2. C  Suppose you have a block of chocolate which you have to eat according to the following rule: Every minute you can eat HALF OF THE REMAINING CHOCOLATE. Topic 7.2 Extended C – Decay Rate and Half Life Perhaps we are in France!  We can represent the amount of chocolate over time with this characteristic curve...  Which is a decaying exponential function that looks like this: C = C o e - t The Chocolate Exponential Decay Function  Where is the decay constant and e = 2.7182... is the base of natural logarithms. FYI: C 0 is the initial amount of chocolate, and C is the amount of chocolate left after time t. 0 t 1/2 2t 1/2 3t 1/2 4t 1/2 5t 1/2 6t 1/2 7t 1/2 8t 1/2 FYI: We call the time it takes to lose half of each remaining piece of chocolate the half-life (t 1/2 ). Question: How long does the chocolate last? t CoCo

3.  Note that this decaying exponential function has the same structure as that of the RC discharge curve of your lab. Topic 7.2 Extended C – Decay Rate and Half Life V = V o e - t where = 1/RC.  We can find the relation between the decay constant and the half-life using the properties of e and ln: C = C o e - t the chocolate decay function C o = C o e - t the definition of half-life (t 1/2 ) 1212 1/2 ln( ) = - t 1/2 1212 ln(x) and e x are inverses -0.693 = - t 1/2 = 0.693 t 1/2 or t 1/2 = 0.693 Relation Between Half- life and Decay Constant What is the decay constant for the chocolate of the previous slide? = 0.693 t 1/2 = 0.693 1 min = 0.693 min -1 = 0.693 60 s = 0.0116 s -1 FYI: You can express the decay constant in any units of time you wish.  Thus C = C o e -0.0116t where t is in seconds.

4.  So what does all of this have to do with radioactive decay? Topic 7.2 Extended C – Decay Rate and Half Life  As you may recall, the probability of any unstable element decaying depends on how high and wide the nuclear potential wall is. That probability is usually very small.  As an analogy, consider the probability that a person you meet in the shopping mall is, or will become, a murderer. This probability is very small.  But if you consider a large population of unstable atoms (or a large population of people) it will increase your chances of seeing an atom that has or will decay (a person who has or will murder).  Thus the decay rate is proportional to the number of remaining atoms... NtNt  -N FYI: The minus sign signifies that the population N is DECREASING. Population GROWTH, on the other hand, would not have the minus sign.  A list of related formulae follows: NtNt = - N Decay Rate or Activity N = N o e - t = 0.693 t 1/2

5. Suppose you have 64 grams of a radioactive material which decays into 1 gram of radioactive material in 10 hours. (a) What is the half-life of this material? Topic 7.2 Extended C – Decay Rate and Half Life  The easiest way to solve this problem is to keep cutting it in half... 64 t half 32 t half 16 t half 8 4 2 1  Note that there are 6 half-lifes in 10 h = 600 min so that t half = 100 min. (b) What is the decay constant ? = 0.693 t 1/2 = 0.693 100 min = 0.00693 min -1 (c) How much radioactive material is left after 4 h? N = N o e - t = 64e -0.00693t but t = 4(60) = 240 min: thus N = 64e -0.00693(240) = 12.13 grams

6.  Decay rates are measured in the SI unit called the becquerel (Bq), which is defined as Topic 7.2 Extended C – Decay Rate and Half Life 1 Bq = 1 decay second  Even though the Bq is the SI unit for radioactive decay rates, another unit, called the curie (Ci) is used. It is defined as 1 Ci = 3.70  10 10 decays second 3.70  10 10 Bq = 1 becquerel 1 curie  Most laboratory specimens are measured in mCi or  Ci since the curie is so large.  Radioactivities of 1 Ci require shielding (or very short times of exposure).

7. Topic 7.2 Extended C – Decay Rate and Half Life Some typical half-lives NuclidePrimary DecayHalf-Life Rubidium-87 --4.7  10 10 y Uranium-238  4.5  10 9 y Plutonium-239  2.4  10 4 y Carbon-14 -- 5730 y Radium-226  1600 y Strontium-90 -- 28 y Cobalt-60 -- 5.3 y Radon-222  3.82 d Iodine-123EC13.3 h Polonium-218 ,  - 3.05 min Oxygen-19 -- 27 s Polonium-213  4  10 -16 s FYI: Carbon, part of the life cycle, can be used to date organisms that have remains containing large amounts of carbon (such as wood at archaeological sites). FYI: Many of these unstable isotopes are the product of nuclear detonations.