The Theory of Radioactive DecayNuclear Physics Lesson 7

HomeworkHomework Read pages 168-171 (Chapter 9.7). Answer Q1-Read pages 168-171 (Chapter 9.7). Answer Q1-

4 by next Monday Period 2. Show all working.4 by next Monday Period 2. Show all working.

HWK days: HWK days: Tues Wk 1 Period 4Tues Wk 1 Period 4Mon Wk 2 Period 2Mon Wk 2 Period 2Thurs Wk 2 Period 4Thurs Wk 2 Period 4

Test - some time before half term.Test - some time before half term.

Learning ObjectivesLearning Objectives Take a look at how we got on last year – Take a look at how we got on last year –

retakes?retakes?

Recap what we learned last year and go Recap what we learned last year and go further further theory of radioactive decay. theory of radioactive decay.

Apply what we know to carbon dating Apply what we know to carbon dating (HWK).(HWK).

AS Physics ResultsAS Physics ResultsA: 4A: 4B: 5B: 5C: 3C: 3D: 1D: 1E: 4E: 4U: 6U: 6Total: 23Total: 23

A2 Physics ResultsA2 Physics Results A*: 4 (Averaging 80% overall, 90%, A*: 4 (Averaging 80% overall, 90%,

Units 4,5,6)Units 4,5,6)A: 4A: 4B: 4B: 4C: 2C: 2D: 1D: 1E: 0E: 0U: 0U: 0Total: 15Total: 15

Mean ScoresMean Scores Mean Unit 1 score: 56%.Mean Unit 1 score: 56%. Mean Unit 2 score: 55%.Mean Unit 2 score: 55%. Mean Unit 3 score: 69%.Mean Unit 3 score: 69%. Last year:-Last year:- Mean Unit 1 score: 67%Mean Unit 1 score: 67% Mean Unit 2 score: 66%Mean Unit 2 score: 66% Mean Unit 3 score: 67%Mean Unit 3 score: 67%

Discussion QuestionsDiscussion Questions What is radioactive decay? Why does it What is radioactive decay? Why does it

happen?happen? What is the definition of half life?What is the definition of half life? How much of the original sample is left How much of the original sample is left

after two half lives?after two half lives? What is activity?What is activity? What is the unit of activity?What is the unit of activity? Are count rate and activity the same Are count rate and activity the same

thing?thing?

Radioactive DecayRadioactive Decay It’s really all about unstable isotopes It’s really all about unstable isotopes

changing to stable isotopes.changing to stable isotopes.

It is random.It is random.

One element is changing into One element is changing into another...and an another...and an αα or a or a ββ particle is particle is emitted.emitted.

Half-LifeHalf-Life The The half-lifehalf-life, T, T1/21/2, of a radioactive isotope , of a radioactive isotope

is the time taken for the mass of the is the time taken for the mass of the isotope to decrease to half the initial mass.isotope to decrease to half the initial mass.

If the half-life is very long then the If the half-life is very long then the substance will remain radioactive for a substance will remain radioactive for a very long time.very long time.

If the half life is short it will decay quickly If the half life is short it will decay quickly

Decay CurveDecay Curve

ActivityActivity The The activityactivity, A ,of a radioactive isotope , A ,of a radioactive isotope

is the number of nuclei of the isotope is the number of nuclei of the isotope that disintegrate per second.that disintegrate per second.

Activity is measured in Becquerels (Bq) Activity is measured in Becquerels (Bq) which is the number of decays per which is the number of decays per second.second.

time(s)decays ofnumber totalq)activity(B

Count Rate versus Count Rate versus ActivityActivity

They not quite the same thing.They not quite the same thing.

Activity is the number of nuclei that Activity is the number of nuclei that decay per second.decay per second.

Count rate is the number of decays Count rate is the number of decays detected detected per second (per second (and includes and includes background countsbackground counts). ). They love to They love to trap you with this!trap you with this!

Theory of Radioactive Theory of Radioactive DecayDecay

Let NLet N00 be the no. of nuclei of a radioactive be the no. of nuclei of a radioactive sample X.sample X.

Let N be the no. of nuclei of X left after time t.Let N be the no. of nuclei of X left after time t.

In time In time ΔΔt, a number of nuclei disintegrate,t, a number of nuclei disintegrate, Δ ΔNN Because Because ΔΔN is proportional to N and N is proportional to N and ΔΔt:-t:-

Where Where λλ is the constant of proportionality and is the constant of proportionality and is known as the decay constant.is known as the decay constant.tNN

The Decay ConstantThe Decay Constant The decay constant is the probability of The decay constant is the probability of

an individual nucleus decaying per an individual nucleus decaying per second.second.

Is has the symbol Is has the symbol λλ, but has nothing to , but has nothing to do wavelength (they just share symbols)do wavelength (they just share symbols)

It has units of sIt has units of s-1-1 (per second). (per second).

Theory of Radioactive Theory of Radioactive DecayDecay

Re-arranging the last equation:-Re-arranging the last equation:-

And recall that And recall that ΔΔN/N/ΔΔt is the rate of t is the rate of disintegration which is the activity.disintegration which is the activity.

So the activity A, of a sample of N So the activity A, of a sample of N nuclei is given by:nuclei is given by:

NtN

NA

Exponential Decay LawsExponential Decay Laws

Where N is the number of nuclei of isotope Where N is the number of nuclei of isotope X after time t and NX after time t and N0 0 is original no. of nuclei.is original no. of nuclei.

Where m is the mass of an isotope X after Where m is the mass of an isotope X after time t and mtime t and m0 0 is the original mass of isotope is the original mass of isotope X.X.

teNN 0

temm 0

Exponential Decay LawsExponential Decay Laws

Where A is the activity of the sample after Where A is the activity of the sample after time t and Atime t and A0 0 is original activity (t=0) of the is original activity (t=0) of the sample. sample.

Where C is the corrected count rate of the Where C is the corrected count rate of the sample after time t and Csample after time t and C0 0 is the original (t=0) is the original (t=0) corrected count rate of the sample. corrected count rate of the sample.

teCC 0

teAA 0

Half-Life IIHalf-Life II The time it takes mass of a radioactive The time it takes mass of a radioactive

sample to decrease by 50%.sample to decrease by 50%.

The time it takes the activity of a The time it takes the activity of a radioactive sample to decrease by 50%.radioactive sample to decrease by 50%.

The time it takes the count rate from a The time it takes the count rate from a radioisotope to decrease by 50%.radioisotope to decrease by 50%.

Learning ObjectivesLearning Objectives Hand in your homework!Hand in your homework!

Define a moleDefine a mole Calculate the number of moles in a Calculate the number of moles in a

sample n=N/Nsample n=N/NAA=m/M=m/MS.S.

Complete questions from last time.Complete questions from last time.

Avogadro’s ConstantAvogadro’s Constant One One molemole of any gas of any gas

contains the same contains the same number of particles. number of particles. This number is called This number is called Avogadro’s constant and Avogadro’s constant and has the symbol Nhas the symbol NAA. The . The value of Nvalue of NAA is 6.02 is 6.02 × 10× 102323 particles per mole.particles per mole.

Calculating the Number of Moles

The number of moles, n, of a gas can be can be calculated using:-

Where N is the total number of molecules and NA is Avogadro’s constant (=6.02 ×

1023)

ANNn

Calculating the Number of Moles

The number of moles can also be The number of moles can also be calculated from the mass:-calculated from the mass:-

Where m is the total number of molecules and MS is the molar mass (the mass of 1

mole of the substance, = nucleon number in grams)

SMmn

Calculating the Number of Moles

You can calculate the number of atoms in You can calculate the number of atoms in the sample from the mass using:-the sample from the mass using:-

You can also go the other way You can also go the other way calculate the mass from the number of calculate the mass from the number of atoms.atoms.

SA Mm

NN

Practice QuestionsPractice Questions 34g of Carbon-14 34g of Carbon-14 number of atoms? number of atoms? 0.534 kg of Uranium-235 0.534 kg of Uranium-235 number of atoms? number of atoms? 234mg of Calcium-40 234mg of Calcium-40 number of atoms? number of atoms?

5.4 × 105.4 × 102323 kg atoms of Oxygen-18 kg atoms of Oxygen-18 mass = ? mass = ? 3.87 × 103.87 × 1020 20 kg atoms of Deuterium kg atoms of Deuterium mass = ? mass = ? 7.84 × 107.84 × 1027 27 kg atoms of Aluminium-27kg atoms of Aluminium-27

mass=?mass=?

Question 0Question 0 0.25 kg radon-226 emits alpha 0.25 kg radon-226 emits alpha

particles at a measured rate of 9 × particles at a measured rate of 9 × 10101212 s s-1-1. What is the decay constant . What is the decay constant of radium? (No of atoms in a mole = of radium? (No of atoms in a mole = 6 × 106 × 102323))

Answer 0Answer 0 Work out the number of particles: Work out the number of particles: 0.2500.250 × 6 × 10 × 6 × 102323 = 6.64 × 10 = 6.64 × 102323 atoms atoms 0.2260.226 We know that the rate of decay is 9 × We know that the rate of decay is 9 ×

10101212 s s-1-1. So we use DN/Dt = -lN . So we use DN/Dt = -lN - 9 × 10- 9 × 101212 s s-1-1 = -l × 6.64 × 10 = -l × 6.64 × 102323 l = 1.36 × 10l = 1.36 × 10-11-11 s s-1-1 (The minus sign indicates a decay)(The minus sign indicates a decay)

Question 1Question 1 A sample of living material contains A sample of living material contains

carbon 14 with an activity of 260 Bq carbon 14 with an activity of 260 Bq kgkg-1-1. What is the decay constant? . What is the decay constant? (The fraction that is made of carbon-(The fraction that is made of carbon-14 is 1.4 ´ 1014 is 1.4 ´ 10-12-12))

Answer 1Answer 1 A sample of living material contains A sample of living material contains

carbon 14 with an activity of 260 Bq kgcarbon 14 with an activity of 260 Bq kg-1-1. . What is the decay constant? (The fraction What is the decay constant? (The fraction that is made of carbon-14 is 1.4 that is made of carbon-14 is 1.4 10 10-12-12) )

No of particles = 1000/12 × 6 × 10No of particles = 1000/12 × 6 × 102323 1.4 1.4 10 10-12-12 = 7 × 10 = 7 × 101313 (P) (P)

Use DN/Dt = -lN (P) Use DN/Dt = -lN (P) -260 = -l × 7 × 10-260 = -l × 7 × 101313 l = 3.7 × 20l = 3.7 × 20-12-12 s s-1-1 (P) (P)

Question 1aQuestion 1a A radiographer has calculated that a A radiographer has calculated that a

patient is to be injected with 1 ´ 10patient is to be injected with 1 ´ 101818 atoms of iodine 131 to monitor thyroid atoms of iodine 131 to monitor thyroid activity. The half-life is 8 days. Calculate: activity. The half-life is 8 days. Calculate:

(a)    the radioactive decay constant (a)    the radioactive decay constant (b)   the initial activity (b)   the initial activity (c)    the number of undecayed atoms of (c)    the number of undecayed atoms of

iodine 131 after 24 days. iodine 131 after 24 days. (d)   The total activity after 3 days.(d)   The total activity after 3 days.

Answer 1aAnswer 1a (a) We need to use (a) We need to use TT1/21/2 = = 0.6930.693

l l we need to convert the 8 days we need to convert the 8 days

into seconds. into seconds. Þ l = Þ l = 0.693 _ 0.693 _ = = 1.00 ´ 101.00 ´ 10-6-6 s s-1-1

8 ´ 864008 ´ 86400 (b) Use D(b) Use DNN = - l = - lNN = 1.00 ´ 10 = 1.00 ´ 10-6-6 s s-1-1 ´ 1 ´ ´ 1 ´

10101818 = = 1 ´ 101 ´ 101212 Bq Bq DDtt

Answer 1aAnswer 1a                   (c) 24 days is 3 half-lives. Therefore the (c) 24 days is 3 half-lives. Therefore the

number atoms remaining undecayed is 1/8 of number atoms remaining undecayed is 1/8 of the original. the original. N N = 1 ´ 10 = 1 ´ 101818 ¸ 8 = ¸ 8 = 1.25 ´ 101.25 ´ 101717

(d) 3 is not so easy. We use (d) 3 is not so easy. We use A = AA = A00ee-lt-lt Þ Þ AA = 1 ´ 10 = 1 ´ 101212 Bq ´ Bq ´ ee-(1.00 -(1.00 10-6 s-1 10-6 s-1 3 3 86400s) 86400s)     Þ Þ AA = 1 ´ 10 = 1 ´ 101212 Bq ´ Bq ´ ee-(0.2592)-(0.2592) = 1 ´ 10= 1 ´ 101212 Bq ´ 0.772 = Bq ´ 0.772 = 7.72 ´ 107.72 ´ 101111 Bq Bq..

Question 2 Question 2 What is the half life of radon-226? l What is the half life of radon-226? l

= 1.36 × 10= 1.36 × 10-11-11 s s-1-1

What is the half life of radon-226? l = 1.36 × What is the half life of radon-226? l = 1.36 × 1010-11-11 s s-1-1

    Formula: N = NFormula: N = N00ee-lt-lt     ½ = e½ = e-1.36 -1.36 10^-11 10^-11 t1/2 (P)   t1/2 (P)  

LogLogee ½ = -1.36 ½ = -1.36 10-11 10-11 t t

1/21/2 (P) (P)

   

t1/2 = -0.693 t1/2 = -0.693 -1.36 -1.36 10-11 = 5.1 10-11 = 5.1 1011 s (= 1600 years) (P) 1011 s (= 1600 years) (P)

Note 10^11 means 1011.  I cannot do a double superscript. Note 10^11 means 1011.  I cannot do a double superscript. 

Question 3Question 3 Strontium-90 is a beta Strontium-90 is a beta emitter. It is one of the radio-emitter. It is one of the radio-nuclides found in the fall out from an nuclides found in the fall out from an atomic bomb explosion. It can be atomic bomb explosion. It can be absorbed into the bone. It emits absorbed into the bone. It emits beta particles and has a half life of beta particles and has a half life of 28 years. What is the time needed 28 years. What is the time needed for the activity to fall to 5 % of the for the activity to fall to 5 % of the original?original?

Strontium-90 is a beta emitter. It is one of the Strontium-90 is a beta emitter. It is one of the radio-nuclides found in the fall out from an radio-nuclides found in the fall out from an atomic bomb explosion. It can be absorbed into atomic bomb explosion. It can be absorbed into the bone. It emits beta particles and has a half the bone. It emits beta particles and has a half life of 28 years. What is the time needed for life of 28 years. What is the time needed for the activity to fall to 5 % of the original? the activity to fall to 5 % of the original?

We need to know how many half lives gives 5 % We need to know how many half lives gives 5 % (0.5)(0.5)yy = 0.05 (P) = 0.05 (P) y y log (0.5) = log (0.05) log (0.5) = log (0.05)

y y -0.3010 = - 1.3010 -0.3010 = - 1.3010 y = 1.3010 y = 1.3010 0.3010 = 4.32 half 0.3010 = 4.32 half

lives (P) lives (P) Time taken = 4.32 Time taken = 4.32 28 (P) 28 (P) Time taken = 121 years (P) Time taken = 121 years (P)

Question 4Question 4 A GM tube placed close to a A GM tube placed close to a radium source gives an initial average radium source gives an initial average corrected count rate of 334 scorrected count rate of 334 s-1-1 (a)     The (a)     The GM tube detects 10 % of the radiation. GM tube detects 10 % of the radiation. What is the initial activity? What is the initial activity?

(b)    Initially there were 1.5 ´ 10(b)    Initially there were 1.5 ´ 1099 nuclei in nuclei in the sample. What is the decay constant? the sample. What is the decay constant?

(c)     What is the half life of the radium in (c)     What is the half life of the radium in days? days?

A GM tube placed close to a radium source A GM tube placed close to a radium source gives an initial average corrected count rate gives an initial average corrected count rate of 334 sof 334 s-1-1

(a)     The GM tube detects 10 % of the (a)     The GM tube detects 10 % of the radiation. What is the initial activity? radiation. What is the initial activity?

(b)    Initially there were 1.5 (b)    Initially there were 1.5 10 1099 nuclei in nuclei in the sample. What is the decay constant? the sample. What is the decay constant?

(c)     What is the half life of the radium in (c)     What is the half life of the radium in days? days?

(a) The activity is 334 (a) The activity is 334 0.1 = 3340 s 0.1 = 3340 s-1-1 (P) (P)

(b) DN/Dt = -lN (P) (b) DN/Dt = -lN (P) -3340 = -l -3340 = -l 1.5 1.5 10 1099 = 2.23 = 2.23 10 10-6-6 s s-1-1

(P) (P) (c) N = No e(c) N = No e-lt-lt ½ = e½ = e-2.23 -2.23 10^-6 10^-6 tt loge ½ = -2.23 loge ½ = -2.23 10 10-6-6 t (P) t (P) t = -0.693 t = -0.693 2.23 2.23 10 10-6-6 = 3.1 = 3.1 10 1055 s s t = 3.6 days (P)t = 3.6 days (P)

Summary Summary

Radioactive decays are random. Radioactive decays are random. Rate of decay depends on the number of Rate of decay depends on the number of

atoms left atoms left The probability of any one nucleus The probability of any one nucleus

decaying in any one second is the decay decaying in any one second is the decay constant constant

Decay constant is given the code Decay constant is given the code ll DN DN = -= -llN N DDt t

Over a longer period of time, decay Over a longer period of time, decay is exponential. is exponential.

N = No e-N = No e-ltlt Half life is the time taken for ½ the Half life is the time taken for ½ the

remaining atoms to decay remaining atoms to decay TT1/21/2 = = 0.6930.693