8.08 Problem Set # 11Problems:1. Rederive an equation for a sound wave, using the Navier-Stokes equation instead of the Eulerequation (used in the lecture). Find the damping coefficient of the sound. (Assume that thedamping has a form e−αt, then α is the damping coefficient.)2. Regard atomic motion in a gas as an random walk due to collisions, give an order-of-magnitudeestimate of the time it would take an air molecule in a room to travel a distance of 1cm. Whatabout 1m? (You need to estimate the mean-free path first.)3. Copper Cusolid has a density 8.9g/cm3and an electric resistivity 1.7 × 10−6Ω cm. Assumethe density of the conduction electrons is the same as the density of the Cuatom and theelectron mass is the same as electron mass in the vacuum. Estimate relaxation time τ andthe mean free path λ of the electrons in Cusolid from the Drude model.4. Silicon Sisolid has a density 2.3g/cm3. The velocity of the sound is 2200m/sec. The thermalconductivity at T = 300K is 1.5Wcm K. Assuming the thermal conductivity is due to thephonons, estimate the mean free path of the phonons. (Hint: you may need to estimate thephonon density first.)5. Thermal insulation of a dilute gas (20 pts.)One way to prevent heat transfer is to create a vacuum. In this problem, we study how heatconductance depends on the density of a gas.Let us consider two square sheets of metal separated by a small distance l. The density ofthe gas between the two metal sheet is n. The heat conductance K between the two metalsheet is defined as the following. If the two metal sheets have a small temperature difference∆T , the energy (heat) transfer rate between the two sheets is P = K∆T .lAHere we assume that the gas particle can be treated as a hard sphere of radius a and massm. The mass is distributed uniformly inside the sphere. The environment temperature is T .The area of the metal sheet is A. The translational motion of the particles is always treatedclassically.A quantitative calculation of a transport property (such as thermal conductivity) can be verycomplicated. Here we will make some assumptions to simplify the calculation. We assumethat the particle can reach thermal equilibration after only one collision with the wall or withanother particle. (ie the average thermal kinetic energy reaches its equilibrium value givenby the local temperature after only one collision with the wall or with other particles.)1In parts (a) and (b) below, answers are only required up to a dimensionless constant. Wealso ignore the rotational motion of the particles in parts (a) and (b).(a) Estimate the heat conductance K when the gas density satisfies1la2 n 1a3.(b) Estimate the heat conductance K when the gas density satisfies n 1la2.(c) Now let us consider the possibility that the particle can rotate, and treat the rotationquantum mechanically. Sketch the temperature dependence of K/Tαand mark the im-portant temperature scales. Here α is the exponent that characterizes the T dependenceof K obtained in (a) and (b): K ∝ Tα. What is the ratioK/TαT →0K/TαT →∞? (Here we again as-sume that the rotational degrees of freedom of a particle can reach thermal equilibrationafter only one collision with the wall or with another particle.)(d) Repeat (a) and (b), but now assume that particles have the following energy-momentumrelation E =