How would one find the final temperature of an air parcel ascending adiabatically in the atmosphere? Like which formula would you use? I used the equation of state for dry air to find the initial volume and density. But I'm not sure how to final the final temperature. My profs notes make no sense.
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$\begingroup$ See also Earth Science. $\endgroup$– gerritCommented Sep 24, 2014 at 14:47
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If you use the barometric equation you find the final pressure. Since for the derivation of the barometric equation: $$ P=P_0 \exp\left(\frac{gmh}{RT}\right) $$ You use the ideal gas equation and assume temperature to be uniform, there really isn't much you can do with that (AFAIK).
However if you account for temperature decline and make the first correction you get this equation: $$ P=P_0 \left(1-\frac{ah}{T_0}\right)^{\frac{gm}{aR}} $$ This will account for temperature linear decline by $T=T_0\left(1-\frac{ah}{T_0}\right)$, where a is an experimental quantity.
If you have the volume AND pressure at the altitude, as well as at sea level you can get the value of a.
However, I just noticed you have the adiabatic condition, so you don't need all of this.... You just use the adiabatic condition of $PV^{\gamma}=const$, as well as the barometric equation to find the new $P$ and $V$ and finally you can use the ideal gas equation to find the new temperature....
