Polonium- 210 , po210 , decays to lead- 206 , pb206 , by alpha emission according to the equation po84210⟶pb82206+he24 if the half-life, /2 , of po210 is 138.4 days , calculate the mass of pb206 produced from a 561.0 mg sample of polonium(iv) chloride, pocl4 , that is left untouched for 333.8 days . assume that the only polonium isotope present in the sample is the po210 isotope. the isotopic molar masses of po210 is 209.98 g/mol and pb206 is 205.97 g/mol .

0.269 g

Explanation:

 ⟶  +  

Data:

Half-life of    (T(1/2)) = 138.4 days

Mass of PoCl4 = 561 mg (0,561 g) and molecular weight of PoCl4 = 350. 79 g/mol  

Time = 338.8 days  

Isotopic masses  

= 209.98 g/mol  

= 205.97 g/mol  

Concepts  

Avogadro’s number: This is the number of constituent particles that are contained in a mol of any substance. These constituted particles can be atoms, molecules or ions). Its value is 6.023*10^23.  

The radioactive decay law is  

N=Noe^(-λt)

Where:  

No = number of atoms in t=0

N = number of atoms in t=t (now) in this case t=338.8 days  

λ= radioactive decay constant  

The radioactive constant is related to the half-life by the next equation  

λ=

so  

λ=  =0,005008 days^(-1)

No (Atoms of    in t=0)

To get No we need to calculate the number of atoms of     in the initial sample. We have a sample of 0,561 g of PoCl4. If we get the number of moles of PoCl4 in the sample, this will be the number of moles of    in the initial sample.  

This is:

= 0,001599 mol of  PoCl4

This is the number of mol of   in the initial sample.

To get the number of atoms in the initial sample we use the Avogadro’s number = 6.023*10^23  

0,001599 mol of   * 6.023*10^23 atoms/ mol of   = 9.632 *10^20 atoms of  

Atoms after 338.8 days

We use the radioactive decay law to get this value  

N=Noe^(-λt)

N=9.632*10^20 e^(-0,005008 days^(-1) * 338.8 days) =1.765*10^20

This is the number of atoms of   in the sample after 338.8 days has passed  

The number of atoms   transformed is equal to the number of atoms of  produced.  

The number of atoms of transformed is No - N  

9.632 *10^20 – 1.765 *10^20 = 7.866*10^20

So, 7.866*10^20 is the number of atoms of produced  

We can get the mass with the Avogadro’s number

(7.866*10^20 atoms of ) / ( 6.023*10^23 atoms of / mol of =  0.001306 moles of

This number of moles have a mass of:

(0,001306 moles of )* (205.97 g of /mol of ) = 0.269 g