Earths Atmosphere. In the troposphere, the part of the

Chapter 18, Problem 18.84

(choose chapter or problem)

Earth’s Atmosphere. In the troposphere, the part of the atmosphere that extends from earth’s surface to an altitude of about 11 km, the temperature is not uniform but decreases with increasing elevation. (a) Show that if the temperature variation is approximated by the linear relationship

\(T=T_{0}-\alpha y\)

where \(T_{0}\) is the temperature at the earth’s surface and T is the temperature at height y, the pressure p at height y is

\(\ln \left(\frac{p}{p_{0}}\right)=\frac{M g}{R \alpha} \ln \left(\frac{T_{0}-\alpha y}{T_{0}}\right)\)

where \(p_{0}\) is the pressure at the earth’s surface and M is the molar mass for air. The coefficient \(\alpha\) is called the lapse rate of temperature. It varies with atmospheric conditions, but an average value is about 0.6 C°/100 m. (b) Show that the above result reduces to the result of Example 18.4 (Section 18.1) in the limit that \(\alpha \rightarrow 0\). (c) With \(\alpha=0.6\mathrm{C}^{\circ}/100\mathrm{\ m}\), calculate p for y = 8863 m and compare your answer to the result of Example 18.4. Take \(T_0=288\mathrm{\ K}\text{ and }p_0=1.00\mathrm{\ atm}\) atm.