bsen-625 advances in food in food engineering. activity

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  • Radioactive Decay

    BSEN-625ADVANCES IN FOOD ENGINEERING

  • Activity

    The rate of decay of a radionuclideIt is the number of atoms that decay per unit timeUnits Bacquarel (Bq): one desintegation/second1 Bq = 1 s-1Curie-(Ci): activity of 1 g of Ra226Ci: 1 Ci = 3.7x1010 Bq

  • Exponential decay

    A/Ao

    The activity of a pure radionuclide decreases exponentially with time

    t

  • Exponential radioactive decay law

    If N = # of atoms of a radionuclide in a sample @ a given time:

    teNoNor

    tNoN

    NotNcNotNoNCI

    =

    =

    +=+=

    ==

    ln

    lnln0ln

    ;0,:..

    ctN

    dtNdN

    NdtdNA

    NdtdN

    +=

    ==

    =

    ln

  • Half-life, T

    693.02ln

    2ln21ln

    21

    ==

    =

    =

    =

    =

    T

    T

    e

    eAoA

    T

    t

    Decay constant

    Time required for the activity of a radionuclide drop by a factor of one-half

  • Exponential decay in term of T

    Tt

    Tt

    AoA

    AoA

    eAoA

    NoN

    Tt

    Tt

    693.02lnln

    21 /

    /63.0

    ==

    =

    == teAoA 63.0=

    2T

    A/Ao1

    T

    0.125

    0.5

    0.25

    3T

    T693.0

    =

    t

  • Example

    Calculate the activity of a 30-MBq source of Na-24 after 2.5 d. What is its decay constantSolution

    T = half-life =15 h (appendix D)

    MBqeAhdhdtMBqAo

    hT

    88.13060/245.2,30

    0462.015693.0693.0

    )600462.0(

    1

    ==

    ===

    ===

  • Mean life,

    The average of all the individual lifetimes that atoms in a sample of the radionuclide experienceThe mean value of t under the exponential curve

    te 1

    A/Ao

    t

  • Mean life,

    It defines a rectangle with area equal to:

    te 1

    T

    T

    edte tt

    >

    ==

    ===

    693.01

    1|11 00 A/Ao

    t

  • Specific Activity, SA

    Activity per unit massBq/gFor a pure radionuclide the SA is determined by its decay constant, , or half-life T, and by its atomic weight M:

    ]/[1017.41002.62323

    gBqMTM

    SA ==

    In [s]# of atoms per gram of nuclide

  • Example

    What is the SA of Ra226 in Bq/g

    gBqgsSAMT

    SA

    AMappendixyT

    /107.31066.3)3600243651600(226

    1017.41017.4226(1600

    101110

    2323

    ==

    =

    =

    ===

  • SA (T,A)

    BqCi

    gCiAT

    SA

    10107.31

    ]/[2261600

    =

    =

    T is expressed in years

  • Serial radioactivity decay

    A sample in which one radionuclide produces one or more radioactive offspring in a chain

    Secular equilibriumTransient equilibriumNo equilibrium

  • Secular equilibrium (T1>>T2)

    At any time a Long-Lived parent (1) decays into a Short-Lived daughter (2), which decays to a stable nuclideT1>>T2A1of the parent is constant (assuming short intervals of time compared to T1)At any time AT = A1 + A2

  • Secular equilibrium (T1>>T2)

    dtudu

    dNduNAuA

    dtNA

    dN

    NAdtdN

    2

    22221

    1

    221

    2

    2212

    ;constant

    =

    ===

    =

    =

    tt eAeAA

    tNANANAc

    tNNCIctNA

    222012

    22021

    221

    2021

    202

    2221

    )1(

    ln

    )ln(0@..

    )ln(

    +=

    =

    ====+=

  • Secular equilibrium

    Activity A2 relatively short-lived radionuclide as function of timeI.C: A20 =0Activity of daughter builds up to that of parent in about 7 half-livesDaughter decays at the same rate it is produced (A2=A1)Secular equilibrium is said to existTotal activity is 2(A1)

    Act

    iviti

    es

    ~7T2

    A1A2=A1

    A2 secularequilibrium

    T1>>T2

    0 t

  • Secular equilibrium

    2211 NN =In terms of numbers of atoms

    A chain of n short-lived radionuclides can all be in secular equilibrium with a long-lived parentThe activity of each member of the chain = activity of parentTotal activity = (n+1)(A of original parent)

  • General Case

    If there is no restriction on the relative magnitudes of T1 and T2:

    m)equilibriusecular a describes (also !!!0

    )(

    0..

    2211

    2012

    12

    1012

    20

    22212

    21

    NNAand

    eeNN

    NCI

    NNdtdN

    tt

    ==>>

    =

    ==

    =

  • Transient equilibrium (T1>T2)

    N20 = 0T1>T2A2 of the daughter initially build-up steadilyWith time, e-2t becomes negligible, since 2>1

  • Transient equilibrium (T1>T2)

    12

    122

    12

    101222

    12

    1012

    12

    1012

    )(

    )(

    0t

    )(

    1

    1

    21

    =

    =

    =

    >>

    =

    AA

    eNN

    eNN

    eeNN

    t

    t

    tt

  • Activities as function of time

    After initially increasing, the daughter activity A2 goes thru a maximum and decreases at the same rate as the parent activityThus, transient equilibrium existThe total activity also reaches a maximum, early than the daughterThe time transient equilibrium is reached depends on T1 & T2

    activ

    ities

    Transientequilibrium

    A1 + A2

    A1

    A2

    A10

    t

    T1 > T2

    0

  • No Equilibrium (T1 < T2)

    When a daughter (N20 = 0) has a longer T2 than the parent T1 its activity build ups a maximum and then declinesThe parent eventually decays away (T1is shorter)Thus, only the daughter is leftNo equilibrium occurs

  • No Equilibrium (T1 < T2)

    Activities as function of time when T2 > T1 and N20 = 0Non equilibrium occursOnly the daughter activity remains

    A1 + A2A1

    A2

    A10

    t

    T2 > T1

    activ

    ities

    0

  • Example

    Starting with a 10 GBq (1010 Bq) sample of pure Sr90 at time t = 0, how long will it take for the total activity (Sr90 + Y90) to build up 17.5 GBq?

  • Solution

    Appendix D38Sr90 - decays with a T = 29.12 y into 39Y90, which - decays into stable 40Zr90 with T = 64 hT1 >> T2Secular equilibrium is reached in about 7T2 = 7x64= 448hAt the end of this time, the Sr90 activity A1 has not diminished appreciablyThe Y90 activity A2 has increased to the level A2=A1=10 GBqTotal activity AT = 20 GBq

  • Solution

    Time at which Y90 reaches 7.5 GBqThe answer will be less than 448 h

    hte

    GBqAGBqAhT

    AeAeAA

    t

    tt

    128)1(105.7

    5.7,10;0108.0/693.0

    0)1(

    0108.021

    122

    20

    201222

    ==

    ====

    =+=

  • Example

    How many gram of Y90 are in secular equilibrium with 1 mg of Sr90?

  • Solution

    The amount of Y90 will be that having the same activity as 1mg of Sr90

    The SA of Sr90 of (T1 = 29.12y) is:

    ggCi

    Cim

    gCi

    dy

    hdh

    yAA

    CigCigA

    gCiSA

    251.0/105.5

    138.0

    /105.590226

    3651

    24164

    1600SA

    m)equilibriusecular (138.0/13810

    /13890226

    12.291600

    5

    52

    21

    31

    =

    =

    =

    =

    ===

    ==

  • Example

    A sample contains 1 mCi of Os191 at time t = 0. The isotope decays by - emission into metastable Ir191m which then decay by emission into Ir191 .

    IrIrOs m 19177191

    7719176

    15.4d 4.94s

  • Example

    (a) how many grams of Os191 are present at t = 0?(b) how many mCi of Ir191m are present at t = 25 d?(c) how many atoms of Ir191m decay between t = 100s and t = 102s?(c) how many atoms of Ir191m decay between t = 30d and t = 40d?

  • Solution

    Secular equilibrium is reached at 7X4.9 = 34 sThus, A1 = A2 at the equilibriumHowever, during the time considered at (b) and (d) A2 will have decayed appreciably (transient equilibrium)

  • Solution

    (a) Grams of Os191

    (b) At t = 25d

    ggCi

    Cim

    gCiSA

    84

    3

    41

    1023.2/1049.4

    10

    /1049.4191226

    4.153651600

    =

    =

    =

    =

    mCieAA 325.01 4.15/25696.021 ===

  • Solution

    (c) Between 100s and 102 s secular equilibrium exists with the osmium source essentially still at its original activity:

    717

    172

    104.7107.32satoms#s 2next theuring107.31

    100@

    ==

    ==

    =

    sd

    smCiAst

  • Solution

    (d) Between 30 and 40s A1 and A2 do not stay constantTransient equilibrium exists, so the # of atoms of Parent and Daughter that decay are equal

    7

    8

    40

    30

    4030

    0450.07

    4.15/693.07

    1073.7)259.0165.0(1022.8

    |0450.0

    107.3107.3

    =

    =

    = tt ee

    Radioactive DecayActivityExponential decayExponential radioactive decay lawHalf-life, TExponential decay in term of TExampleMean life, tMean life, tSpecific Activity, SAExampleSA (T,A)Serial radioactivity decaySecular equilibrium (T1>>T2)Secular equilibrium (T1>>T2)Secular equilibriumSecular equilibriumGeneral CaseTransient equilibrium (T1>T2)Transient equilibrium (T1>T2)Activities as function of timeNo Equilibrium (T1 < T2)No Equilibrium (T1 < T2)ExampleSolutionSolutionExampleSolutionExampleExampleSolutionSolutionSolutionSolution