# Molecular Level Presentation

## Introduction

The concepts associated with phases of matter at microscopic and macroscopic levels are important in the study of heat and mass transfer, and therefore they are briefly reviewed here. For this purpose, consider a lump of ordinary sugar. When the lump is broken into smaller pieces, each of these smaller pieces is still identifiable as a particle of sugar based on properties such as its color, density, and crystalline shape. If one continues to grind the sugar to a finer powder, the basic properties of the material remain the same except that the size of the particles is reduced. When the very fine powder sugar is dissolved in water, the particles are too small to be seen with a microscope, yet the taste of sugar persists. Evaporating the water from the sugar solution restores the original distinguishing properties of the solid sugar mentioned above. This simple experiment shows that matter is composed of particles which are extremely small. The smallest particle of sugar that retains any identifying properties of the substance is a molecule of sugar. A molecule is the smallest chemical unit of a substance that is capable of stable, independent existence; however, not all substances are composed of molecules. Some substances are composed of electrically-charged particles known as ions. To get an idea of the extremely small size of molecules, we can consider that a molecule of water is about $3 \times {10^{ - 10}}{\rm{m}}$ (3 Angstroms, Å) in diameter. On the other hand, molecules of more complex substances may have sizes of more than 200 Å. If a molecule of sugar is analyzed further, it is found to consist of particles of three simpler kinds of matter: carbon, hydrogen and oxygen. These simpler forms of matter are called elements. An atom is the smallest unit of an element that can exist either alone or in combination with other atoms of the same or different elements. The smallest atom, an atom of hydrogen, has a diameter of 0.6 Å. The largest atoms are slightly larger than 6 Å in size. The atomic mass of an atom is expressed in atomic mass units. One atomic mass unit is equal to ${\rm{1}}{\rm{.6605402}} \times {\rm{1}}{{\rm{0}}^{ - 27}}{\rm{ kg}}{\rm{.}}$ This mass is 1/12 the mass of the carbon-12 atom. The integer nearest to the atomic mass is called the mass number of an atom. The mass number for a hydrogen atom is 1; for a common uranium atom, one of the heaviest atoms, it is 238. Under normal conditions at the macroscopic level, there are three phases (or states) of matter: solid phase, liquid phase, and gaseous phase. Plasma is sometimes called the fourth phase of matter. In the description of matter, phase indicates how particles group together to form a substance. The structure of a substance can vary from compactly-arranged particles to highly-dispersed ones. In a solid, the particles are close together in a fixed pattern, while in a liquid the particles are almost as close together as in a solid but are not held in any fixed pattern. In a gas, the particles are also not held in any fixed pattern, but the average distance between particles is large. Both liquids and gases are called fluids. A gas that is capable of conducting electricity is called plasma. Gases do not normally conduct electricity, but when they are heated to high temperatures or collide intensely with each other, they form electrically-charged particles called ions. These ions give the plasma the ability to conduct an electrical current. Because of the high temperatures that prevail in the sun and other stars, their constituent matter exists almost entirely in the plasma phase.

## Kinetic Theory

According to the elementary kinetic theory of matter, the molecules of a substance are in constant motion. This motion depends on the average kinetic energy of molecules, which depends in turn on the temperature of the substance. Furthermore, the collisions between molecules are perfectly elastic except when chemical changes or molecular excitations occur. The concept of heat as the transfer of thermal energy can be explained by considering the molecular structures of a substance. At the standard reference state (25 ˚C at 1 atm), the density of a typical gas is about 1/1000 of that of the same matter as a liquid. If the molecules in the liquid are closely packed, the distance between gas molecules is about 10001 / 3 = 10 times the size of the molecules. Since the size of molecules is on the order of 10 − 10m, the distance between the molecules of gas is on the order of 10 − 9m. Therefore, a gas in the standard reference state can be viewed as a set of molecules with large distances between them. Since the distance between gas molecules is so large, the intermolecular forces are very weak, except when molecules collide with each other. The distance that a molecule travels between two collisions is on the order of 10 − 7 m, and the average velocity of molecules is about 500 m/s, which means that the molecules collide with each other every 10 − 10s, or at the rate of 10 billion collisions per second. The duration of each collision is approximately 10 − 13s, a much shorter interval than the average time between two collisions. The movement of gas molecules can therefore be characterized as frequent collisions between molecules, with free movement between collisions. For any particular molecule, the magnitude and direction of the velocity changes arbitrarily due to frequent collision. The free path between collisions is also arbitrary and difficult to trace. Although the motion of the individual molecule is random and chaotic, the movement of the molecules in a system can be characterized using statistical rules. The following assumptions about the structure of the gases are made in order to investigate the statistical rules of the random motion of the molecules:

1. The size of the gas molecules is negligible compared with the distance between gas molecules.

2. The molecules collide infrequently because the collision time is much shorter than the free motion time.

3. The effects of gravity and any other field force are negligible, thus the molecules move along straight lines between collisions. The motion of gas molecules obey Newton’s second law.

4. The collision of gas molecules is elastic, which means that the kinetic energy before and after a collision is the same.

Therefore, the gas can be viewed as a set of elastic molecules that move freely and randomly. Any gas that satisfies the above assumptions is referred to as an ideal gas. At thermodynamic equilibrium, the density of the gas in a container is uniform. Therefore, it is reasonable to assume that the gas molecules do not prefer any particular direction over other directions. In other words, the average of the square of the velocity components of the gas in all three directions should be the same, i.e., $\overline {{u^2}} = \overline {{v^2}} = \overline {{w^2}} .$ Information concerning mean molecular velocity, frequency, mean free path, and density number with the above assumptions can be obtained using simple kinetic theory (Berry et al, 2000). The average magnitude of the molecular velocity is given by simple kinetic theory

$\bar c = \sqrt {\frac{{8{k_b}T}}{{\pi m}}} \qquad \qquad(1)$

where kb is the Boltzmann constant, and m is the mass of the molecule. For any stationary surface exposed to the gas, the frequency of the gas molecular bombardment per unit area on one side is given by

$f= \frac{1}{4} \mathfrak{N} \overline{c} \qquad \qquad (2)$

where $\mathfrak{N}$ is the number density of the molecules, defined as number of molecules per unit volume ($\mathfrak{N} =N/V$). The mean free path, defined as average distance traveled by a molecule between collisions, is

$\lambda = \frac{1}{\sqrt{2} \pi {{\sigma}^2}\mathfrak{N}} \qquad \qquad (3)$

where σ is the molecular diameter. The relaxation time, τ which is the average time between two subsequent collisions, is:

$\tau = \frac{\lambda }{{\bar c}}\qquad \qquad(4)$

The collision rate, τ − 1, is the average number of collisions an individual particle undergoes per unit time. After the last collision with other molecules, the molecule travels an average distance of 2λ / 3 before it collides with the plane. A table for the particle diameters, mean free path, mean velocity and relaxation time (τ) between molecular collisions is given in Table 1 for some gases at 25° C and atmospheric pressure.

Table 1 Kinetic properties of gases at 25 °C and atmospheric pressure (Lide, 2004)*

 Gas $\sigma \times {10^{10}}(m)$ $\lambda \times {10^8}(m)$ $\bar {c_m}(m/s)$ τ(ps) Air 3.66 6.91 467 148 Ar 3.58 7.22 397 182 CO2 4.53 4.51 379 119 H2 2.71 12.6 1769 71 He 2.15 20.0 1256 159 Kr 4.08 5.58 274 203 N2 3.70 6.76 475 142 NH3 4.32 4.97 609 82 Ne 2.54 14.3 559 256 O2 3.55 7.36 444 166 Xe 4.78 4.05 219 185
• Reproduced by permission of Routledge/Taylor & Francis Group, LLC.

The mass flux of molecules in one direction at a point in a gas is given as (Tien and Lienhard, 1979):

${m_{molecules}}^{''} = \frac{\mathfrak{N} \overline{c} m}{4} = \mathfrak{N} m {(\frac{{k_b}T}{2 \pi m})}^{1/2} = \mathfrak{N} {(\frac{{k_b}Tm}{2 \pi })}^{1/2} \qquad \qquad (5)$

The pressure of gas in a container results from the large number of gas molecules colliding with the container wall. Although each molecule in the container collides with the container wall randomly and discontinuously, the collisions of a large number of molecules with the container wall impart a constant and continuous pressure on the wall. As expected, the pressure in a container is related to the number, and average velocity, of the molecules by

$p = \frac{1}{3}Nm\frac{{\overline {{c^2}} }}{V}\qquad \qquad(6)$

where N is the number of molecules in the container, m is the mass of each molecule, V is the volume of the container, and $\overline {{c^2}}$ is the average of the square of the molecular velocity.

$\overline {{c^2}} = \frac{1}{N}\sum\limits_{n = 1}^N {c_n^2} \qquad \qquad(7)$

The average of the square of the molecules’ velocity is related to its three components by

$\overline {{u^2}} = \overline {{v^2}} = \overline {{w^2}} = \frac{1}{3}\overline {{c^2}}\qquad \qquad (8)$

The average kinetic energy of a molecule is defined as

$\bar E = \frac{1}{2}m{\bar c^2}\qquad \qquad(9)$

Substituting eq. (9) into eq. (6) yields

$pV = \frac{2}{3}N\bar E\qquad \qquad(10)$

The monatomic ideal gas also satisfies the ideal gas law, i.e.,

$pV = n{R_u}T\qquad \qquad(11)$

where Ru = 8.3143kJ / kmol − K is the universal gas constant, which is the same for all gases. Combining eqs. (10) and (11) yields

$\bar E = \frac{3}{2}\frac{{{R_u}}}{{{N_A}}}T\qquad \qquad(12)$

where NA = N / n is the number of molecules per mole, which is a constant that equals $6.022 \times {10^{23}}$ and is referred to as Avogadro’s number. Equation (12) can also be rewritten as

$\bar E = \frac{3}{2}{k_b}T\qquad \qquad(13)$

where the Boltzmann constant is

${k_b} = \frac{{{R_u}}}{{{N_A}}} = \frac{{8.3143}}{{6.022 \times {{10}^{23}}}} = 1.38 \times {10^{ - 23}}{\rm{J/K}}\qquad \qquad(14)$

The specific heat at constant volume, cv, is given by kinetic theory as

${c_v} = \frac{3}{2}\frac{{{k_b}}}{m}\qquad \qquad(15)$

From eq. (12) it is evident that the average kinetic energy of molecules increases with increasing temperature. In other words, the molecules in a high-temperature gas have more kinetic energy than those in a low-temperature gas. When two objects at different temperatures come into contact, the higher-kinetic-energy molecules of the high-temperature object collide with the lower-kinetic-energy molecules of the low-temperature object. During these molecular collisions, some of the molecular kinetic energy of the high-temperature object is transferred to the molecules of the low-temperature object. Consequently, the molecules of the low-temperature object gain kinetic energy and its overall temperature increases. In the experiment conducted by Joule (see Fig. 1), the paddle wheel collides with the water molecules and kinetic energy is transferred from the wheel to the water molecules, causing the water temperature to rise. Another important concept that can be illustrated using kinetic theory is internal energy, E, defined as the sum total of all the energy of all the molecules in an object. The internal energy of an ideal gas equals the sum of all the kinetic energies of all its atoms. This sum can be expressed as the total number of molecules, N, times the average kinetic energy per atom, i.e.,

$E = N\bar E = \frac{3}{2}N{k_b}T \qquad \qquad (16)$

which shows that the internal energy of an ideal gas is only a function of mole number and temperature. Internal energy is also sometimes called thermal energy. It is very important to distinguish between temperature, internal energy, and heat. Temperature is related to the average kinetic energy of individual molecules [see eq. (12)], while internal, or thermal, energy is the total energy of all of the molecules in the object [see eq. (16)]. If two objects with equal mass of the same material and the same temperature are joined together, the temperature of the combined objects remains the same, but the internal energy of the system is doubled. Heat is a transfer of thermal energy from one object to another object at lower temperature. The direction of heat transfer between two objects depends solely on relative temperature, not on the amount of internal energy contained within each object. Similarly, viscosity, μ, thermal conductivity, k, and mass self-diffusion coefficient, D11, can be obtained using simple kinetic theory. Following are the results:

$\mu = \frac{2}{{3{\pi ^{3/2}}}}\frac{{\sqrt {m{k_b}T} }}{{{\sigma ^2}}}\qquad \qquad(17)$

$k = \frac{1}{{{\pi ^{3/2}}{\sigma ^2}}}\sqrt {\frac{{k_b^3T}}{m}} \qquad \qquad(18)$

${D_{11}} = \frac{2}{{3{\pi ^{3/2}}{\sigma ^2}P}}\sqrt {\frac{{k_b^3{T^3}}}{m}} \qquad \qquad(19)$

Equations (17) – (19) can be used for binary systems with components 1 and 2, if σ and m are replaced by 1 + σ2) / 2 and m1m2 / (m1 + m2), respectively. The significance of the above results should not be overlooked even though some simplified assumptions were used in their developments. Equations (17) and (18) for μ and k are independent of pressure for a gas. This is proven experimentally for pressure up to 10 atmospheric pressures. According to this prediction, viscosity and thermal conductivity are proportional to 1/2 power of absolute temperature while the diffusion coefficient is proportional to 3/2 power of absolute temperature. To better model the temperature effects, one needs to replace the rigid sphere model and the mean free path concepts and use the Boltzmann equation to describe the nonequilibrium phenomena accordingly. It is important to point out that equations described above using simple kinetic theory are valid only for an ideal monatomic gas. For ideal gas molecules containing more than one atom, the molecules can rotate and the different atoms

Figure 1 Modes of molecular kinetic energy: (a) rotational energy, (b) vibrational energy.

in the molecule can vibrate around their equilibrium position (see Fig. 1). Therefore, the kinetic energy of molecules with more than one atom must include both rotational and vibrational energy, and their internal energy at a given temperature will be greater than that of a monatomic gas at the same temperature. The internal energy of an ideal gas with a molecule containing more than one atom still depends solely on mole number and temperature. The internal energy of a real gas is a function of both temperature and pressure, which is a more complex condition than that of an ideal gas. The internal energies of liquids and solids are much more complicated, because the interactive forces between atoms and molecules also contribute to their internal energy. As noted above the simple kinetic theory of ideal gas was based on the mean free path concept. While it provides the first order of magnitude approximation for several key transport phenomena properties, the simple kinetic theory is limited to local equilibrium and therefore it is for time durations much larger than relaxation time. The advanced kinetic theory is based on the Boltzmann transport equation, which is presented in the next section.

## Intermolecular Forces and Boltzmann Transport Equation

A keen understanding of intermolecular forces is imperative for discussing the different phases of matter. In general, the intermolecular forces of a solid are greater than those of a liquid. This trend can be observed when looking at the force it takes to separate a solid as compared to that required to separate a liquid. Also, the molecules in a solid are much more confined to their position in the solid’s structure as compared to the molecules of a liquid, thereby affecting their ability to move. Most solids and liquids are deemed incompressible. The underlying reason for their “incompressibility” is that the molecules repel each other when they are forced closer than their normal spacing; the closer they become, the greater the repelling force (Tien and Lienhard, 1979). A gas differs from both a solid and a liquid in that its kinetic energy is great enough to overcome the intermolecular forces, causing the molecules to separate without restraint. The intermolecular forces in a gas decrease as the distance between the molecules increases. Both gravitational and electrical forces contribute to intermolecular forces; for many solids and liquids, the electrical forces are on the order of 1029 times greater than the gravitational force. Therefore, the gravitational forces are typically ignored. To quantify the intermolecular forces, a potential function φ(r) is defined as the energy required to bring two molecules, which are initially separated by an infinite distance, to a finite separation distance r. The form of the function always depends on the nature of the forces between molecules, which can be either repulsive or attractive depending on intermolecular spacing. When the molecules are close together, a repulsive electrical force is dominant. The repulsive force is due to interference of the electron orbits between two molecules, and it increases rapidly as the distance between two molecules decreases. When the molecules are not very close to each other, the forces acting between molecules are attractive in nature and generally fall into one of three categories. The first category is electrostatic forces, which occur between molecules that have a finite dipole moment, such as water or alcohol. The second category is induction forces, which occur when a permanently-charged particle or dipole induces a dipole in a nearby neutral molecule. The third category is dispersion forces, which are caused by transient dipoles in nominally-neutral molecules or atoms. An accurate representation of the intermolecular potential function φ(r) should account for all of the forces discussed above. It should be able to reflect repulsive forces for small spacing and attractive forces in the intermediate distance. When the distance between molecules is very large, there should be no intermolecular forces. While the exact form of φ(r) is not known, the following Lennard-Jones 6-12 potential function provides a satisfactory empirical expression for nonpolar molecules:

$\phi (r) = 4\varepsilon \left[ {{{\left( {\frac{{{r_0}}}{r}} \right)}^{12}} - {{\left( {\frac{{{r_0}}}{r}} \right)}^6}} \right]\qquad \qquad(20)$

where $\varepsilon$ is a constant and r0 is a characteristic length. Both of them depend on the type of the molecules. Figure 2 shows the Lennard-Jones 6-12 potential as a function of distance between two molecules. When the distance between molecules is small, the Lennard-Jones potential decreases with increasing distance until a point is reached after which the repulsive force dominates, and it is necessary to add

Figure 2 Lennard-Jones 6-12 potential vs. distance between two spherical, nonpolar molecules.

energy to the system in order to bring the molecules any closer. As the molecules separate, there is a distance, rmin, at which the Lennard-Jones potential becomes minimum. As the molecules move further apart, the Lennard-Jones potential increases with increasing distance between molecules and the attractive force dominates. The Lennard-Jones potential approaches zero when the molecular distance becomes very large. When the Lennard-Jones potential is at minimum, the following condition is satisfied:

$\frac{{d\phi ({r_{\min }})}}{{dr}} = 0\qquad \qquad(21)$

Substituting eq. (20) into eq. (21), one obtains,

${r_{\min }} = {2^{1/6}}{r_0} \approx 1.12{r_0}\qquad \qquad(22)$

For typical gas molecules, r0 ranges from 0.25 to 0.4 nm, which result a range from 0.28 to 0.45 nm for rmin. The Lennard-Jones potential at this point is

$\phi ({r_{\min }}) = - \varepsilon \qquad \qquad(23)$

When looking at the three phases of matter in the context of Fig. 1.5, some relationships can be described. For the solid state, the atoms are limited to vibrating about the equilibrium position, because they do not have enough energy to overcome the attractive force. The molecules in a liquid are free to move because they have a higher level of vibrational energy, but they have approximately the same molecular distance as the molecules of a solid. The energy required to overcome the attractive forces in a solid, thus allowing the molecules to move freely, corresponds to the latent heat of fusion. In the gaseous phase, on the other hand, the molecules are so far apart that they are virtually unaffected by intermolecular forces. The energy required to create vapor by separating closely-spaced molecules in a liquid corresponds to the latent heat of vaporization. Simulation of phase change at the molecular level is not necessary for many applications in macro spatial and time scales. For heat transfer at micro spatial and time scales, the continuum transport model breaks down and simulation at the molecular level becomes necessary. One example that requires molecular dynamics simulation is heat transfer and phase change during ultrashort pulsed laser materials processing (Wang and Xu, 2002). This process is very complex because it involves extremely high rates of heating (on the order of 1016 K/s) and high temperature gradients (on the order of 1011 K/m). The motion of each molecule (i) in the system is described by Newton’s second law, i.e.,

$\sum\limits_{j = 1(j \ne i)}^N {{{\mathbf{F}}_{ij}}} = {m_i}\frac{{{d^2}{{\mathbf{r}}_i}}}{{d{t^2}}}\qquad \qquad(24)$, i = 1,2,3...n

where mi and ${{\mathbf{r}}_i}$ are the mass and position of the ith molecule in the system. In arriving at eq. (24), it is assumed that the molecules are monatomic and have only three degrees of freedom of motion. For molecules with more than one atom, it is also necessary to consider the effect of rotation. The Lennard-Jones potential between the ith and jth molecules is obtained by

${\phi _{ij}} = 4\varepsilon \left[ {{{\left( {\frac{{{r_0}}}{{{r_{ij}}}}} \right)}^{12}} - {{\left( {\frac{{{r_0}}}{{{r_{ij}}}}} \right)}^6}} \right]\qquad \qquad(25)$

The force between the ith and jth molecules can be obtained from

${{\mathbf{F}}_{ij}} = - \nabla {\phi _{ij}} = \frac{{24\varepsilon }}{{{r_o}}}\left[ {{{\left( {\frac{{{r_o}}}{{{r_{ij}}}}} \right)}^{13}} - {{\left( {\frac{{{r_o}}}{{{r_{ij}}}}} \right)}^7}} \right]\frac{{{{\mathbf{r}}_{ij}}}}{{{r_{ij}}}}\qquad \qquad(26)$

where rij is the distance between the ith and jth molecules. The transport properties can be obtained by the kinetic theory in the form of very complicated multiple integrals that involve intermolecular forces. For a non-polar substance that Lennard-Jones potential is valid, these integrals can be evaluated numerically. For a pure gas, the self-diffusivity, D, viscosity, μ, and thermal conductivity, k, are (Bird et al., 2002)

$D = \frac{3}{8}\frac{{\sqrt {\pi m{k_b}T} }}{{\pi {\sigma ^2}{\Omega _D}}}\frac{1}{\rho }\qquad \qquad(27)$

$\mu = \frac{5}{{16}}\frac{{\sqrt {\pi m{k_b}T} }}{{\pi {\sigma ^2}{\Omega _\mu }}}\qquad \qquad(28)$

$k = \frac{{25}}{{32}}\frac{{\sqrt {\pi m{k_b}T} }}{{\pi {\sigma ^2}{\Omega _k}}}{\bar c_v}\qquad \qquad(29)$

where σ is collision diameter, and ${\bar c_v}$ in is the molar specific heat under constant volume. The dimensionless collision integrals are related by ${\Omega _\mu } = {\Omega _k} \approx 1.1{\Omega _D}$ and are slow varying functions of ${k_b}T/\varepsilon$ ($\varepsilon$ is a characteristic energy of molecular interaction). If all molecules can be assumed to be rigid balls, all collision integrals will become unity. It follows from eqs. (27) – (29) that the Prandtl and Schmidt numbers are, respectively, 0.66 and 0.75, which is a very good approximation for monatomic gases (e.g., helium in Table C.1). The transport properties at system length scales of less than 10λ will be different from the macroscopic properties, because the gas molecules are not free to move as they naturally would. While the transport properties discussed here are limited to low-density monatomic gases, the discussion can also be extended to polyatomic gases, and monatomic and polyatomic liquids. For a nonequilibrium system, the mean free path theory is no longer valid, and the Boltzmann equation should be used to describe the molecular velocity distribution in the system. For low-density nonreacting monatomic gas mixtures, the random molecular movement can be described by the molecular velocity distribution function ${f_i}({\mathbf{c}},{\mathbf{x}},t)$, where ${\mathbf{c}}$ is the particle velocity and ${\mathbf{x}}$ is the position vector in the mixture. At time t, the probable number of molecules of the ith species that are located in the volume element dx at position x and have velocity within the range $d{\mathbf{c}}$ about ${\mathbf{c}}$ is ${f_i}({\mathbf{c}},{\mathbf{x}},t)d{\mathbf{c}}d{\mathbf{x}}$. The evolution of the velocity distribution function with time can be described using the Boltzmann equation

$\frac{{D{f_i}}}{{Dt}} = \frac{{\partial {f_i}}}{{\partial t}} + {\mathbf{c}} \cdot {\nabla _{\mathbf{x}}}{f_i} + {\mathbf{a}} \cdot {\nabla _{\mathbf{c}}}{f_f} = {\Omega _i}(f)\qquad \qquad(30)$

where ${\nabla _{\mathbf{x}}}$ and ${\nabla _{\mathbf{c}}}$ are the $\nabla$ operator with respect to x and c, respectively (see Appendix G), a is the particle acceleration (m/s2), and Ωi is a five-fold integral term that accounts for the effect of molecular collision on the change of velocity distribution function fi. The Boltzmann equation can also be considered to be a continuity equation in a six-dimensional position-velocity space (${\mathbf{x}}{\rm{ and }}{\mathbf{c}}$). The velocity distribution function is related to the number density (number of particles per unit volume) by

$\int {f_i} (c, x, t) dc = {\mathfrak{N}}_i (x, t) \qquad \qquad (31)$

Note that the density is $\rho (x, t) = m \mathfrak{N} (x, t)$ where m is the mass of the particle. The total number of particles N inside the volume V as a function of time is

$N(t) = \int_V {\int_c {{\rm{f}}({\mathbf{c}},{\mathbf{x}},t)d{\mathbf{c}}} } d{\mathbf{x}}\qquad \qquad(32)$

In thermodynamic equilibrium, f is independent of time and space, i.e., $f({\mathbf{c}},{\mathbf{x}},t) = f({\mathbf{c}})$. It can be demonstrated that the stress tensor, heat flux, and diffusive mass flux can be obtained from the solution of the velocity distribution function fi. More detailed information about the Boltzmann equation and its applications related to transport phenomena in multiphase systems can be found in Chapter 3. Most macroscopic transport equations, such as Fourier’s law of conduction, Navier-Stokes equation for viscous flow, or the equation of radiative transfer for photons and phonons can be developed from the Boltzmann transport equation using local equilibrium assumptions. More detailed information concerning formulation and solution techniques of the Boltzmann equation can be found in Tien and Lienhard (1985) and Ceracignani (1988). In addition, the Boltzmann equation can also be used to describe transport of electrons and electron-lattice interaction and to yield the two-step heat conduction model for electron and lattice temperatures for nanoscale and microscale heat transfer (Qiu and Tien, 1993; Chen, 2004; Zhang and Chen, 2007; Zhang, 2007).

Cohesion is the intermolecular attractive force between molecules of the same kind or phase. For a solid, cohesion is significant only when the molecules are extremely close together; for instance, once a crack forms in a metal structure, the two edges of the crack will not rejoin even if pushed together. The rejoining can not occur because gas molecules attach to the fractured surface, preventing the cohesive intermolecular attraction from occurring. The fundamental basis for viscosity observed in fluids is cohesion within the fluids. Viscosity is the resistance of a liquid or a gas to shear forces; it can be measured as a ratio of shear stress to shear strain. As a significant factor in the analysis of fluid flows, viscosity depends on temperature. Generally, as temperature increases the viscosity of a gas increases, while that of a liquid decreases. Adhesion is the intermolecular attractive force between molecules of a different kind or phase. An example of adhesion is the phenomenon of water wetting a glass surface. Intermolecular forces between the water and the glass cause the wetting. In this case, the adhesive force between the water and the glass is greater than the cohesive forces within the water. The opposite case can also occur, where the liquid is repelled from the surface, indicating that cohesion in the liquid is greater than adhesion between the liquid and the solid. For example, when a freshly waxed car sits in the rain, the raindrops bead on the surface and then easily flow off.

## Enthalpy and Energy

Phase change processes are always accompanied by a change of enthalpy, which we will now consider at the molecular level. Phase change phenomena can be viewed as the destruction or formation of intermolecular bonds as the result of changes in intermolecular forces. The intermolecular forces between the molecules in a solid are greater than those between molecules in a liquid, which are in turn greater than those between molecules in a gas. This reflects the greater distance between molecules in a gas than in a liquid, and the greater distance between molecules in a liquid than in a solid. As a result, the intermolecular bonds in a solid are stronger than those in a liquid. In a gas, which has the weakest intermolecular forces of all three phases, intermolecular bonds do not exist between the widely separated molecules.

Table 2 Energies of the H2O molecule in the vicinity of 273 K

 Types of energy Approximate magnitude per molecule (eV) Lattice vibration 0.0054 Intermolecular hydrogen bond breaking 0.58 Enthalpy of melting 0.06 Enthalpy of vaporization 0.39 Enthalpy of sublimation 0.49

When the intermolecular bonds between the molecules in a solid are completely broken, sublimation occurs. The approximate energy levels identified for the H2O molecule near 273 K are summarized in Table 2. Since it takes 0.29 eV (electron-volts) to break a hydrogen bond in the ice lattice, and there are two bonds per H2O molecule, the energy required to completely free a H2O molecule from its neighbor should be 0.58 eV. The energy required to break two hydrogen bonds, 0.58 eV, should be of the same order of magnitude as the enthalpy of sublimation, which is 0.49 eV, as shown in Table 1.4. The energy required to melt ice is 0.06 eV per molecule, while it takes 0.39 eV per molecule to vaporize liquid water. The difference between the enthalpy of melting and vaporization at the molecular level explains the difference in the latent heats of fusion and of vaporization. The internal energy of a substance with molecules containing more than one atom (such as H2O) is the sum of the kinetic, rotational, and vibrational energies. Since the molecules in a solid are held in a fixed pattern and are not free to move or rotate, the lattice vibrational energy is the primary contributor to the internal energy of the ice. As can be seen from Table 1.4, the enthalpy of melting and vaporization are much larger than the lattice vibrational energy, which explains why the latent heat is usually much greater than the sensible heat. Different phases are characterized by their bond energy and their molecular configurations. For example, the intermolecular bonds in a solid are very strong, and thus able to hold the molecules in a fixed pattern. The intermolecular bonds in a liquid are strong enough to hold the molecules together but not strong enough to hold them in a fixed pattern. The intermolecular bonds in a gas are completely broken and the molecules can move freely. Therefore, phase change can be viewed as conversion from one type of intermolecular ordering to another, i.e., a reordering process. Sublimation, melting, and vaporization are all processes that increase the randomness of the system and therefore produce increases of entropy. Since phase changes occur at constant temperature, one can express the increases of entropy in these processes as:

$\Delta s = \frac{{\Delta h}}{T} = {\mathop{\rm constant}} \qquad \qquad(33)$

where Δh is the change of enthalpy during phase change, i.e., the latent heat, and T is the phase change temperature. The constant in eq. (33) depends on the particular phase change process but is independent of the substance. For vaporization and condensation, Trouton’s rule is applicable

${s_v} - {s_\ell } = \frac{{{h_{\ell v}}}}{{{T_{sat}}}} \simeq 83.7{\rm{J/(mol - K)}}\qquad \qquad(34)$

while Richards’ rule is valid for melting and solidification

${s_\ell } - {s_s} = \frac{{{h_{s\ell }}}}{{{T_m}}} \simeq 8.37{\rm{J/(mol - K)}}\qquad \qquad(35)$

$\mathfrak{N} = \frac{p}{{k_b}T} = {1.013 \times {{10}^5} Pa}{(1.381 \times {{10}^{-23}} J/K) \times (298.15K)} = 2.46 \times {{10}^{25}} molecules/{m^3}$

The mean distance between molecules is

$\overline{L} ={ \mathfrak{N}}^{-1/3} = 3.4 \times {{10}^{-9}}m = 3.4nm$

The diameter of molecules, according to Table 1.3, is

$\sigma = 0.366{\rm{nm}} = 3.7 \times {10^{ - 10}}{\rm{m}}$

The mean free path can be obtained from eq. (17), i.e.

$\lambda = \frac{1}{\sqrt{2} \pi {{\sigma}^2} \mathfrak{N}} = \frac{1}{\sqrt{2} \pi \times {{(3.7 \times {{10}^{-10}})}^2} \times 2.46 \times {{10}^{25}}}= 66 \times {{10}^{-9}}m = 66nm$

The average magnitude of the molecular velocity can be obtained from eq. (1), i.e.,

$\bar c = \sqrt {\frac{{8{k_b}T}}{{\pi m}}} = \sqrt {\frac{{8 \times 1.381 \times {{10}^{ - 23}} \times 298.15}}{{\pi \times 28.97 \times 1.66{\rm{ }} \times {\rm{ }}{{10}^{ - 27}}}}} = 465{\rm{m/s}}$

where the molecular mass of air is 28.97 atomic mass units (each atomic mass unit is $1.66{\rm{ }} \times {\rm{ }}{10^{ - 27}}$ kg). The relaxation time is

$\tau = \frac{\lambda }{{\bar c}} = 0.14{\rm{ns}}$

The speed of sound in air is

${V_{sound}} = \sqrt {\gamma {R_g}T} = 345{\rm{m/s}}$

where γ = cp / cv = 1.4 for air. The speed of sound is less than the molecular velocity of 465m/s. The number of collisions for each molecule per second is

τ − 1 = 7billion / s

Since the average speed of molecules is higher than the speed of sound, and the number of collisions that each molecules experiences is high, it does not take very long for the nose to detect odor.

## References

Faghri, A., and Zhang, Y., 2006, Transport Phenomena in Multiphase Systems, Elsevier, Burlington, MA.

Tien, C.L., Lienhard, J.H., 1985, Statistical Thermodynamics, Revised Sub. Edition, Hemisphere Publishing Corp., New York, NY.