Chapter 21, Problem 21.43P

The law of atmospheres states that the number density of molecules in the atmosphere depends on height y above sea level according to

$n_V(y)=n_0{e}^{-n_{0}gy/k_{\text{a}}T}$

where n₀ is the number density at sea level (where y = 0). The average height of a molecule in the Earth’s atmosphere is given by

$y_{\text{avg}}=\frac{{\displaystyle {\int }_0^{\infty }y{n}_V(y)dy}}{{\displaystyle {\int }_0^{\infty }{n}_V(y)dy}}=\frac{{\displaystyle {\int }_0^{\infty }y{e}^{-n_{0}gy/k_{\text{a}}T}dy}}{{\displaystyle {\int }_0^{\infty }{e}^{-n_{0}gy/k_{\text{a}}T}dy}}$

(a) Prove that this average height is equal to k_BT/m₀g. (b) Evaluate the average height, assuming the temperature is 10.0°C and the molecular mass is 28.9 u, both uniform throughout the atmosphere.