It can be proven5 that the half-life and the decay constant are related by the equation: Note: ln2 is the natural log of 2; ln2 loge2 0.693. The decay constant has a unit of per second, s1.
RELATIONSHIP BETWEEN HALF-LIFE AND DECAY CONSTANT
T1/2 T1/2
In2 l
half-life decay constant SAMPLE PROBLEM 22A
The half-life of carbon-14 is 5 730 years. (i) How much of a carbon-14 sample remains after 11 460 years? (ii) Calculate the decay constant of carbon-14. (iii) What is the rate of decay for a sample containing 5.2 1011 atoms of carbon-14?
SAMPLE ANSWER 22A (i) A quarter of the original sample remains.
Explanation: (ii) T1/2
In2 l
5 730(365)(24)(60)(60) In2 l
l 1.807 1011 3.8 1012 In2
(iii) The decay constant of carbon-14 is 3.8 1012 s1. The rate of decay is 2.014 Bq for the sample.
A lN (3.8 1012)(5.3 1011) 2.014
11 460 5 730 2
, which implies the carbon has been decaying for two half-lives.This means that of the original sample remains. 1 22
Decay chains
Unstable nuclei decay into their daughter nuclei. These in turn may be unstable and decay further.This list of isotopes from the parent isotope down to a stable isotope is called a decay chain. An example (which doesn’t need to be learnt off!) is the decay chain of Uranium-238.
238 92U:234
:222 :210
90Th a:234 86Rn a:218 82Pb a:210
91Pa b :234 84Po a:214 83Bi b :210
92U b :230 82Pb a:214 84Po b:206
90 Th a:
226 88Ra a
83Bi b :214 82Pb a
5This proof is available on www.gillmacmillan.ie for illustrature purposes and is not examinable. THE NUCLEUS 407 84Po b
