 # Two different radioactive isotopes decay to 10% of their respective original amounts. Isotope A does this in 33 days, while isotope B does this in 43 days. What is the approximate difference in the half-lives of the isotopes? A. 3 B. 10 C. 13 D. 33 Decay equation is given by---------------------(1)where A is amount at time t, P is original amount, k is decay constant, t is time So using this on isotope A first. It decays to 10% of original. So lets say its original amount was , so now its amount will be 10% of which will be So plug in A place, in P place, then 33 as time in t place and let its decay constant be . So plug all these in equation (1)Divide both sides by Solve for for that take ln on both sides-------------(2)Similarly for isotope B, if its original amount was then 10 % of will be . And if its decay constant be and its time is 43so plug these in equation (1)Divide both sides by ------------------------(3)again solve for -------------(3)Now half life means amount is 1/2 of original amount. So that means if P is original amount then A will become or 0.5PUsing this in equation (1) we getDivide both sides by P------------------------(3) -----------(4)thats the equation for half lifeSo plug result of as -0.069775 in k place in equation (4) we get its half life as ----------------------(5)So life for isotope A is 9.933989Similarly find half life of isotope B.For that plug as -0.053548 in k place in equation (4) we get its half life as -----------(6)so half life of isotope B is 12.9443Now find difference betwen half lives of two isotopes by subtracting results (5) and (6)so we get final answer as 12.9443- 9.933989 = 3.01So choice A as 3 is the right answer.

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