Question

With the assumption that the air temperature is a uniform 0.0°C, what is the density of the air at an altitude of 1.00 km as a percentage of the density at the surface?

Short answer

The density of the air at an altitude of 1.00 km as a percentage of the density at the surface is $\rho =0.833{\rho }_0$

Step-by-step solution

Step 1:

For density, the function of pressure is

$\rho =\frac{P.M}{R.T}$

As$\rho \propto P$

$$\begin{gathered} \frac{\rho }{{\rho }_0}=\frac{P}{P_0} \\ \frac{P}{P_0}=e^{\raisebox{1ex}{$-M.g\times 1000$}\!\left/ \!\raisebox{-1ex}{$RT$}\right.} \\ =e^{\raisebox{1ex}{$-28.8\times {10}^{-3}\times 9.8\times 1000$}\!\left/ \!\raisebox{-1ex}{$8.315\times 273$}\right.} \\ =0.883 \end{gathered}$$

Therefore, the density of the air at an altitude of 1.00 km as a percentage of the density at the surface is$\rho =0.883{\rho }_0$.