3. To a good approximation the density, p, of gas in a thin, isothermal atmosphere (i.e.an atmosphere in thermal equilibrium and whose height is such that the value of theacceleration due to gravity, g, can be considered to be constant throughout it) may berepresented by \rho=\rho_{0} e^{-z / H}, \quad H=\frac{k_{\mathrm{B}} T}{\mu g} where Po is the gas density at the surface, z is the height above the surface, H is the scale height of the atmosphere, and is the average mass of a molecule in the gas making up the atmosphere. \text { (a) Using } \rho=\rho_{0} e^{-z / H} \text { show that } n=n_{0} e^{-z / H} \text {, where } n \text { is the number density of } molecules at height (b) Determine the ratio of the number density of air molecules at an altitude of 10 km to the number density at the surface for the Earth's atmosphere. The average mass of an air molecule can be taken to be μ = 29 atomic mass units (29 amu).