Is this correct or completely wrong?

Question

I've bumped onto the problem described below and I couldn't tell if it is wrong. It looks like it could be correct but it makes no physical sense.

Thank you! Marcelo!

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I begin with $$\frac{p_2}{p_1} = \left(\frac{T_1}{T_2}\right)^{\frac{n}{1-n}\cdot \frac{-1}{-1}} = \left(\frac{T_1}{T_2}\right)^\frac{-n}{n-1}$$ and since $a^{-1} = \frac{1}{a}$, $$\frac{p_2}{p_1} = \left(\frac{T_2}{T_1}\right)^{\frac{n}{1-n}}$$ but this does not seem correct.

Answer 1

In the final step, why have you taken -1 out of the denominator of the power as well? at the end it should be

$$\frac{n}{n-1}$$ not $$\frac{n}{1-n}$$

I don't see the point downvoting things like this so much either. Bit confused why the answers have been downvoted too

Answer 2

In Physics, we often write $P^{1-n}T^n = constant$. That is

$$P_1^{1-n}T_1^n = P_2^{1-n}T_2^n $$

or equivalently

$$P_1T_1^{n/(1-n)} = P_2T_2^{n/(1-n)}$$

or equivalently

$${P_2 \over P_1} = \left( {T_1 \over T_2} \right)^{n/(1-n)}$$ or equivalently

$${P_2 \over P_1} = \left( {T_2 \over T_1} \right)^{n/(n-1)}$$

For the last equivalence and the nub of your question, $$\left( {T_1 \over T_2} \right)^{n/(1-n)} = \left(\left( {T_2 \over T_1} \right)^{-1}\right)^{n/(1-n)} = \left( {T_2 \over T_1} \right)^{-n/(1-n)} = \left( {T_2 \over T_1} \right)^{n/(n-1)}$$

Hopefully that much is now clear.

This is the pressure-temperature relationship for an adiabatic process, where $n$ (also written as $\gamma$) is a physical constant for the system known as the heat capacity ratio. The relation $P^{1-n}T^n = constant$ differs from the $P/T = constant$ law you might be thinking of, because in the latter case, work (positive or negative) must be done to change the system that way.