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ems to decay. The rate of decay continuously decreases as the sample gets smaller; maller. The mass, A of a radioactive material with an initial sample mass of Ao is

to time, t and half-life, h This can be represented by the following equation:I .2 EARN ING TT he H al f- L if e E q ua ti o ntT h e e x po n en t h is tim e d iv id ed byhal f - l ife . T his q uo tie nt re pre se nts th en um be r o f h alf-liv es , w hic h is th en um be r o f tim es the in itia l a mo un t isr e c l u c e c l o o n e - h a l ten using tbis equation it is important to measure tbe masses A anQAa using tbe

e units. The same is true for t and In the following Tutorial, you will apply thistion to solve problems involving the half-life of radioactive isotopes.

m p l e P ro b l e m 1n-1 9 has a ha lf-life o f 17 .22 s . W ha t m ass o f neo n-1 9 w illa in from a 100 m g in it ia l sam p le a fte r 30 s?

n : = 100 m g; h = 17 .22 s ; t= 30 sir e d :

1 )1ly sis: A = A o 2 h

m p le P ro b l e m 2100 m g sam ple o f m agnesium -27 decays by 7 o f itsv ious m ass eve ry m inu te . D ete rm ine its ha lf - life and s ta te the-life d ec ay eq uation.

1. T he d ec ay o f m ag ne siu m-27 can be m ode lled us ing atab le o r g raph . If 7 decays du ring each m inu te ,then 93 r ema ins. C rea te a tab le s im ila r to T a b l e 2t o d et er mine the m ass re ma in in g a fte r e ach m inu te .

l e 2 Mass o f Magne siu m-2 7 R em aini ngl i m e m in ) In i t i a l m as s m g ) F i n al m a s s m g )

0 100 0.93 100) = 93 1 93 0.9393) = 86.49

2 86.49 0.93 86.49) = 80.443 80.44 0.93 80.44 ) = 74.814 74.81 69.575 69 .57 64.706 64 .70 60.177 60.17 55 .968 55 .96 52 .049 52 .04 48.40

i 10 48.40 45.01

1 )1S o l u t i o n : o 2 h 1 30 g= (100 mg) 2 17.22~1 1 . 7 4 2 2

100 m g) 2 = 30 m g

S t a t e m e n t: The re w ill b e 30 m g o f neon -19 rem ain ing a fte r 30 s .

S t e p 2 . U se the da ta in Tab le 2 to crea te a graph o f m assre m ain in g v ersus tim e F i g u r e 2 ).

M a s s o f M a g n e s i u m -2 7 v . l i m e10 090

l 80.. . . 70NE 60: :: :I ij 50lC I : J 40E- 30mmC I: J 20: iE

1000 2 4 6 8 10

T i m e m in )F i g u r e 2

S t e p 3 . U se the da ta in Tab le 2 and the grap h in F ig ure 2to d ete rm ine the ha lf- life o f m agnes ium -27 . H alfo f the in it ia l m ass o f m ag ne siu m -2 7 h as d ec ay edapprox im ate ly ha lfw ay be tw een 9 m in and 10 m in .S o , th e h alf-life o f m agnes ium -27 is abou t 9 .5 m in .

7 .3 H alf-L ife 331

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The half-life of a radioactive isotope is the amount of time required for it todecay to one-half of its original mass.

Half-lives can vary from a tiny fraction of a second to millions of years. The decay of a radioactive isotope can be mathematically modelled using atable, a graph, or an equation.

Some isotopes like carbon-14 and aluminum-26 have useful applications duein part to their particular half-lives.

Carbon -14 is a useful isotope for dating fossils and other archaeologicalobjects.

. Ch lo r ine -38, w hich u nd ergo es be ta -n eg ative de ca y, h as ah alf-life o f 3 7.24 m in . ttmlla ) C ons tru ct a tab le tha t com pares the m ass o f C I-38r em a in in g a ft er t m in ute s to r s eve ra l va lue s o f t.

b) D raw a g raph tha t il lu str ate s th is re la t ionsh ip .c) W ha t iso tope does C I-38 decay in to?

2 . Go ld -198, w ith a h alf-life o f 2 .6 days , is u se d to d iag no sea nd tr ea t live r d isease. IlDEla ) W rite a ha lf- life decay equation tha t re la te s the m ass o f

A u-19 8 re main ing to tim e in days .b ) W hat pe rcen tage o f a sam ple o f A u -198 w ou ld rem ain

afteri) 1 da y?ii) 1 w ee k?

3 . C ob alt-6 0, w ith a ha lf- life o f 5 .3 y ea rs , has a num be r o fappl ications , in c lud ing m edi ca l the rapy and the s te ri li za ti ono f m e dic al tools . De te rm ine the m ass o f a 50 g sam ple tha tw ou ld re m ain a fte r ) 6 m onths ) 5 yea rs m

4 . W hat type o f rad io ac tiv e d ec ay is in vo lv ed in c arb ond ating ? E xp la in th e pro ces s o f ca rb on da ting. ttm El

5. A foss il con ta in s 70 o f the ca rbon -14 it once had asa liv in g crea tu re . U se th e ha lf- life de ca y eq uation tod ete rm in e w hen the c re atu re d ie d. m

Alum inum-26, w h ic h d ec ay s in to m ag ne siu m -2 6, has a ha lf- lif eo f a p pr ox im ate ly 720 000 years. U se th is in fo rm atio n to a ns we rQ ues tions 6 and 7 .6. a ) W ha t type o f decay does A I- 2 6 u nd erg o?b) D oe s A I-2 6 d ec ay in the sam e way as C -14?

Exp lain. ttm El7. A m oon rock has 3 of it s o rig in a l A I -2 6 m ass. ttmll

a ) D ete rm ine the age o f th e m oo n rock .b ) D is cu ss a ny a ss um ptions tha t m us t be m ade w hen

using th is m ethod o f da t ing .8 . Take a re gu la r s he et o f p ap er. M ea su re its leng th and

w id th and dete rm ine the a rea . Fo ld th e p ap er n ea tly in ha lf .D ete rm in e th e n ew area . R epe at un til yo u c an not fo ld thep ap er a ny lo ng er. E xp la in how th is m ode l can be used todescri be h a lf -lif e. ttm l

7 3 Half Life