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In an effort to understand more about the first law I was wondering about the potential energy term and how it would influence the gas inside the box I made this question and tried to answer it.

I will assume that the air follows the ideal gas law, there is no air resistance and the box stops moving immediately when it reaches the rigid ground.

The first law: $$ 0 = \Delta PE + \Delta U\\ gH = c_v (T' - T)\\ T' = gH/c_v + T $$

would this be correct or missing something? (i.e. invalid assumptions..)

I am also wondering if the energy is fully transferred to the ground instead..

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    $\begingroup$ Gravitational potential energy is converted into kinetic energy, not heat. $\endgroup$
    – user122423
    Jun 14 '17 at 2:59
  • $\begingroup$ Related? physics.stackexchange.com/q/302585/104696 $\endgroup$
    – Farcher
    Jun 14 '17 at 6:33
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$Energy\space of\space the\space system\space (E) = Internal\space energy\space (U) + PE_{system} + KE_{system}$,

$\Delta E =\Delta U+\Delta{PE} + \Delta{KE}$,

The condition for $\Delta E =\Delta{U}$ is $\Delta{PE}=\Delta{KE}=0$.

The first law of thermodynamics gives us the relation between the change in energy of the system and heat and work.

$\Delta E + PdV=Q$,

$\Delta U+\Delta {PE} + PdV=Q$ $\tag1$

In your question, $\Delta {PE}=-mgh$ and $PdV=0 $ $(as\space dV=0)$.

Your equation is correct.

Check this out : What is the meaning of internal Energy?

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Exactly, $dU=\delta Q -\delta W$, and here $\delta W=mgh$ and $\delta Q=0$. You can run some numbers to see that the value is very small for realistic settings.

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