In Exercises 14 - 18, we list some radioactive isotopes and their associated hal

Question

In Exercises 14 - 18, we list some radioactive isotopes and their associated half-lives. Assume that each decays according to the formula Alt) - Apekt where A, is the initial amount of the material and k is the decay constant. For each isotope: . Find the decay constant k. Round your answer to four decimal places. ·Find a function which gives the amount of isotope A which remains after time t. (Keep the units of A and t the same as the given data.) Determine how long it takes for 90% of the material to decay. Round your answer to two decinal places. (HINT: If 90% of the material decays, how mach is left?) Uranium 235, used for nuclear power, initial amount 1 kg grams, half-life 704 million years.

Explanation / Answer

A = A0ekt
half life = 704 millon years = 704*106 years;
At half life, A = A0/2
substituting it in the equation,
A0/2 = A0 ekt
ekt = 1/2
kt = ln (1/2)
k = ln (1/2) / t = -0.693/ 7.04*108 = -0.0984375 * 10-8 = -9.84375 * 10-10
Thus, decay constant k = -9.84375 * 10-10 per year

Amount remaining = A = A0 e-9.84375*10^(-10) t
This expression gives the amount of isotope remaining (A) after a time period of 't' years;

If 90% of the isotope decays then what's left is 100-90 = 10% ;
substituting in the equation :
A = A0/10 = A0 e-9.84375*10^(-10) t
e-9.84375*10^(-10) t = 1/10
-9.84375*10-10 * t  = ln (1/10) = -2.302585
t = -2.302585/ (-9.84375*10-10) = 0.23391* 1010 years = 2.3391* 109 years or 2.3391 billion years;

Thus, after ~ 2.3391 billion years, only 10% of the isotope will be left;

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