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##\frac {t_{1/2}}{t_{3/4}}## for an nth order reaction

  1. May 4, 2015 #1
    1. The problem statement, all variables and given/known data
    The question:
    For an nth order reaction ##\frac {t_{1/2}}{t_{3/4}}## depends on (n isn't equal to 1) :
    (A) initial concentration only
    (B) 'n' only
    (C) initial concentration and 'n' both
    (D) sometimes 'n' and sometimes initial concentration.

    2. Relevant equations
    ##\frac{dc}{dt}=-kc^n##

    3. The attempt at a solution
    Integrating this equation:
    (from ##C_0## to ##C## and ##0## to ##t##)
    ##\frac {1}{n-1}## ##[\frac {1}{C^{n-1}}-\frac {1}{(C_{0})^{n-1}}]=kt##
    So when ##t=t_{1/2}## then ##C=C_0/2## and when ##t=t_{3/4}## then ##C=C_0/4##
    And when we put this in the equation and devide then there is no ##C_0## in that. So why is (C) correct? (B) should be correct. Where am I wrong?
     
    Last edited: May 5, 2015
  2. jcsd
  3. May 5, 2015 #2

    Borek

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    Staff: Mentor

    I am not convinced that is correct.

    What is [itex]t_{\frac 3 4}[/itex]?
     
  4. May 5, 2015 #3
    The time in which 3/4th of the reactant is used up. Is it wrong?
     
  5. May 5, 2015 #4

    Borek

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    Staff: Mentor

    Used, or left? I am honestly not sure. Perhaps that's because I am still sleepy.
     
  6. May 5, 2015 #5

    epenguin

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    Homework Helper
    Gold Member

    So far as I can see you are not wrong and there is a cancellation such that only n appears in the ratio.

    Where are you getting these problems from? It would be useful if your personal info included where you are.
     
  7. May 19, 2015 #6

    James Pelezo

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    Gold Member

    I've seen reports on decay times referred to as t3/4 that refer to the time for completion of two half-lives. That is, t1/2(1) + t1/2(2) = t3/4, time required to consume 3/4 of the initial amount of reagent. However, for C = Co/4 which is the sum of two half-lives, but holds only for 1st order reactions. (for example, radioactive decay is 1st order) For n ≠ 1 each subsequent t1/2 calculated compresses or expands depending on the order of reaction being studied. Here are some trends of half-lives relative to order of reaction. For 'dependence on initial concentration', and in the light of the fact that none of the half-life equations contain a concentration term, one would have to assume (for the sake of answering the above question) that to consider a time as t1/2, one has to assume (arbitrarily) that an initial concentration value must exist so as to give meaning to the term half-life, which, I must admit, is a stretch, but is the only logical inference I could relate to the question given.

    For a zero-order reaction:
    zero_order_half-life.jpg

    For a 1st order reaction
    first_order_half-life.jpg

    For a 2nd order reaction:
    Screen_shot_2011-03-14_at_2.02.16_PM.png
     
    Last edited: May 19, 2015
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