1. 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 -

$\displaystyle p1V1^n = p2V2^n$

where n is the polytropic index

if p1V1/T1 = p2V2/T2

Determine an expression for p1 in terms of p2,t1,t2 and n.

2. Well, we are given:

$\displaystyle p_1V_1^n = p_2V_2^n$

So, to say:

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

Therefore:

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

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

Now, using the equivalency before:

$\displaystyle p_1V_1^n = p_2V_2^n$

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

So:

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

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

We know that:

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

So therefore:

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