Question

Complete the table for the radioactive isotope. (Round your answers to two decimal places.)

Isotope 226Ra
Half-Life = 1599
Initial amount = 250 g

Find the decay after 1000 years
Fine the amount after 2,000 years

Answer 1

The amount of substance at any time t is given by the equation,
                                At = A1 x e^-kt
From the given half-life,
                           At/A1 = 0.5 = e^-k(1599)
The value of k is 4.3349x10^-4
Using the same equation for the next items
 (1000 years)      At = (250 g) x e^(-4.3349x10^-4)(1000) = 162.06 grams
 (2000 years)      At = (250 g) x e^(-4.3349x10^-4)(2000) = 105.05 grams

Answer 2

Answer: The amount of Ra-226 isotope after 1000 years is 162.14 grams and after 2000 years are 105.16 grams

Explanation:

The equation used to calculate half life for first order kinetics:

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

where,

k = rate constant of the reaction = ?

$t_{1/2}$ = half life of the reaction = 1599 years

Putting values in above equation, we get:

$k=\frac{0.693}{1599yrs}=4.33\times 10^{-4}yr^{-1}$

Integrated rate law expression for first order kinetics is given by the equation:

$N=N_oe^{-kt}$

where,

N = amount left after time 't'

$N_o$ = initial amount = 250 grams

t = time taken =

k = rate constant = $4.33\times 10^{-4}yr^{-1}$

• When t = 1000 years

Putting values in above equation, we get:

$N=250\times e^{-(4.33\times 10^{-4}\times 1000)}\\\\N=162.14g$

• When t = 2000 years

Putting values in above equation, we get:

$N=250\times e^{-(4.33\times 10^{-4}\times 2000)}\\\\N=105.16g$

Hence, the amount of Ra-226 isotope after 1000 years is 162.14 grams and after 2000 years are 105.16 grams