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$\pu{1 mol}$ of an ideal gas at $\pu{400 K}$ and $\pu{10 atm}$ is allowed to expand adiabatically against $\pu{2 atm}$ external pressure. Find the final temperature of the gas if it is diatomic.

While looking for the solution of the problem above, I came across the formula for adiabatic compression in a reference book, and I was wondering how I could derive it:

$$T_2 = T_ 1 \left(\frac {C_V + P_\text{ex}\frac{R}{P_1} } { C_V + R} \right)$$

First I tried using the enthalpy equation $$∆H = ∆W + ∆nRT,$$

but it proved to be futile. Then I tried using the definition of an adiabatic process, $PV^\gamma$, but that too didn't yield the desired result. Could someone kindly help me solve this question?

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    $\begingroup$ Chem+Math Expression formatting reference: MathJax Basics / Chem+Math expressions/formulas/equations / Upright vs italic / Math SE Mathjax tutorial // MathJax is preferred not to be used in CH SE Q titles. $\endgroup$
    – Poutnik
    Sep 6, 2023 at 15:05
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    $\begingroup$ For plain text formatting by StackExchange markdown, see as inspiration SE meta - formatting // SE meta - tables // Advanced formatting $\endgroup$
    – Poutnik
    Sep 6, 2023 at 15:05
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    $\begingroup$ The definition of adiabatic process is "no thermal exchange with environment" and not what you said. $PV^\gamma$ is derived from that. So you are not against memorizing formulae when they are nice. (Same with me.) Now to the point: what is $P_{ex}$? $\endgroup$ Sep 6, 2023 at 15:48
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    $\begingroup$ Imagine that you gave me a problem on calculating a potential energy of an apple. Now imagine that I come up with an answer that involves the mass of the apple, the gravity of Earth, the distance to the Moon, and the pressure at the bottom of the Mariana trench. Without looking at the actual expression, do you think it may be correct? Why or why not? $\endgroup$ Sep 7, 2023 at 12:01
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    $\begingroup$ This is an irreversible adiabatic change there is already an answer here chemistry.stackexchange.com/questions/97677/… $\endgroup$
    – porphyrin
    Sep 7, 2023 at 17:07

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