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Compressing and cooling air in a Coolgardie Safe, then decompressing

I'm a 10th grader so it would help to explain how, in the answer to the link above, $$T_{cool}\left(\frac{V_{final}}{V}\right)\left(\frac{P_{ambient}}{P_{cool}}\right)$$ became... $$T_{cool}\left(\frac{P_{ambient}}{P_{cool}}\right)^{2/7}$$

Where did 2/7 come from?

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  • $\begingroup$ The 2/7 comes from analyzing the process of adiabatic reversible expansion or compression of an ideal gas. It is related to the ratio of the heat capacity at constant pressure to the heat capacity at constant volume. $\endgroup$
    – Chet Miller
    Commented May 22, 2020 at 11:43
  • $\begingroup$ Does it have anything to do with the 7/5 gamma value? $\endgroup$
    – El Flea
    Commented May 22, 2020 at 12:10
  • $\begingroup$ Yes it does. $$1-1/\gamma$$ $\endgroup$
    – Chet Miller
    Commented May 22, 2020 at 14:07
  • $\begingroup$ If you use $T_{cool}\left(\frac{V_{final}}{V}\right)\left(\frac{P_{ambient}}{P_{cool}}\right)$ with input values, it doesn't give the same answer as $T_{cool}\left(\frac{P_{ambient}}{P_{cool}}\right)^{2/7}$. There's contradiction here?! $\endgroup$
    – El Flea
    Commented May 24, 2020 at 5:30

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