Rearranging formula and substitution question?

**chunkylumber111** · September 20th 2011, 08:34 AM

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!...

**Siron** · September 20th 2011, 09:02 AM

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 $p_1=...$

**chunkylumber111** · September 22nd 2011, 11:43 AM

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?

**Siron** · September 22nd 2011, 12:31 PM

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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