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Math Help - help substituting one formula into another

  1. #1
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    help substituting one formula into another

    First of all apologies if this is in the wrong section, I really wasn't sure where to put it. I'm a physics student in my 2nd year at university and came across the following in a text book:

    Note that all derivatives are partial

    (1) U=(1/C)*(P - T*(dP/dT))

    (2) P*v = RT(1+B/v)

    Substitute (2) into (1) gives:

    U= (-1/C) * ((RT^2)/v^2) * (dB/dT)

    The book just gives the result and misses out all the mathematical steps and when I try it myself I just can't get that result.

    from (2) P= ((RT)/v) + ((RTB)/v^2)

    dp/dt = R/v + (RB)/v^2

    Substitute in gives:

    U = (1/c) * [ { ((RT)/v) + ((RTB)/v^2) } - { T*(R/v + (RB)/v^2) } ]

    Which just equals zero?

    I'm sure I must be doing something obvious wrong, but can't see it.. any help would be appreciated!
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  2. #2
    Forum Admin topsquark's Avatar
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    Re: help substituting one formula into another

    Quote Originally Posted by dexev View Post
    First of all apologies if this is in the wrong section, I really wasn't sure where to put it. I'm a physics student in my 2nd year at university and came across the following in a text book:

    Note that all derivatives are partial

    (1) U=(1/C)*(P - T*(dP/dT))

    (2) P*v = RT(1+B/v)

    Substitute (2) into (1) gives:

    U= (-1/C) * ((RT^2)/v^2) * (dB/dT)

    The book just gives the result and misses out all the mathematical steps and when I try it myself I just can't get that result.

    from (2) P= ((RT)/v) + ((RTB)/v^2)

    dp/dt = R/v + (RB)/v^2

    Substitute in gives:

    U = (1/c) * [ { ((RT)/v) + ((RTB)/v^2) } - { T*(R/v + (RB)/v^2) } ]

    Which just equals zero?

    I'm sure I must be doing something obvious wrong, but can't see it.. any help would be appreciated!
    Here's a rundown for you.

    Step by step.
    Pv = RT \left ( 1 + \frac{B}{v} \right )

    P = RT \left ( \frac{1}{v} + \frac{B}{v^2} \right )

    Now, v is an independent variable here, so no T dependence.

    P_T = R \left ( \frac{1}{v} + \frac{B}{v^2} \right ) + \frac{RT}{v^2} \cdot B_T

    Putting this into the U equation:
    U = \frac{1}{C} \cdot \left [ RT \left ( \frac{1}{v} + \frac{B}{v^2} \right ) - T \left ( R  \left ( \frac{1}{v} + \frac{B}{v^2} \right ) + \frac{RT}{v^2}B_T \right ) \right ]

    Upon simplifying (Hah!) you get the book's answer.

    -Dan
    Last edited by topsquark; November 30th 2012 at 08:23 PM.
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