1.

Try to derive the barometrical formula for an atmosphere in which the temperature decreases linearly with the altitude.

Answer»


Solution :The force ACTING on a vertical gas column are the force of gravity `RHO gS dh` and force of pressure `F = -S dp`. Since these forces are in equillibrium `rho GH - dh`. We can elimnate the density, using the Mendelleev-Clapeyron equation : `p = 1M rho RT`. Fromrelation
`T = T_(0) (1 - alpha h)` we obtain `dT = -alpha T_0 dh`. Hence
`(dp)/(p) = (Mg)/(alpha RT_0) . (dT)/(T)`
Integrating , we obtain
In `p = (Mg)/(alpha RT_0) ln T` + const.
Since the equation is valid at any point of the gravitaional field, we have on the planet.s surface
`ln p_0 = (Mg)/(alpha RT_0) ln T_0 + "const"`
SUBTRACTING this force the preceding equation, we eliminate the integration constant to obtain
`ln p - ln p_0 = (Mg)/(alpha RT_0) (ln T - ln T_0)`
Whence the formula sought for the barometric destribution.


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