From the gas-equation in general, in the atmosphere
| 1 | dp | = | 1 | dp | − | 1 | dθ | = | ρ | − | 1 | dθ | = | 1 | − | 1 | dθ | , | |||||
| ρ | dz | p | dz | θ | dz | p | θ | dz | k | θ | dz |
(8)
which is positive, and the density ρ diminishes with the ascent, provided the temperature-gradient dθ/dz does not exceed θ/k.
With uniform temperature, taking k constant in the gas-equation,
dp/dz = ρ = p/k, p = p0ez/k,
(9)
so that in ascending in the atmosphere of thermal equilibrium the pressure and density diminish at compound discount, and for pressures p1 and p2 at heights z1 and z2
(z1 − z2)/k = loge (p2/p1) = 2.3 log10 (p2/p1).
(10)