Radon (222Rn) is a radioactive, -emitting gas which constitutes a hazard in uran

Question

Radon (222Rn) is a radioactive, -emitting gas which constitutes a hazard in uranium mines. Its half-life is 3.8 days, but it is constantly generated from the long lived parent nuclei.

a) How many -particles are emitted in 1 minute from 5.0 cm of pure Radon gas at room temperature and standard atmospheric pressure? The density of Radon under these conditions is 9.7Kg/m3.

b) How many -particles will have been emitted from that amount of Radon over the course of 1 month? Take into account the follow up decays of the daughter nuclei. You can find the decay chain at https://en.wikipedia.org/wiki/Radon. You can simplify the calculations if they don’t change the results by more than 1%.

c) After this month all of the radon will have transformed into 2 different elements. Which elements are these and how many grams will have been produced of each?

Explanation / Answer

volume of the gas Rn-222   - 5.0 cu.m     ( given as 5 cm does not make senese, assumed a printing mistake)

mass Rn-222 = 5.0 *9.7 = 48.5 kg

1 mole of Rn-222 (222 gm )contains 0.602e+24 atoms

No. of Rn-222 atoms in the gas = 48.5/0.222 *0.602e+24

                                                     = 131.52 e+24

Half life   T1/2   = 3.8 dy = 3.2832e+5 s

number of atoms decaying in 1 minute

     = 131.52 e+24 * ( 1- exp ( -60*0.693/3.2832e+5)

                    =1.67e+22 , alpha aprticle emitted in 1 minitue, for every atom decaying an alpha particle is emitted

  atoms decaying in 1 month

               = 131.52 e+24 * ( 1- exp ( -3600*24*30* 0.693/3.2832e+5)

               = 1.309e+26

Rn- 222 daughter products , Po-218 and Po-214 are also alpha -emitter with short half life,

For every Rn-222 decay there will be 3 - alpha particle, we ingore the nuclie in the chain which have very long life of years.

Total alpha partice = 3*1.309e+26

                                 = 3.93e+26

All of Rn-222 decyas to Pb-210 which has a half life of 22.3 yrs which further decays to Bi-210

For simplicity we assume we have

131.52 e+24 atoms of Pb-210 (48.5 kg)

every Pb-210 decayed produces Bi-210 which has an half-life of 5 dys and decyas to P0-210

Pb-210 atoms present after 1- month

            = 131.52e+24 * exp(- 0.693/22.3*12)

              = 131.18e+24  

              = 45.76 kg of Pb-210

Po-210 produced = 0.34 e+24 atoms

                             = 117.24 gms

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