Serway & Jewett — The Kinetic Theory of Gases: Problem 26.

The law of atmospheres states that the number density of molecules in the atmosphere depends on height \(y\) above sea level according to \[ n_V(y) = n_0 e^{-m_0 gy / k_B T} \] where \(n_0\) is the number density at sea level (where \(y = 0\)). The average height of a molecule in the Earth’s atmosphere is given by \[ y_{\text{avg}} = \frac{\int_0^{\infty} y n_V(y) dy}{\int_0^{\infty} n_V(y) dy} = \frac{\int_0^{\infty} y e^{-m_0 gy / k_B T} dy}{\int_0^{\infty} e^{-m_0 gy / k_B T} dy} \] (a) Prove that this average height is equal to \(k_B T / m_0 g\). (b) Evaluate the average height, assuming the temperature is \(10.0^\circ\text{C}\) and the molecular mass is \(28.9\text{ u}\), both uniform throughout the atmosphere.