A radioactive isotope has a half-life of 30 days . The time required for 7 8 of its original mass to decay is:

Step-by-step solution

Radioactive decay follows first-order kinetics. Let the initial mass of the radioactive isotope be N₀ . The fraction of mass that has decayed is 7 8 . Therefore, the fraction of mass remaining ( N ) is 1 - 7 8 = 1 8 . The relationship between the remaining amount and the number of half-lives ( n ) is given by: N N₀ = ( 1 2 )^n Substituting the values: 1 8 = ( 1 2 )^n ( 1 2 )^3 = ( 1 2 )^n Thus, n = 3 . The total time required is the number of half-lives multiplied by the half-life period: t = n t_ 1/2 t = 3 30 days