Question

The activity of a radioactive sample is the number nuclear disintegrations per second, which is equal to the first-order rate constant times the number of radioactive nuclei present. The fundamental unit of radioactivity is the curie (Ci), where 1 Ci corresponds to exactly 3.70 × 1010 disintegrations per second. This decay rate is equivalent to that of 1 g of radium−226. The molar mass of Ra−226 is 226.03 g/mol.

a.Calculate the rate constant for the radium decay.

b.Calculate the half-life for the radium decay.

c.Starting with 2.7 g of the radium sample, what is the activity after 450 yr?

answer in scientific notation

Answer 1

The radioactive disintegration follows first order rate law.

rate constant = [2.303/t] log (a/a-x)

And rate of reaction = rate constant x[reactant]

For radioactivity

decay rate = k [N]

a) decay rate = 1 .00 Ci

= 3.70x10¹⁰ dps

[N] = (1g/226g/mol) x6.023 x 10²³

thus rate constant k = rate /[N]

= 3.7x10¹⁰ x226/6.023x10²³

   = 1.388x10^-11/s

b)Thus half life of first order reactiion = 0.693/k

= 0.693/1.388x10^-11/s

=4.99x10¹⁰s

= 1.582 x10³ yrs

= 1582 yrs

c) starting with 2.7g

N₀₌ a= 2.7/226 moles and N =(a-x) = xg/226 moles

we know

k = [2.303/t] log (a/a-x)

0.693/1582Yrs = [2.303 /450yrs ] log (2.7/x)

solving for x ,

x=2.21 g

Thus activity after 450 yrs = rate constant [N]

= (1.388x10^-11/sx 2.21x6.023x10²³)/226

= 8.174x10¹⁰ dps