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Math Help - Help understanding an integration in the derivation of an adsorption model (see pic)

  1. #1
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    Help understanding an integration in the derivation of an adsorption model (see pic)

    Hi,

    Could someone be so kind and show be how equation (3) in integrated step by step to give (4) (please enlarge attached picture). I am stumped. Also equally frustrating is the fact that I cant even follow the step between (4) to give (5). I would love to hear back from someone to enlighten me.

    Thanks very much in advance,

    Best,

    Renaud

    PS. important i think: Xr(t) represents the concentration of species (X) on resin (r) (physical adsorption beads) which are thrown into a solution at t=0. therefore I know that Xr(t) should be 0 at time equals 0. Hope this helps too.



    Help understanding an integration in the derivation of an adsorption model (see pic)-derivation.png
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  2. #2
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    Re: Help understanding an integration in the derivation of an adsorption model (see p

    Hey renocchi.

    Hint: Use separation of variables to get all the X_r terms on the LHS and everything else on the right hand side. (Also note that the anti-derivative of -1/(a - y)dy = ln(a-y))
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  3. #3
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    Re: Help understanding an integration in the derivation of an adsorption model (see p

    yeah think i did take that into consideration:

    ln[-Xr(t) + d] = kt + C

    -Xr(t) + d = exp(kt) + d

    Xr(t) = exp(kt)

    which is not what the solution is..... grr.
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  4. #4
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    Re: Help understanding an integration in the derivation of an adsorption model (see p

    You need to adjust for the minus sign which will) which gives

    -ln(Xr(t) + d) = kt + C or
    ln(Xr(t) + d) = -kt + C or
    Xr(t) + D = e^C*e^(-kt) or
    Xr(t) = e^c*e^(-kt) - D
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  5. #5
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    Re: Help understanding an integration in the derivation of an adsorption model (see p

    f dXr(t)/-Xr(t) + d] = k f dt
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    Re: Help understanding an integration in the derivation of an adsorption model (see p

    Xr(t) = e^c*e^(-kt) - D woa...

    a^n a^m = a^n+m?!?

    product rule?
    but i thought taking the exponent on both sides would give:

    ln(Xr(t) + d) = -kt + C
    Xr(t) + d = e^-kt + e^C
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  7. #7
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    Re: Help understanding an integration in the derivation of an adsorption model (see p

    Taking exp(a+b) = e(a+b) = e^a*e^b (standard properties of exponents).
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  8. #8
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    Re: Help understanding an integration in the derivation of an adsorption model (see p

    Quote Originally Posted by chiro View Post
    Also note that the anti-derivative of -1/(a - y)dy = ln(a-y))
    Where do i find this identity? Im looking everywhere and cannot find it. the minus sign is throwing me off big time.

    thanks for giving me hints.... but for some reason I still cannot come to terms with this integral yet.
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  9. #9
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    Re: Help understanding an integration in the derivation of an adsorption model (see p

    Its easier to do it the other way: differentiate 1/(a-x) with respect to x using the chain rule and by noting that d/dx (1/x) = log(x).
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