A rate of decay of a radioactive element is independent of temperature, pressure, Infact for any external conditions. A simple law discovered by Rutherford states that a certain fraction of any sample of radioactive element undergoes change in a unit time. If we start with  10mg   of the radioactive substance  (t1/2  =  1day)  , only  5mg will be left at the end of the first day. At the end of 2nd day  2.5mg will be left and so on. λ  =  2.303tlogaa − x A radioactive element has atomic mass 90 amu and a half – life of 28 years. The number of disintegrations per second per gm of the element is

The decay constant (λ) of a radioactive element can be calculated using the formula:

λ = 0.693 / t1/2

where t1/2 is the half-life of the element. Given that the half-life of the element is 28 years, we can substitute this into the formula to find λ:

λ = 0.693 / 28 years = 0.02475 years^-1

The number of atoms (N) in 1 gram of the element can be calculated using the formula:

N = Avogadro's number (6.022 x 10^23 atoms/mole) / atomic mass

Given that the atomic mass of the element is 90 amu, we can substitute this into the formula to find N:

N = (6.022 x 10^23 atoms/mole) / 90 amu = 6.69 x 10^21 atoms

The number of disintegrations per second (A) can be calculated using the formula:

A = λN

Substituting the values we found for λ and N into this formula gives:

A = (0.02475 years^-1)(6.69 x 10^21 atoms) = 1.65 x 10^20 disintegrations/year

Since there are approximately 3.15 x 10^7 seconds in a year, we can convert this to disintegrations per second by dividing by the number of seconds in a year:

A = (1.65 x 10^20 disintegrations/year) / (3.15 x 10^7 seconds/year) = 5.24 x 10^12 disintegrations/second

So, the number of disintegrations per second per gram of the element is approximately 5.24 x 10^12.