91Ó°ÊÓ

A crude model for the molecular distribution of atmospheric gases above Earth's surface (denoted by height \(h\) ) can be obtained by considering the potential energy due to gravity: $$P(h)=e^{-m g h} / k T$$ In this expression \(m\) is the per-particle mass of the gas, \(g\) is the acceleration due to gravity, \(k\) is a constant equal to \(1.38 \times 10^{-23} \mathrm{JK}^{-1},\) and \(T\) is temperature. Determine \(\langle h\rangle\) for methane \(\left(\mathrm{CH}_{4}\right)\) using this distribution function.

In nonlinear optical switching devices based on dye-doped polymer systems, the spatial orientation of the dye molecules in the polymer is an important parameter. These devices are generally constructed by orienting dye molecules with a large dipole moment using an electric field. Imagine placing a vector along the molecular dipole moment such that the molecular orientation can be described by the orientation of this vector in space relative to the applied field ( \(z\) direction) as illustrated here: For random molecular orientation about the \(z\) axis, the probability distribution describing molecular orientation along the \(z\) axis is given by \(P(\theta)=\sin \theta d \theta / \int_{0}^{\pi} \sin \theta d \theta .\) Orientation is quantified using moments of \(\cos \theta\). a. Determine \(\langle\cos \theta\rangle\) for this probability distribution. b. Determine \(\left\langle\cos ^{2} \theta\right\rangle\) for this probability distribution.