Answer to Question #166578 in Atomic and Nuclear Physics for areeba

Question

A few days before the Christmas of 1898, Pierre Curie scrawled the word 'radium' in his notebook as the name for a new element he and his wife Marie had discovered in their laboratory in Paris. Radium is a brilliant white, luminescent, rare and highly radioactive metallic element. They used the radiation emitted from radium 226 to define the unit the curie (Ci). 1 curie = 3.7 x 10¹⁰ decays per second.

a.     A sample of 3.2 x 10²² radium 226 atoms have an activity of 12 Ci. Determine the decay constant for radium 226.    

b.     Radium 226 is a solid at room temperature and it decays to radon 222 which is a colourless and odourless radioactive gas at room temperature. Radon 222 has a half life of 3.8 days and also decays via alpha decay. 

Compare the dangers of exposure to samples of radium 226 and radon 222, assuming each sample initially has the same number of atoms of each element.  

c.     During the decay of radium 226, nuclear equilibrium of radon 222 can occur at which the rates of production and decay of radon 222 are equal. 

(i)             What does this tell you about the number of radon 222 atoms at any point in time?  

(ii)            Determine the number of radon 222 atoms that must be present in the sample in part (a) for this nuclear equilibrium to occur.

Answer 1

a)

$$(3.2\cdot10^{22})\lambda=12\cdot3.7\cdot10^{16}\\\lambda=1.39\cdot10^{-5}\ s^{-1}$$

b)

$$A=(3.2\cdot10^{22})\frac{\ln{2}}{3.8(24)(3600)}=6.8\cdot10^{16}\ Bq$$

c) i) It is constant.

ii)

$$(1.39\cdot10^{-5})N=6.8\cdot10^{16}\\N=4.9\cdot10^{21}$$