount of the reactant = 8 g

a - x = amount left after decay process = ?

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

`1000=(2.303)/(4.33* 10^(-4))\log(8)/(a-x)`

`a-x=5.19g`

Thus, the amount after 1000 years will be, 5.19 grams.

Now we have to calculate the amount after 10000 years.

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

`t=(2.303)/(k)\log(a)/(a-x)`

where,

k = rate constant =
`4.33* 10^(-4)\text{ years}^(-1)`

t = time passed by the sample = 10000 years

a = initial amount of the reactant = 8 g

a - x = amount left after decay process = ?

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

`10000=(2.303)/(4.33* 10^(-4))\log(8)/(a-x)`

`a-x=0.105g`

Thus, the amount after 10000 years will be, 0.105 grams.