The Mystery ofThe Mystery ofHalf- Life and Rate of Half- Life and Rate of

DecayDecay

The truth is out there...The truth is out there...

BY CANSU TÜRKAY

10-N

Before we start....Before we start....At the end of this presentation, you will be a At the end of this presentation, you will be a genious about these fallowing issues (at least genious about these fallowing issues (at least I hope so ) :I hope so ) :

- Conservation of Nucleon NumberConservation of Nucleon Number

- Radioactive (a type of exponentional) Decay Radioactive (a type of exponentional) Decay Law and its ProofLaw and its Proof

- Concept of Half- life- Concept of Half- life

- How to solve half-life problems How to solve half-life problems

Conservation of....Conservation of....

All three types of radioactive decays All three types of radioactive decays (Alfa, beta and gamma) hold classical (Alfa, beta and gamma) hold classical conservation laws.conservation laws.

Energy, linear momentum, angular Energy, linear momentum, angular momentum, electric charge are all momentum, electric charge are all conservedconserved

Conservation of...Conservation of...

The law of conservation of nucleon The law of conservation of nucleon number states that the total number of number states that the total number of nucleons (A) remains constant in any nucleons (A) remains constant in any process, although one particle can process, although one particle can change into another ( protons into change into another ( protons into neutrons or vica versa). This is accepted neutrons or vica versa). This is accepted to be true for all the three radioactive to be true for all the three radioactive decays.decays.

Radioactive Decay Law Radioactive Decay Law and its Proof and its Proof

Radioactive decay is the spontaneous Radioactive decay is the spontaneous release of energy in the form of release of energy in the form of radioactive particles or waves.radioactive particles or waves.

It results in a decrease over time of the It results in a decrease over time of the original amount of the radioactive original amount of the radioactive material.material.

Radioactive Decay Law Radioactive Decay Law and its Proofand its Proof

Any radioactive isotope consists of a vast Any radioactive isotope consists of a vast number of radioactive nuclei.number of radioactive nuclei.

Nuclei does not decay all at once.Nuclei does not decay all at once. Decay over a period of time.Decay over a period of time. We can not predict when it will decay, its We can not predict when it will decay, its

a random process but...a random process but...

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... We can determine, based on ... We can determine, based on probability, approximately how many probability, approximately how many nuclei in a sample will decay over a given nuclei in a sample will decay over a given time period, by asuming that each time period, by asuming that each nucleus has the same probability of nucleus has the same probability of decaying in each second it exists. decaying in each second it exists.

RadioactiveRadioactive Decay Law Decay Law and its Proofand its Proof

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Exponentional DecayExponentional Decay

A quantity is said to be subject to A quantity is said to be subject to exponentional decay if it decreases at a exponentional decay if it decreases at a rate proportional to its value.rate proportional to its value.

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Exponentional DecayExponentional Decay

Symbolically, this can be expressed as Symbolically, this can be expressed as the fallowing differential equation where the fallowing differential equation where N is the quantity and N is the quantity and λλ is a positive is a positive number called the decay constant:number called the decay constant:

∆∆NN = - = - λλNN

∆ ∆ tt

Relating it to radioactive Relating it to radioactive decay law:decay law:

The number of decays are represented The number of decays are represented by by ∆N∆N

The short time interval that ∆N occurs is The short time interval that ∆N occurs is represented by ∆trepresented by ∆t

N is the number of nuclei presentN is the number of nuclei present λλ is the decay constant is the decay constant

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Here comes our first equation AGAIN, try Here comes our first equation AGAIN, try to look it with the new perspective:to look it with the new perspective:

∆∆NN = - = - λλNN

∆ ∆ tt

Relating it to radioactive Relating it to radioactive decay law:decay law:

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What was that?!!!What was that?!!!

In the previous equation you have seen a In the previous equation you have seen a symbol like: symbol like: λλ

λλ is a constant of proportionality, called the is a constant of proportionality, called the decay constant.decay constant.

It differs according to the isotope it is in.It differs according to the isotope it is in. The greater The greater λλ is, the greater the rate of decay is, the greater the rate of decay This means that the greater This means that the greater λλ is, the more is, the more

radioactive the isotope is said to be.radioactive the isotope is said to be.

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Still confused about the Still confused about the equation...equation...

Don’t worry! If you are still confused about Don’t worry! If you are still confused about why this equation is like this, here is some why this equation is like this, here is some of the important points....of the important points....

Confused Minds...Confused Minds...

With each decay that occurs (With each decay that occurs (∆N) in a ∆N) in a short time period (∆t),a decrease in the short time period (∆t),a decrease in the number N of the nuclei present is number N of the nuclei present is observed.observed.

So; the minus sign indicates that N is So; the minus sign indicates that N is decreasing.decreasing.

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Got it!!!!Got it!!!!

Now, here is our little old equation:Now, here is our little old equation:

∆∆NN = - = - λλNN

∆ ∆ ttPOF!!!

Now it has become the radioactive decay law!Now it has become the radioactive decay law!(yehu)(yehu)

What was that???What was that???

NN00 is the number of nuclei present at time is the number of nuclei present at time

t = 0t = 0 The symbol e is the natural expoentional (as The symbol e is the natural expoentional (as

we saw in the topic logarithm)we saw in the topic logarithm)

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So what?So what?

Thus, the number of parent nuclei in a Thus, the number of parent nuclei in a sample decreases exponentionally in sample decreases exponentionally in timetime

If reaction is first order with respect to If reaction is first order with respect to [N], integration with respect to time, [N], integration with respect to time, tt, , gives this equation.gives this equation.

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As seen in the figure As seen in the figure below…below…

Please just focus on how it decays exponetionally. Half-life will be discussed soon…

HALF-LIFEHALF-LIFE

The amount of time required for one-half The amount of time required for one-half or 50% of the radioactive atoms to or 50% of the radioactive atoms to undergo a radioactive decay.undergo a radioactive decay.

Every radioactive element has a specific Every radioactive element has a specific half-life associated with it.half-life associated with it.

Is a spontaneous process.Is a spontaneous process.

HALF-LIFEHALF-LIFE

Ooops!!!Ooops!!!

Remember the first few slides? We Remember the first few slides? We stated that we can not predict when stated that we can not predict when particular atom of an element will decay. particular atom of an element will decay. However half-life is defined for the time at However half-life is defined for the time at which 50% of the atoms have decayed. which 50% of the atoms have decayed. Why can’t we make a ratio and predict Why can’t we make a ratio and predict when all will decay???when all will decay???

AnswerAnswer The concept of half-life relies on a lot of The concept of half-life relies on a lot of

radioactive atoms being present. As an example, radioactive atoms being present. As an example, imagine you could see inside a bag of popcorn as imagine you could see inside a bag of popcorn as you heat it inside your microwave oven. While you heat it inside your microwave oven. While you could not predict when (or if) a particular you could not predict when (or if) a particular kernel would "pop," you would observe that after kernel would "pop," you would observe that after 2-3 minutes, all the kernels that were going to 2-3 minutes, all the kernels that were going to pop had in fact done so. In a similar way, we pop had in fact done so. In a similar way, we know that, when dealing with a lot of radioactive know that, when dealing with a lot of radioactive atoms, we can accurately predict when one-half atoms, we can accurately predict when one-half of them have decayed, even if we do not know of them have decayed, even if we do not know the exact time that a particular atom will do so. the exact time that a particular atom will do so.

HALF-LIFEHALF-LIFE

Range fractions of a second to billions of Range fractions of a second to billions of years.years.

Is a measure of how stable the nuclei is.Is a measure of how stable the nuclei is. No operation or process of any kind (i.e., No operation or process of any kind (i.e.,

chemical or physical) has ever been chemical or physical) has ever been shown to change the rate at which a shown to change the rate at which a radionuclide decays.radionuclide decays.

How to calculate half-How to calculate half-life?life?

The half life of first order reaction is a The half life of first order reaction is a constant, independent of the initial constant, independent of the initial concentration. concentration.

The decay constant and half-life has the The decay constant and half-life has the relationship :relationship :

hlhl = ln(2) / = ln(2) / λλ

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Calculations for half-lifeCalculations for half-life

As an example, Technetium-99 has a As an example, Technetium-99 has a half-life of 6 hours.This means that, if half-life of 6 hours.This means that, if there is 100 grams of Technetium is there is 100 grams of Technetium is present initially, after six hours, only 50 present initially, after six hours, only 50 grams of it would be left.After another 6 grams of it would be left.After another 6 hours, 25 grams, one quarter of the initial hours, 25 grams, one quarter of the initial amount will be left. And that goes on like amount will be left. And that goes on like this. this.

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Bye!

Calculating Half-LifeCalculating Half-Life

R (original amount)R (original amount) n (number of half-lifes)n (number of half-lifes)

R . (1/2)n

Try it!!!Try it!!!

Now lets try to solve a half-life calculation Now lets try to solve a half-life calculation problem…problem…

64 grams of Serenium-87, is left 4 grams 64 grams of Serenium-87, is left 4 grams after 20 days by radioactive decay. How after 20 days by radioactive decay. How long is its half life?long is its half life?

SolutionSolution

Initially, Sr is 64 grams, and after 20 Initially, Sr is 64 grams, and after 20 days, it becomes 4 grams.The arrows days, it becomes 4 grams.The arrows represent the half-life.represent the half-life.

64 g64 g 64 . ½ 64 . ½ 64 . ½ . ½ …64 . ½ . ½ …

It goes like this till it reaches 4 grams, in 20 It goes like this till it reaches 4 grams, in 20 days. days.

1/2 1/2

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SolutionSolution

We have to find after how many We have to find after how many multiplications by ½ does 64 becomes 4.multiplications by ½ does 64 becomes 4.

We can simply state that, We can simply state that,

Where n is the number of half lifes it has Where n is the number of half lifes it has experienced.experienced.

64 . (1/2)n

SolutionSolution

n = 4 half-lifes

And as we are given the information that this process happened in 20 days ;

4 half-lifes = 20 days1 half life = 5 daysTataa!!! We have found it really easily!

64 . (1/2)n = 4

26-n = 22

QuestionsQuestions

Explain the reason for why can’t we predict Explain the reason for why can’t we predict when/if a nucleus of a radioactive isotope with when/if a nucleus of a radioactive isotope with a known- half life would decay?a known- half life would decay?

Define half-life briefly.Define half-life briefly.

QuestionsQuestions

Explain the law of conservation of Explain the law of conservation of nucleon number.nucleon number.

Does nuclei decay all at once/ how does Does nuclei decay all at once/ how does it decay?it decay?

A quantity is said to be subject to A quantity is said to be subject to exponentional decay if…?exponentional decay if…?

THE END!!!THE END!!!

Resources:Resources: http://http://cathylawcathylaw.com/.com/imagesimages//halflifebarhalflifebar..jpgjpg http://burro.astr.cwru.edu/Academics/Astr221/HW/HW3http://burro.astr.cwru.edu/Academics/Astr221/HW/HW3

/noft.gif/noft.gif http://www.http://www.chemchem..oxox..acac..ukuk//vrchemistryvrchemistry//ConservationConservation

/page35./page35.htmhtm www.gcse.comwww.gcse.com/ radio/halflife3.htm/ radio/halflife3.htm www.nucmed.buffalo.eduwww.nucmed.buffalo.edu/.../ sld003.htm/.../ sld003.htm http://www.iem-inc.com/prhlfr.htmlhttp://www.iem-inc.com/prhlfr.html http://www.math.duke.edu/education/ccp/materials/diffchttp://www.math.duke.edu/education/ccp/materials/diffc

alc/raddec/raddec1.htmlalc/raddec/raddec1.html http://www.http://www.mrgalemrgale.com/.com/onlhlponlhlp//nucpartnucpart//halflifehalflife..htmhtm