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In meteorology, the Earth's atmosphere up to the stratosphere is subject to processes that are, to a good approximation, adiabatic in nature.

Based on the state equation of an ideal gas, the adiabatic temperature laps rate with altitude is usually derived to be

$$\frac{d T}{d z} = -g/c_p$$

Starting at ~300K at sea level, temperature decreases linearly with altitude z. But if we increase altitude more and more to infinity (towards space), temperature would become negative at some height.

Of course I know, that this not the case of all and that stratosphere has again positive lapse rate.

However, if one considers an idealized atmosphere of ideal gas from a purely theoretical point of view, the derivation of the lapse rate must contain a significant error due to the discrepancy mentioned.

What is the point in particular where the adiabatic model breaks down and loses its validity? So far I was not able to identify some kind of approximation in the derivation of lapse rate which would be violated at higher altitudes . Is upper atmosphere not adiabatic anymore? Why? What are the prerequisites for an atmosphere to be adiabatic? This is always more or less assumed in introductory books, but I have still to find a proper justification.

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    $\begingroup$ In the stratosphere (>10 km), the air temperature actually increases with altitude as a result of absorption of energetic solar radiation by oxygen and ozone at wavelengths < 305 nm. $\endgroup$
    – Chet Miller
    Mar 8 at 22:13

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