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Math Help - Rearranging formula and substitution question?

  1. #1
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    Rearranging formula and substitution question?

    Hi, I have the following problem:


    The pressure, p, and volume, V, of a gas undergoing a polytropic process are related by the equation


    p1V1^n = p2 V2^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.

    (1 and 2 should be lower case) formatting didn't quite work!...
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  2. #2
    MHF Contributor Siron's Avatar
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    Re: Rearranging formula and substitution question?

    This is what I would do, first reform:
    p_1\cdot (V_1)^n =p_2\cdot (V_2)^n  \Leftrightarrow \left(\frac{V_2}{V_1}\right)^{n}=\frac{p_1}{p_2} \Leftrightarrow \frac{V_2}{V_1}=\sqrt[n]{\frac{p_1}{p_2}} (1)
    Also rewrite:
    \frac{p_1\cdot V_1}{T_1}=\frac{p_2\cdot V_2}{T_2} \Leftrightarrow p_1=\frac{p_2\cdot V_2\cdot T_1}{T_2\cdot V_1}=\frac{p_2\cdot T_1}{T_2}\cdot \frac{V_2}{V_1} (2)

    Substitute (1) in (2) and rewrite it as p_1=...
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  3. #3
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    Re: Rearranging formula and substitution question?

    Thanks for the help. So after substitution p1 = (p2.T1)/T2 x nthroot(p1/p2)

    I'm still not sure how I go about rearranging to get rid of the nthroot(p1) term on the right hand side though, as ideally we just want p1 = .... without another p1 term in the equation, if you know what I mean?
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  4. #4
    MHF Contributor Siron's Avatar
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    Re: Rearranging formula and substitution question?

    If you do the substitution I suggested then you get:
    p_1=\frac{p_2\cdot T_1}{T_2}\cdot \sqrt[n]{\frac{p_1}{p_2}}
    \Leftrightarrow \frac{p_1}{p_1^{\frac{1}{n}}}=\frac{p_2\cdot T_1}{T_2}\cdot \frac{1}{\sqrt[n]{p_2}}
    \Leftrightarrow p_1^{1-\frac{1}{n}}=\frac{p_2\cdot T_1}{T_2\cdot \sqrt[n]{p_2}}
    \Leftrightarrow \left(p_1^{1-\frac{1}{n}}\right)^{\frac{n}{n-1}}=\left(\frac{p_2\cdot T_1}{T_2\cdot \sqrt[n]{p_2}}\right)^{\frac{n}{n-1}}
    \Leftrightarrow p_1=\left(\frac{p_2\cdot T_1}{T_2\cdot \sqrt[n]{p_2}}\right)^{\frac{n}{n-1}} (and suppose n \neq 1)
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