The radioactive decay of carbon-14 is first order and the half-life is 5800 years. While a plant or animal is living, it has a constant proportion of carbon-14 (relative to carbon-12) in its composition. When the organism dies, the proportion of carbon-14 decreases as a result of radioactive decay and the age of the organism can be determined if the proportion of carbon-14 in its remains is measured. If the proportion of carbon-14 in an ancient piece of wood is found to be one quarter that in living trees, how old is the sample?

Question

anonymous · Answer

i know this formula: $$   t _{1/2} = {\ln(2)}/{k}$$   but what is k?

anonymous · Answer

k is constant of decay and it is sometimes written as lambda

anonymous · Answer

but how do i calculate it? i have no other information besides that paragraph

anonymous · Answer

ok ill try to solve it, if i remember correctly your formula is correct, furthermore you can write it like this:
t(1/2)=ln(2) / k = T ln(2) 
where T (tau) is mean lifetime
so it says in your task that 1/4 of element remains so this stands that 2 half-life's have passed because one lifetime is 1/2 of elements and two lifetimes is 1/4 of elements
do you have choices for answers cause im a bit rusty with decays...

ghazi · Answer

\[just solve \it by K= 2.303/t * log a/(a-x) where x is remaining amount of carbon ..now you can solve for 't' that is how old is the sample..since you have obtained rate constant by half life equation

anonymous · Answer

Since this is a radioactive disintegration and it is stated, the reaction is first order. For a first order reaction, the half life (t subscript 1/2)should be equal to ln(2) / k. Since the half life is given, k can be calculated. :) Okay, then for the amount of carbon-14 remaining, the equation would be C = Co exp (-kt) where k is determined before. using the relationship C = 1/4 of Co C now is one-quarter that of the one in living trees, t can be calculated, which is now 11,600 years. This would actually make sense since the half life is 5800 and two half-lives have passed already, thus. two 5800 years (which is 11600 :)))