# Algebraic transposition

• May 26th 2009, 09:44 AM
Algebraic transposition
I have this homework question that I'm having difficulties with. Any help would be most welcome.

The pressure, p, and volume, v, of a gas undergoing a polytropic process are related by the equation, $\displaystyle p_1 v_1^n=p_2 v_2^n$ where n is the polytropic index.

If $\displaystyle \frac{p_1 v_1}{T_1}= \frac{p_2 v_2}{T_2}$ determine the expression for $\displaystyle p_1$, in terms of $\displaystyle p_2$, $\displaystyle T_1$, $\displaystyle T_2$, and n.
• May 26th 2009, 01:03 PM
Isomorphism
Quote:

The pressure, p, and volume, v, of a gas undergoing a polytropic process are related by the equation, $\displaystyle p_1 v_1^n=p_2 v_2^n$ where n is the polytropic index.
If $\displaystyle \frac{p_1 v_1}{T_1}= \frac{p_2 v_2}{T_2}$ determine the expression for $\displaystyle p_1$, in terms of $\displaystyle p_2$, $\displaystyle T_1$, $\displaystyle T_2$, and n.
$\displaystyle (*) \quad p_1 v_1^n=p_2 v_2^n \implies p_1 =p_2 \left(\frac{v_2}{v_1}\right)^n$
$\displaystyle (*) \quad \frac{p_1 v_1}{T_1}= \frac{p_2 v_2}{T_2} \implies \frac{v_2}{v_1}= \frac{p_1 T_2}{p_2 T_1}$
Thus $\displaystyle p_1 =p_2 \left(\frac{v_2}{v_1}\right)^n = p_2 \left(\frac{p_1 T_2}{p_2 T_1}\right)^n \implies p_1 = p_2 \left(\frac{T_2}{T_1}\right)^{\frac{n}{n-1}}$