Different isotopes of the same element can have different ha

Different isotopes of the same element can have different half lives. With t in years, the decay of plutonium-240 is given by:

Q=Qe^-0.00011t

Where the decay of the plutonium-242 is given by:

Q=Qe^-0.00018t

a) Without a detailed computation, is the half life of plutonium-240 greater or less than a half life of plutonium-242?

b) Compete the half lives, to three significant figures, of each of the isotopes.

c) Over a time scale of 3 to 4 half lives of plutonium-242, sketch the percentage remaining of each isotope (on a single graph).

Solution

a> The term half-life is defined as the time it takes for one-half of the atoms of a radioactive
material to disintegrate.

For plutonium-240 the decay is :
   Q=Qe^-0.00011t = Q/e^(.00011t)   -----> (1)

Now for half life Q = Q/2 on the left hand side
=> Q/2=Qe^-0.00011t = 1/2 =1/e^(.00011t)
e^(.00011t) = 2
take log both sides
=> .00011t = ln(2)
t(half life)= ln(2)/.00011     ---------(2)

For plutonium-242 the decay is :
   Q=Qe^-0.00018t= Q/e^(.00018t)   -----> (3)
likewise (2) becomes
t(half life)= ln(2)/.00018   -----(4)

now the denominator of (4) is greater than that of (2)

t(half life) for plutonium-240 is > t(half life) for plutonium-242

b> t(half life) for plutonium-240

t(half life)= ln(2)/.00011 = 6301.33 years

t(half life) for plutonium-242

t(half life)= ln(2)/.00018 = 3850.82 years