5.4: Transport of Matter and Energy in Perfect Gases

Introduction

In this section we will consider the transport of matter and energy in perfect gases to develop a microscopic particulate picture of the processes. Additionally, we will use this model to develop some quantitative predictions on how rates of transport depend on conditions.

By transport we mean the movement of the matter or energy from an initial location in a sample to somewhere else. Two common examples of this are diffusion, where a high concentration of material spreads out until it is evenly distributed, and thermal conduction, where heat energy moves from a hot region to increase the temperature of cooler regions. The rate of this migration through a surface separating one region from the next is quantified by the flux (usually symbolized by \(J\)). In the simplest form appropriate for transport fluxes, J is a vector perpendicular to the planar surface through which the migration is occurring. The magnitude of the vector is the amount transported per unit time per unit area. In the case of particles using SI units the units would be s^-1m^-2 (particles/s/m²). See figure \(\PageIndex{1}\).

Diffusion (matter transport)

Consider the system sketched in figure \(\PageIndex{1}\). Call the axis perpendicular to the grey plane (parallel to the long sides of the box) z  with the positive direction in the direction of the arrow. At any given time particles will be traveling in the direction of the arrow and the opposite direction. In a homogeneous system the number of particles of a given type moving through the plane will be the same each direction. Symbolically this can be represented as a flux in the z direction of:

\[J_z = \left|{J_{+z}}\right|- \left|J_{-z}\right| = 0,\nonumber\]

where the signs on the subscript indicate the direction of the flux. In this formulation we are looking at only at the magnitude in each direction.

Figure \(\PageIndex{1}\): Cartoon of flux through a plane surface of area A. The length of the vector represents the magnitude of the flux through the plane per unit time per unit area.

To make the mathematics more straightforward it is best to have a positive flux \(J_z\) indicate net flux in the +z direction (direction of arrow). 

If the number density (concentration) varies with position (e.g. \(\rho\) decreases as we move towards +z) the two fluxes will not balance out. Because the particles are undergoing collisions that might change their trajectory so that even with an initial velocity that would allow them to cross the plane they might not. The maximum average rate they could cross the plane is going to be limited by how many molecules are close enough to the plane that they will not suffer a collision before they cross it. So we need to estimate the number density on each side within one mean free path (\(\lambda = (\sqrt{2}\rho\sigma)^{-1}\)) of the plane.

First remember that we know how to calculate the rate of collision per unit area with a plane within the kinetic molecular theory. This is just the rate of collision with a wall per unit area:

\[z_w = \dfrac{1}{4} \dfrac{N}{V} \langle v \rangle =\dfrac{\langle v \rangle}{4} \rho = \rho \sqrt {\dfrac{k_B T}{2 \pi m}},\label{z_w}\]

where <v> is the average velocity, \(k_B\) is the Boltzmann constant and m is the particle mass. The upper limit to the flux from one side is \(z_w\), when \(\rho\) is the number density at distance \(\lambda\) from the plane.

If the rate of change of the number density (\(\rho\)) at the plane is \(\frac{d\rho}{dz}|_o\) then the number density at distance \(\lambda\) (one mean free path) to the left of the plane is:

\[\rho_l = \rho_o -\lambda\frac{d\rho}{dz}|_o \]

and to the right is:

\[\rho_l = \rho_o +\lambda\frac{d\rho}{dz}|_o. \]

Substituting these number densities into equation \(\ref{z_w}\) we get the following two expressions:

\[J_{+z} = \frac{\langle v \rangle}{4}\left( \rho_o -\lambda\frac{d\rho}{dz}|_o\right)\]

and

\[J_{-z} = \frac{\langle v \rangle}{4}\left( \rho_o +\lambda\frac{d\rho}{dz}|_o\right).\]

As the fluxes are both positive, the total flux is:

\[J_z = J_{+z} - J_{-z} = \frac{\langle v \rangle}{4}\left( \rho_o -\lambda\frac{d\rho}{dz}|_o - \rho_o -\lambda\frac{d\rho}{dz}|_o\right) = \frac{\langle v \rangle}{4}\left( -2\lambda\frac{d\rho}{dz}|_o\right) = -\frac{1}{2}\langle v \rangle\lambda\frac{d\rho}{dz}|_o.\label{J_z_big}\]

This is an overestimate because some of the particles will have most of their velocity parallel to the plane. This means that some particles will travel so far before crossing the plane that they will undergo a collision that prevents them from crossing the plane. A more careful and much more tedious analysis leads to a factor of 2/3 decrease in the estimate of \(J_z\):

\[J_z =  -\frac{1}{3}\langle v \rangle\lambda\frac{d\rho}{dz}|_o.\label{J_z_better}\]

Fick's law of diffusion states that the observed flux of particles is proportional to the number density gradient and the proportionality constant (the diffusion constant) is symbolized by D:

\[J_z = -D \frac{\partial \rho}{\partial z}. \label{ficks1}\]

Thus for our simple gas case:

\[D = \frac{1}{3}\langle v \rangle\lambda = \frac{1}{3}\lambda\sqrt{\frac{8k_B T}{\pi m}} = \frac{1}{3\sqrt{2}\rho\sigma}\sqrt{\frac{8k_B T}{\pi m}}.\label{gas_D}\]

So the diffusion constant increases with increased temperature, but decreases as the particles become more massive or the mean free path decreases (higher density or larger particles). In high density systems intermolecular interactions become important and the trends are maintained, but do not exactly follow this simple model.

Thermal conductivity

The coefficient of thermal conductivity \(\kappa\) is the proportionality constant between the temperature gradient and thermal energy flux as defined by the equation:

\[J_z(thermal) = -\kappa \frac{dT}{dz}.\label{J_z_therm1}\]

The energy is transported by moving particles, but in this case there is no gradient in the density of particles just the amount of energy the particles have. Thus this time the energy flux \(J_z(thermal)\) is:

\[J_z (thermal)=  -\frac{1}{3}\langle v \rangle\lambda\rho\frac{d\epsilon}{dz},\label{J_z_therm2}\]

where \(\epsilon\) is the thermal energy content of a particle and \(\rho\epsilon\) is the energy density per unit volume. We can express equation \(\ref{J_z_therm2}\) in terms of \(\frac{dT}{dz}\) by using the chain rule:

\[\frac{d\epsilon}{dz} = \frac{d\epsilon}{dT}\frac{dT}{dz}.\nonumber\]

Substituting this in to equation \(\ref{J_z_therm2}\) yields:

\[J_z (thermal)=  -\frac{1}{3}\langle v \rangle\lambda\rho\frac{d\epsilon}{dT}\frac{dT}{dz}.\label{J_z_therm3}\]

This can be converted to a more useful form by using the definition of molar heat capacity:

\[C_{V,m} = \left(\frac{dE_m}{dT}\right)_V = N_A\left(\frac{d\epsilon}{dT}\right)_V.\label{CVm}\]

Thus, by substituting \(C_{V,m}/N_A\) for \(\frac{d\epsilon}{dT}\) in equation \(\ref{J_z_therm3}\)  and expanding \(\rho\) to (N/V) we get:

\[J_z (thermal)=  -\frac{1}{3}\langle v \rangle\lambda\frac{N}{V}\frac{C_{V,m}}{N_A}\frac{dT}{dz} = -\frac{1}{3}\langle v \rangle\lambda\frac{n}{V}{C_{V,m}}\frac{dT}{dz} =  -\frac{1}{3}\langle v \rangle\lambda[X]{C_{V,m}}\frac{dT}{dz},\label{J_z_therm4}\]

where in the last form of the equation [X] is the molar concentration of the gas. Comparing equation \(\ref{J_z_therm4}\) to equation \(\ref{J_z_therm1}\) we see that:

\[\kappa = \frac{1}{3}\langle v \rangle\lambda[X]{C_{V,m}}.\label{kappa1}\]

Expanding this equation out we get:

\[\kappa = \frac{1}{3\sqrt{2}\rho\sigma}\sqrt{\frac{8k_B T}{\pi m}}[X]C_{V,m} =\frac{1}{3\sqrt{2}\sigma}\sqrt{\frac{8k_B T}{\pi m}}\frac{C_{V,m}}{N_A}\label{kappa2}\]

using the fact that \(\frac{n}{V}C_{V,m} = \rho\frac{C_{V,m}}{N_A}\).

The physical interpretation of equation \(\ref{kappa2}\) is that the thermal conductivity increases with increasing molar heat capacity and temperature, while decreasing as the mass of the particle or their sizes increase. Notice that for a fixed temperature the thermal conductivity does not depend on the gas density (or equivalently pressure in the ideal gas case). Since gases generally have low molar heat capacities they are better insulators than liquids or solids. This is why we make double-paned windows by separating two sheets of glass by dry air or sometimes pure N₂. The lowest heat capacity gases are the noble gases because they are monatomic, meaning they only have kinetic energy contributions to their heat capacity near ambient temperatures. Xe is the best choice for insulation because it is the most massive and the largest diameter of the non-radioactive noble gases. Unfortunately, Xe is relatively rare and thus too expensive to use as the insulating gas in windows and insulators such as expanded foams. Gases only work well as insulators if they are prevented from developing convection currents (wind), so that the energy is only transported diffusively. This is why most gas based insulators (fiberglass, expanded foams, etc) use solids to block convection of the gases.

Equation \(\ref{kappa2}\) shows no pressure dependence for thermal conductivity, yet we know that vacuum jackets (dewar's) are very good insulation. Why? The discrepancy is due to the fact that the preceding analysis ignores the fact that when the pressure is low enough that the mean free path is larger than the gap between the walls, the rate of energy transport becomes proportional to the number of particles available to transport energy across the gap. Thus once the pressure is low enough that the mean free path is greater than the distance between the walls a pressure decrease leads to a decrease in thermal conductivity. An alternative way of thinking about this is that matter is required to transport the thermal energy. When there is no matter no energy can be transported. Dewars work well when gas pressures are < 10^-9 atm. Dewar performance can be further enhanced by making the interior walls reflective so that less energy is transmitted across the gap by thermal blackbody radiation photons.

Viscosity

Most people are familiar with viscosity from handling liquids of different viscosities. For example, the viscosity of honey is higher than water, meaning the honey resists flowing more than water (colloquially honey is "thicker" than water). Empirically, the relative viscosity of two fluids can be determined by measuring the relative amount of time it takes for the same amount of each fluid to flow non-turbulently through a tube. Viscosity is fundamentally the proportionality constant relating the transfer of momentum perpendicular to the direction of flow. Consider the diagram below of a non-turbulent (Newtonian) flow through a tube in the x direction. If the bottom of the figure represents a wall of the tube the fluid in immediate contact with the wall is not flowing while each layer as we move towards the center of the tube flows at a faster rate as indicated by the length of the arrows.

Figure \(\PageIndex{2}\): Cartoon of Newtonian flow of a fluid through a tube in the x direction. The length of the arrows indicate the relative velocities of layers within the tube. At the center of the tube the fluid has the highest velocity.

In each of the layers the particles have different velocities in the x direction (\(v_x(z)\)). Each \(v_x(z)\) has a corresponding x-component of momentum \(p_x(z) = mv_x(z)\). Because of thermal motion particles also move randomly in the z direction. Each time a particle moves in the z direction it carries the \(p_x\) it had in one layer to a neighboring layer. In this way the x momentum of a layer is reduced by transfer from slower layers and increased by transfer from faster layers. The viscosity \(\eta\) is the proportionality constant between the momentum flux and the change in velocity with z:

\[J_z (x\,momentum)=  -\eta\frac{dv_x}{dz}.\label{J_z_visc1}\]

As in the case of thermal transfer the momentum is transferred with the matter, but there is no gradient in the amount of matter just the x-component of the momentum (\(p_x\)). The amount of matter moving is still the same so:

\[J_z (x\,momentum)=  -\frac{1}{3}\langle v \rangle\lambda\rho\frac{dp_x}{dz} = -\frac{1}{3}\langle v \rangle\lambda\rho m \frac{dv_x}{dz},\label{J_z_visc2}\]

where \(\rho p_x\) is the momentum density per unit volume. Comparing equations \(\ref{J_z_visc1}\) and \(\ref{J_z_visc2}\) we see that:

\[\eta = \frac{1}{3}\langle v \rangle\lambda\rho m = \frac{1}{3}\sqrt{\frac{8 k_B T}{\pi m}}\frac{m}{\sqrt{2}\rho\sigma}\rho = \frac{1}{3}\sqrt{\frac{8  m k_B T}{\pi}}\frac{1}{\sqrt{2}\sigma}.\label{eta}\]

This equation implies the viscosity of an idealized gas is independent of the number density, inversely dependent on the size of the particles and proportional to \(\sqrt{T}\) or \(\sqrt{m}\). Thus, the prediction from this model is that the viscosity of gases increases with temperature! This is observed and is the opposite of what is seen in liquids, where the viscosity decreases with increasing temperature. For gases the viscosity increases with temperature because increasing temperature increases the rate at which particles exchange between layers carrying the momentum of flow between layers, which increases the tendency of the layers to maintain the same velocity. In condensed phases the viscosity is primarily due to intermolecular attractions restricting the flow of particles past each other. Increased temperature disrupts the attractions, thus lowering the viscosity.

References

The contents of this section are based on the following two references:

1. P.W. Atkins, J. de Paula, Physical Chemistry, 7th Ed. (W.H. Freeman and Co. New York, 2002) Pp. 826 - 832.
2. A. Cooksy, Physical Chemistry: Thermodynamics (Pearson Ed. Inc. Columbus, 2014) Pp. 184 - 190.

Contributors

• Jonathan Gutow (UW Oshkosh)