Answer
$-1.65\times10^{-24}$
Work Step by Step
We differentiate equation 15.2 to obtain:
$dp = -\rho g dy$
$ dy = \frac{dp}{-\rho g }$
We know that for an adiabatic process, $ PV^{\gamma}$ and $TV^{\gamma-1}$ are both constant, so we can set their derivatives equal to each other. Thus, we find:
$\frac{\gamma}{\gamma-1}\cdot\frac{dP}{V}=dT$
Thus, we find:
$\frac{dT}{dy}=\frac{\frac{\gamma}{\gamma-1}\cdot\frac{dP}{V}}{\frac{dp}{-\rho g }}$
$\frac{dT}{dy}=\frac{\frac{\gamma}{\gamma-1}\cdot\frac{dP}{V}}{\frac{dp}{-\frac{m}{V} g }}$
$\frac{dT}{dy}=\frac{\frac{\gamma}{\gamma-1}\cdot dP}{\frac{dp}{-m g }}$
$\frac{dT}{dy}=\frac{\frac{1.4}{1.4-1}}{\frac{1}{-(4.81556\times10^{-26}) (9.81) }}=-1.65\times10^{-24}$
