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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

Verified Answer :

meerkat18

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: 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

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