Many properties of a gas of atoms can be estimated using a...

Many properties of a gas of atoms can be estimated using a simple model of the gas as an assembly of colliding hard spheres. The purpose of this problem is to derive approximate expressions for a number of coefficients that are used to quantitatively describe various phenomena. For each of the coefficients below state your answer in terms of: $k=$ Boltzmann's constant, $T=$ temperature, $R=$ radius of atom, $m=$ mass of atom, $c=$ heat capacity per gram, $\rho=$ density. You may neglect factors of order unity. (a) Derive the coefficient of thermal conductivity, $\kappa$ (units: $\mathrm{g} \cdot \mathrm{cm} / \mathrm{s}^3$. $\mathrm{K})$. This occurs in the relation between the heat flux and the temperature gradient. (b) Derive the coefficient of viscosity, $\eta$ (units: $\mathrm{g} / \mathrm{cm} \cdot \mathrm{s}$ ). This occurs in the relation between the tangential force per unit area and the velocity gradient. (c) Derive the diffusion coefficient, $D$ (units: $\mathrm{cm}^2 / \mathrm{s}$ ). This characterizes a system containing gases of two species. It relates the time rate of change of the density of one species to its inhomogeneity in density.

Many properties of a gas of atoms can be estimated using a simple model of the gas as an assembly of colliding hard spheres. The purpose of this problem is to derive approximate expressions for a number of coefficients that are used to quantitatively describe various phenomena. For each of the coefficients below state your answer in terms of: $k=$ Boltzmann's constant, $T=$ temperature, $R=$ radius of atom, $m=$ mass of atom, $c=$ heat capacity per gram, $\rho=$ density. You may neglect factors of order unity.
(a) Derive the coefficient of thermal conductivity, $\kappa$ (units: $\mathrm{g} \cdot \mathrm{cm} / \mathrm{s}^3$. $\mathrm{K})$. This occurs in the relation between the heat flux and the temperature gradient.
(b) Derive the coefficient of viscosity, $\eta$ (units: $\mathrm{g} / \mathrm{cm} \cdot \mathrm{s}$ ). This occurs in the relation between the tangential force per unit area and the velocity gradient.
(c) Derive the diffusion coefficient, $D$ (units: $\mathrm{cm}^2 / \mathrm{s}$ ). This characterizes a system containing gases of two species. It relates the time rate of change of the density of one species to its inhomogeneity in density.

Key Concepts

Kinetic Theory of Gases

This concept underlies the microscopic description of gases by treating them as a collection of individual atoms or molecules in constant, random motion. The kinetic theory connects macroscopic thermodynamic quantities such as temperature and pressure to the average kinetic energy of particles, typically proportional to kT, where k is Boltzmann's constant. It forms the foundation for deriving transport properties of gases by relating molecular motion and collisions to observable phenomena.

Mean Free Path and Collision Cross Section

The mean free path is the average distance a particle travels between successive collisions, and it is determined by the density of the gas and the effective collision cross section of the particles. In the hard sphere model, the collision cross section is approximated using the physical size of the particles (proportional to the square of the radius), allowing one to estimate how frequently particles interact. This concept is crucial for understanding how momentum and energy are transferred across a gas.

Transport Coefficients

Transport coefficients, including thermal conductivity, viscosity, and diffusion coefficients, describe how momentum, energy, and mass are transferred through a medium. These coefficients are derived by combining ideas from the kinetic theory with the specifics of particle collisions, such as the mean free path and average particle velocity. They provide a quantitative link between microscopic dynamics (such as collisions and particle motion) and macroscopic transport phenomena like heat conduction, momentum diffusion (viscosity), and species mixing (diffusion).

Hard Sphere Model

The hard sphere model simplifies the interaction between atoms or molecules by representing them as rigid, non-deforming spheres that collide elastically. This idealized model allows for tractable calculations of the mean free path and collision frequency, which are essential for estimating transport coefficients. Although it neglects more complex interparticle forces, the hard sphere approximation captures the basic geometry and collision dynamics in dilute gases.

Dimensional Analysis

Dimensional analysis is a powerful tool used to derive scaling relations and approximate formulas by ensuring consistency of physical units. In the context of deriving transport coefficients, it helps in combining physical quantities such as temperature, particle size, and mass in a way that produces the correct units for properties like thermal conductivity, viscosity, and diffusivity. This method is invaluable for estimating the magnitude of physical quantities when detailed numerical factors are neglected or unknown.

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