In 1908 Rutherford together with H- Geiger measured the rate of emission of `alpha` particles (x) by radius (in the nature this element is represented by a single nuclide `._(88)^(226)Ra`) and found that `1.00 g` of radius emits `x=3.42xx10^(10) alpha`- particle per second.
In 1911 Ritherford and American physical chemist B. Boltwood measured the rate of formation of helium from radius. This experiment permits to obtain the most accurate value of Avogadro's number avaliable at the time, given that the value of molar volume of ideal gas was well established. To achieve this goal a sample of radium salt purified from decays products and contaning `m=192` mg of Re was put into a device an the volume of the evolved helium was measured. After 83 days `(t=83.0 "days")` of the experiment `6.58 mm^(3)` of He was collected `(V_(He)=6.58 mm^(3) "corrected to" 0^(@)C "and" 1 atm)`.
To inderstand the results of this experiment we shall need the kinetic scheme of redioactive decay of Re which is given below (half-lives are the arrows, the type of decay is below the arrows).
`Ra underset(alpha)overset( gt 1500 "years")(rarr)Rn underset(alpha)overset(3.83 "days")(rarr)RaAunderset(alpha)overset(3.05 "min")(rarr)RaBunderset(beta)overset(26.8)(rarr)RaCunderset(beta)overset(17.7 "min")(rarr)`
`rarrRaC'underset(alpha)overset(1.63xx10^(-4) S)(rarr)RaDunderset(beta)overset(27.1 "years")(rarr)RaEunderset(beta)overset(5 "days")(rarr)Pounderset(alpha)overset(138 "days")(rarr)Pb` (stable)
(RaA-RaE are intermediate products of radon decay).
Write the first six radioactive dacays using a modern natation showing atomic and mass numbers of all nuclei involved.
As a rough first approximation half-lives of all radius decay products, except those of RaD and Po, may be assumed to be negligible compared to the time of measurement t. Using this approximation perform the following calculations.

Answer

`._(88)^(226)Ra rarr ._(86)^(222)Rn rarr ._(2)^(4)He`
`._(33)^(222)Rn rarr ._(84)^(218)Po rarr ._(2)^(4)He`
`._(88)^(218)Po rarr ._(82)^(214)Pb rarr ._(2)^(4)He`
`._(82)^(214)Pb rarr ._(83)^(214)Bi rarr e`
`._(83)^(214)Bi rarr ._(84)^(214)Po rarr e`
`._(84)^(214)Po rarr._(82)^(210)Pb rarr ._(2)^(4)He`