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Homework Help: Thermo. Relative Error

  1. May 3, 2010 #1
    1. The problem statement, all variables and given/known data
    In a few words, show that

    [tex]\frac{dT_i}{T_i} = 3,73\frac{dr_s}{r_s}[/tex]

    Where,

    [tex]r_s = \lim_{P_i\rightarrow 0}\frac{P_s}{P_i} [/tex]

    There's a picture below for more details.


    2. Relevant equations

    [tex]T_i = \frac{100}{r_s - 1} [/tex]


    3. The attempt at a solution

    So I tried this,

    [tex]T_i = \frac{100}{r_s - 1} [/tex]

    Applying 'ln' to both sides,

    [tex]ln(T_i) = ln(\frac{100}{r_s - 1}) [/tex]

    And then deriving both sides by [tex]\frac{d}{dr_s}[/tex], as [tex]T_i[/tex] is a function of [tex]r_s[/tex] (right?) I get to

    [tex]\frac{1}{T_i}\frac{dT_i}{dr_s} = \frac{r_s - 1}{100}\frac{-100}{(r_s - 1)^2}\frac{dr_s}{dr_s} [/tex]

    [tex]\frac{1}{T_i}\frac{dT_i}{dr_s} = \frac{-1}{r_s - 1} [/tex]

    [tex]\frac{1}{T_i}\frac{dT_i}{dr_s} = \frac{1}{1 - r_s} [/tex]

    [tex]\frac{dT_i}{T_i} = \frac{dr_s}{1 - r_s} [/tex]

    Where

    [tex]r_s = \lim_{P_i\rightarrow 0} (\frac{P_s}{P_i}) [/tex]


    But this looks completely nonsense when comparing to what I was asked to prove in the first place.

    I should have gotten to something like

    [tex]\frac{dT_i}{T_i} = 3,73\frac{dr_s}{r_s}[/tex]

    But instead I got to

    [tex]\frac{dT_i}{T_i} = \frac{dr_s}{1 - r_s} [/tex]

    Thoughts?
     

    Attached Files:

    Last edited: May 3, 2010
  2. jcsd
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