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Math Help - Quantifying a series using dose and half-life

  1. #1
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    Quantifying a series using dose and half-life

    Please move if this is the wrong forum - and this is not homework but a SciFi concept I am needing to explain for a story.

    There is an artificial environment (say some kind of capsule) which requires the daily dosing of a chemical compound with a half life of a day, such that its presence in the environment receeds like this if a dose of say 100units is introduced on day one:

    1 Sat 4 100 2 50.000000
    2 Sun 5 25.000000
    3 Mon 6 12.500000
    4 Tue 7 6.250000
    5 Wed 8 3.125000
    6 Thu 9 1.562500
    7 Fri 10 0.781250
    8 Sat 11 0.390625
    9 Sun 12 0.195313
    10 Mon 13 0.097656
    11 Tue 14 0.048828
    12 Wed 15 0.024414
    13 Thu 16 0.012207
    14 Fri 17 0.006104
    15 Sat 18 0.003052
    16 Sun 19 0.001526 proximal point of trace presence
    17 Mon 20 0.000763
    18 Tue 21 0.000381
    19 Wed 22 0.000191
    20 Thu 23 0.000095
    21 Fri 24 0.000048
    22 Sat 25 0.000024
    23 Sun 26 0.000012
    24 Mon 27 0.000006
    25 Tue 28 0.000003
    26 Wed 29 0.000001
    27 Thu 30 0.000001
    28 Fri 31 0.000000
    29 Sat 1 0.000000
    30 Sun 2 0.000000
    31 Mon 3 0.000000

    A you can see we end up with a point at which only trace elements exist for practical purposes.

    Now, if that dosage is introduced daily at a rate of 100units per day what actually happens to the accumulation within the environment?

    The 100 units introduced on day one represents 50 remaining on day two when another 100 is introduced, which means that the environment contains 150 on day two.
    On day three, when another 100 is introduced the 150 from day 2 is reduced to 75 and there is therefore 175, Day four's 100 consequently joins 175/2 leaving 187.5. Day 5 is 193.75, Day 6 196.875 and we begin to see a series ultimately leading to an approximately stable amount of this element (given a half life of a day) in the artifical system...

    What I seem unable to do in a spreadsheet or otherwise with any certainty is:

    Properly express the series,
    understand the nature of this series.

    Can anyone assist - will it ultimately infinitely approach 200???


    This is all I can come up with in my spreadsheet>

    100
    100 50 150
    100 75 175
    100 87.5 187.5
    100 93.75 193.75
    100 96.875 196.875
    100 98.4375 198.4375
    100 99.21875 199.21875
    100 99.609375 199.609375
    100 99.8046875 199.8046875
    100 99.90234375 199.9023438
    100 99.95117188 199.9511719
    100 99.97558594 199.9755859
    100 99.98779297 199.987793
    100 99.99389648 199.9938965
    100 99.99694824 199.9969482
    100 99.99847412 199.9984741
    100 99.99923706 199.9992371
    100 99.99961853 199.9996185
    100 99.99980927 199.9998093
    100 99.99990463 199.9999046
    100 99.99995232 199.9999523
    100 99.99997616 199.9999762
    100 99.99998808 199.9999881
    100 99.99999404 199.999994
    100 99.99999702 199.999997
    100 99.99999851 199.9999985
    100 99.99999925 199.9999993
    100 99.99999963 199.9999996
    100 99.99999981 199.9999998
    100 99.99999991 199.9999999

    Is this right? That an element with a half life of one day when dosed at rate of 100 units per day will ultimately stabilise at twice the daily dose?

    Will
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  2. #2
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    earboth's Avatar
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    You are dealing with a geometric serie. (Geometric Series -- from Wolfram MathWorld)

    The level of the dose at day n is:

    s_n=100+50+25+12.5+ ... + \dfrac{100}{2^{n-1}}

    The ration between to summands is r=\dfrac12.

    Since r < 1 you'll get:

    \lim_{n \to +\infty} = \dfrac{a_0}{1-r}

    In your case:

    \lim_{n \to +\infty} = \dfrac{100}{1-\dfrac12} = 200
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  3. #3
    Grand Panjandrum
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    Quote Originally Posted by stardotstar View Post
    Please move if this is the wrong forum - and this is not homework but a SciFi concept I am needing to explain for a story.

    There is an artificial environment (say some kind of capsule) which requires the daily dosing of a chemical compound with a half life of a day, such that its presence in the environment receeds like this if a dose of say 100units is introduced on day one:

    1 Sat 4 100 2 50.000000
    2 Sun 5 25.000000
    3 Mon 6 12.500000
    4 Tue 7 6.250000
    5 Wed 8 3.125000
    6 Thu 9 1.562500
    7 Fri 10 0.781250
    8 Sat 11 0.390625
    9 Sun 12 0.195313
    10 Mon 13 0.097656
    11 Tue 14 0.048828
    12 Wed 15 0.024414
    13 Thu 16 0.012207
    14 Fri 17 0.006104
    15 Sat 18 0.003052
    16 Sun 19 0.001526 proximal point of trace presence
    17 Mon 20 0.000763
    18 Tue 21 0.000381
    19 Wed 22 0.000191
    20 Thu 23 0.000095
    21 Fri 24 0.000048
    22 Sat 25 0.000024
    23 Sun 26 0.000012
    24 Mon 27 0.000006
    25 Tue 28 0.000003
    26 Wed 29 0.000001
    27 Thu 30 0.000001
    28 Fri 31 0.000000
    29 Sat 1 0.000000
    30 Sun 2 0.000000
    31 Mon 3 0.000000

    A you can see we end up with a point at which only trace elements exist for practical purposes.

    Now, if that dosage is introduced daily at a rate of 100units per day what actually happens to the accumulation within the environment?

    The 100 units introduced on day one represents 50 remaining on day two when another 100 is introduced, which means that the environment contains 150 on day two.
    On day three, when another 100 is introduced the 150 from day 2 is reduced to 75 and there is therefore 175, Day four's 100 consequently joins 175/2 leaving 187.5. Day 5 is 193.75, Day 6 196.875 and we begin to see a series ultimately leading to an approximately stable amount of this element (given a half life of a day) in the artifical system...

    What I seem unable to do in a spreadsheet or otherwise with any certainty is:

    Properly express the series,
    understand the nature of this series.

    Can anyone assist - will it ultimately infinitely approach 200???


    This is all I can come up with in my spreadsheet>

    100
    100 50 150
    100 75 175
    100 87.5 187.5
    100 93.75 193.75
    100 96.875 196.875
    100 98.4375 198.4375
    100 99.21875 199.21875
    100 99.609375 199.609375
    100 99.8046875 199.8046875
    100 99.90234375 199.9023438
    100 99.95117188 199.9511719
    100 99.97558594 199.9755859
    100 99.98779297 199.987793
    100 99.99389648 199.9938965
    100 99.99694824 199.9969482
    100 99.99847412 199.9984741
    100 99.99923706 199.9992371
    100 99.99961853 199.9996185
    100 99.99980927 199.9998093
    100 99.99990463 199.9999046
    100 99.99995232 199.9999523
    100 99.99997616 199.9999762
    100 99.99998808 199.9999881
    100 99.99999404 199.999994
    100 99.99999702 199.999997
    100 99.99999851 199.9999985
    100 99.99999925 199.9999993
    100 99.99999963 199.9999996
    100 99.99999981 199.9999998
    100 99.99999991 199.9999999

    Is this right? That an element with a half life of one day when dosed at rate of 100 units per day will ultimately stabilise at twice the daily dose?

    Will
    In the long term (after a large enough number of days so that we have a sort of equilibrium) consider the environmental load just after dosing, call this x, next day just before the next dosing this will have fallen to x/2, but after dosing will return to x, so:

    x=x/2+100

    and a bit of algebra shows that x=200.

    CB
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  4. #4
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    Joined
    May 2008
    Posts
    10
    Quote Originally Posted by CaptainBlack View Post
    In the long term (after a large enough number of days so that we have a sort of equilibrium) consider the environmental load just after dosing, call this x, next day just before the next dosing this will have fallen to x/2, but after dosing will return to x, so:

    x=x/2+100

    and a bit of algebra shows that x=200.

    CB

    Thank you both - the elaboration of the series and link from earboth is most informative and the simplicity of your response CB is quite brilliant (at least to my fumbling mathematical aparatus) thank you very much guys.
    Will
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