Question

I'm working problem set 2 - 1H-1. I get part a, but cannot understand part b. I don't understand where the equation for lamda came from in the answer key.

Answer 1

Part a) is asking you for an equation for lamda.
Lamda is the TIME taken for \[y _{0}\] to decay to half its amount
\[1/2(y _{0})\]
You are given the equation for decay that is
\[y = y _{0}e ^{-kt}\]
where y = is the amount left after time 't', so setting \[y = 1/2(y_{0})\] and then solving for 't' (time) will give you the equation for the time taken to decrease to half its initial amount ie it's half life.
\[1/2(y_{0}) = y_{0}e ^{-kt}\]
\[y_{0}/y_{0}2= e ^{-kt}\] cancel
\[1/2 = e ^{-kt}\] take the natural log of each side
\[\ln 1/2 = -kt\] divide by -k
\[(\ln1/2)/-k = t \] simplify
\[t = -\ln2/-k\] here's the time taken to decay to half the initial amount
\[t = \ln2/k\]\[\lambda = \ln2/k\] change the t to lambda like the initial statement and question.