Nuclear Chemistry: Rates of Radioactive Decay

Transcription

1 Activity 48 Nuclear Chemistry: Rates f Radiactive Decay What D Yu Think? Radiactive idine-131 is used t treat thyrid cancer. A patient is given 20.0 mg f idine-131 in the frm f NaI, and after 8 days nly 10 mg remain. Hw lng, frm the time f the initial dse, d yu think it will it take until nly 5 mg remain? (A) 12 days (B) 16 days (C) 20 days (D) cannt be determined frm the infrmatin given Learning Objectives Understand hw the level f radiactivity decreases with time Understand hw radiactivity can be used t determine the age f materials Success Criteria Relate the amunt f radiactive material remaining after sme perid f time t the half-life f the radiactive istpe and the rate cnstant fr the decay Estimate the age f materials frm their radiactivity Prerequisites Activity 28: Rates f Chemical Reactins Activity 46: Nuclear Chemistry: Radiactivity Activity 48 Nuclear Chemistry: Rates f Radiactive Decay 325

2 Tasks 1. Examine the mdel and cmplete Table 48.1 by entering N, the number f unstable nuclides remaining at each pint in time, and ln(n), in the last tw clumns, respectively. Mdel In the chart belw, ten unstable nuclides are represented by white circles. They decay spntaneusly by sme mechanism t prduce stable nuclides, which are indicated by dark circles. Table 48.1 Time Evlutin f Nuclear Decay Time (mins) Original and Prduct Nuclides N ln(n) Make tw graphs f the data (see the next page): Graph 1 with N pltted n the y-axis and time pltted n the x-axis, and Graph 2 with ln(n) pltted n y-axis and time pltted n the x-axis. Quality graphs will have titles, labels n the axes, data pint shwn with small circles arund them, and a smth line drawn thrugh the data pints. 3. On yur graphs mark the pints n the x-axis where the fractin remaining is equal t 1, 1, and Fundatins f Chemistry

3 Graph 1: Graph 2: Activity 48 Nuclear Chemistry: Rates f Radiactive Decay 327

4 Key Questins 1. Frm Graph 1, hw lng des it take fr half f the radiactive nuclides in the mdel t decay? This time is called the half-life. 2. Des the time it takes fr half f the radiactive nuclides t decay depend n the pint in time taken as the starting pint? Explain hw the infrmatin in Graph 1 supprts yur answer. 3. Hw can the data in the mdel be used t identify radiactive decay as ne f the fllwing types f reactins: zer rder, first rder, r secnd rder? 4. What is the mathematical equatin that represents the straight line that can be drawn thrugh the data pints in Graph 2? 5. Hw can the rate cnstant fr this radiactive decay be btained frm Graph 2? Exercises Use the fllwing infrmatin fr Exercises 1 5: Tw radiactive istpes, A and B, have decay rate cnstants k A and k B, respectively, where k A is larger than k B. N A and N B are the number f nuclides present at t 0 fr each f these istpes. The decay rate R is the number f decay events per secnd, where R N/ t k N 328 Fundatins f Chemistry

5 1. Sketch ne graph with a curve fr N A (t) and a curve fr N B (t) t cmpare the number f radiactive nuclides present as a functin f time fr bth f these istpes. Start at t 0 with the same number f A istpes and B istpes. A B 2. Sketch ne graph with a curve fr ln [N A (t)] and a curve fr ln [N B (t)] as a functin f time t cmpare these functins fr the tw istpes. Again, start at t 0 with same number f A istpes and B istpes. B A 3. Identify the istpe A r B with the lnger half-life. 4. Identify the istpe A r B with the faster decay rate when the amunts f A and B present are equal. Activity 48 Nuclear Chemistry: Rates f Radiactive Decay 329

6 5. Using yur answer t Key Questin 4, derive the fllwing relatinship between the half-life and the rate cnstant: t ½ ln(2)/k. ln(n) ln(n ) -kt when t t then N 1/2 ln ln(n ) - kt1/2 t 1/2 N 2 N ln N 2 ln(2) k k N 2 6. Derive yur answer t Key Questin 4 frm the fact that the data in Graph 1 is described by an expnential functin: N N e kt N ln(n) N e kt ln(n )- kt 7. Idine-123 has a half-life f 13.3 hurs. Using yur answer t Exercise 5, calculate a value fr the decay rate cnstant, k, in units f s 1. k ln(2) 13.3 hr hr 1min k s hr 60 min 60 s Determine the half-life f a radiactive nuclide if, after 2.5 hrs, nly 1/32 f the initial amunt remains unchanged. 9. Estimate hw many half-lives must pass until less than 1% f a radiistpe remains. Nte: After the students make an estimatin withut ding the calculatin, yu culd have them calculate exactly when 1% f the material remains. Yu might pint ut that knwing hw lng it takes fr the level f radiactivity t decrease t 1% r 0.1% f the initial value in a patient is imprtant in medical applicatins. Calculatin: x x x lg(2) lg(100) x 6.64 S 6.64 half-lives are needed until lg(2) % f the material remains. 330 Fundatins f Chemistry