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Math Help - Algebraic Methods 2

  1. #1
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    Algebraic Methods 2

    Any help would be really apreciated with the question

    the pressure , p, and volume, v, of a gas undegoing a polytropic process are related by the equation -

    p1V1^n = p2V2^n

    where n is the polytropic index

    if p1V1/T1 = p2V2/T2

    Determine an expression for p1 in terms of p2,t1,t2 and n.
    Last edited by gav1981; February 28th 2008 at 12:17 PM.
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  2. #2
    Super Member Aryth's Avatar
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    Well, we are given:

    p_1V_1^n = p_2V_2^n

    So, to say:

    \frac{p_1V_1}{t_1} = \frac{p_2V_2}{t_2}

    Therefore:

    p_1V_1 = t_1\frac{p_2V_2}{t_2}

    p_1 = p_2\frac{V_2}{V_1}\frac{t_1}{t_2}

    Now, using the equivalency before:

    p_1V_1^n = p_2V_2^n

    \sqrt[n]{\frac{p_1}{p_2}} = \frac{V_2}{V_1}

    So:

    p_1 = p_2\sqrt[n]{\frac{p_1}{p_2}}\left(\frac{t_1}{t_2}\right)

    \frac{p_1}{p_1^{\frac{1}{n}}} = p_2\sqrt[n]{\frac{1}{p_2}}\left(\frac{t_1}{t_2}\right)

    We know that:

    \frac{p_1}{p_1^{\frac{1}{n}}} = p_1^{\frac{n-1}{n}}

    So therefore:

    p_1 = \sqrt[\frac{n-1}{n}]{p_2\sqrt[n]{\frac{1}{p_2}}\left(\frac{t_1}{t_2}\right)}
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