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For adiabatic process pv^2 = a const: p (R T/p)^2 = a const: p^1-r T^r = a const: T^2 (1 - r) p^-r dp + r T^r - 1 p^1 - r dT = 0 (1 - r) dp + r p/T dt = 0 or, r p/T dT = (r - 1) dp dT/dx = T/p (r - 1)/r dp/dx we have r = c_p /w implies r - 1/r = 1 - 1/r = 1 - c_v/c_p = c_p - c_v/c_p = R/c_p therefore dT/dx = R T/p c_p (dp/dx) = v/c_p dp/dx, where p v = +R T we have, dp/dx = -row g rho = n/v dT/dx = -v/c_p rho g = -Mg/c_p dT/dn = -g/c_p, for unit mass
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