Kinetic theory

From Thermal-FluidsPedia

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 1000^1/3=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^-7m, 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 $k b$ 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

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

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

${\dot m''_{molecules}} = \frac{{\mathfrak{N}\bar cm}}{4} = \mathfrak{N}m{\left( {\frac{{{k_b}T}}{{2\pi m}}} \right)^{1/2}} = \mathfrak{N}{\left( {\frac{{{k_b}Tm}}{{2\pi }}} \right)^{1/2}}\qquad \qquad(5)$

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

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

Kinetic properties of gases at 25 °C and atmospheric pressure

Gas	σ (m)	$\lambda \times {10^8}{\rm{ (m)}}$	${\bar c_m}$ (m/s)	$τ$ (ps)
Air	3.66	6.91	467	148
Ar	3.58	7.22	397	182
CH₄	3.82	5.25	627	84
C₂H₄	4.52	3.74	474	78
C₆H₆	5.27	16.2	284	569
CO₂	4.53	4.51	379	119
C_l2	4.40	2.99	298	100
H₂	2.71	12.6	1769	71
He	2.15	20.0	1256	159
N₂	3.70	6.76	475	142
NH₃	4.32	4.97	609	82
O₂	3.55	7.36	444	166
SO₂	4.29	2.99	313	96

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 $R u = 8.3143kJ / kmol - K$ 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 $N A = N / n$ 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,

c v

, 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, $D 11$ , can be obtained using simple kinetic theory. Following are the results:

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

$\mu = \frac{2}{{3{\pi ^{3/2}}}}\frac{{\sqrt {m{k_b}T} }}{{{\sigma ^2}}}\qquad \qquad(17)$ $k = \frac{1}{{{\pi ^{3/2}}{\sigma ^2}}}\sqrt {\frac{{k_b^3T}}{m}}\qquad \qquad(18)$ ${D_{11}} = \frac{2}{{3{\pi ^{3/2}}{\sigma ^2}P}}\sqrt {\frac{{k_b^3{T^3}}}{m}}\qquad \qquad(19)$

Equations (17) – (19) can be used for binary systems with components 1 and 2, if σ and $m$ are replaced by $(σ 1 + σ 2) / 2$ and $m 1 m 2 / (m 1 + m 2)$ , 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 in the molecule can vibrate around their equilibrium position (see figure on the right). 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.

References

Berry, R.S., Rice S.A., and Ross J., 2000, Physical Chemistry, 2^nd edition, Oxford University Press.

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

Faghri, A., Zhang, Y., and Howell, J. R., 2010, Advanced Heat and Mass Transfer, Global Digital Press, Columbia, MO.

Tien, C.L., and Lienhard, J.H., 1979, Statistical Thermodynamics, Hemisphere, Washington, D.C.

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Kinetic theory

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