Kinetic theory: 07/06/11

Kinetic theory

From Wikipedia, the free encyclopedia

The temperature of an ideal monatomic gas is a measure of the average kinetic energy of its atoms. The size of helium atoms relative to their spacing is shown to scale under 1950 atmospheres of pressure. The atoms have a certain, average speed, slowed down here two trillion fold from room temperature.

This article applies to gases; see also Kinetic theory of solids

The kinetic theory of gases describes a gas as a large number of small particles (atoms or molecules), all of which are in constant, random motion. The rapidly moving particles constantly collide with each other and with the walls of the container. Kinetic theory explains macroscopic properties of gases, such as pressure, temperature, or volume, by considering their molecular composition and motion. Essentially, the theory posits that pressure is due not to static repulsion between molecules, as was Isaac Newton's conjecture, but due to collisions between molecules moving at different velocities.
While the particles making up a gas are too small to be visible, the jittering motion of pollen grains or dust particles which can be seen under a microscope, known as Brownian motion, results directly from collisions between the particle and gas molecules. As pointed out by Albert Einstein in 1905, this experimental evidence for kinetic theory is generally seen as having confirmed the existence of atoms and molecules.

History

In 1738 Daniel Bernoulli published Hydrodynamica, which laid the basis for the kinetic theory of gases. In this work, Bernoulli positioned the argument, still used to this day, that gases consist of great numbers of molecules moving in all directions, that their impact on a surface causes the gas pressure that we feel, and that what we experience as heat is simply the kinetic energy of their motion. The theory was not immediately accepted, in part because conservation of energy had not yet been established, and it was not obvious to physicists how the collisions between molecules could be perfectly elastic.
Other pioneers of the kinetic theory (which were neglected by their contemporaries) were Mikhail Lomonosov (1747),^[3] Georges-Louis Le Sage (ca. 1780, published 1818),^[4] John Herapath (1816)^[5] and John James Waterston (1843),^[6] which connected their research with the development of mechanical explanations of gravitation. In 1856 August Krönig (probably after reading a paper of Waterston) created a simple gas-kinetic model, which only considered the translational motion of the particles.^[7]
In 1857 Rudolf Clausius, according to his own words independently of Krönig, developed a similar, but much more sophisticated version of the theory which included translational and contrary to Krönig also rotational and vibrational molecular motions. In this same work he introduced the concept of mean free path of a particle. ^[8] In 1859, after reading a paper by Clausius, James Clerk Maxwell formulated the Maxwell distribution of molecular velocities, which gave the proportion of molecules having a certain velocity in a specific range. This was the first-ever statistical law in physics.^[9] In his 1873 thirteen page article 'Molecules', Maxwell states: “we are told that an 'atom' is a material point, invested and surrounded by 'potential forces' and that when 'flying molecules' strike against a solid body in constant succession it causes what is called pressure of air and other gases.”^[10] In 1871, Ludwig Boltzmann generalized Maxwell's achievement and formulated the Maxwell–Boltzmann distribution. Also the logarithmic connection between entropy and probability was first stated by him.
In the beginning of twentieth century, however, atoms were considered by many physicists to be purely hypothetical constructs, rather than real objects. An important turning point was Albert Einstein's (1905)^[11] and Marian Smoluchowski's (1906)^[12] papers on Brownian motion, which succeeded in making certain accurate quantitative predictions based on the kinetic theory.

Postulates

The theory for ideal gases makes the following assumptions:

The gas consists of very small particles, all with non-zero mass.
The number of molecules is so large that statistical treatment can be applied.
These molecules are in constant, random motion. The rapidly moving particles constantly collide with the walls of the container.
The collisions of gas particles with the walls of the container holding them are perfectly elastic.
Except during collisions, the interactions among molecules are negligible (they exert no forces on one another).
The total volume of the individual gas molecules added up is negligible compared to the volume of the container. This is equivalent to stating that the average distance separating the gas particles is large compared to their size.
The molecules are perfectly spherical in shape, and elastic in nature.
The average kinetic energy of the gas particles depends only on the temperature of the system.
Relativistic effects are negligible.
Quantum-mechanical effects are negligible. This means that the inter-particle distance is much larger than the thermal de Broglie wavelength and the molecules are treated as classical objects.
The time during collision of molecule with the container's wall is negligible as compared to the time between successive collisions.
The equations of motion of the molecules are time-reversible.

More modern developments relax these assumptions and are based on the Boltzmann equation. These can accurately describe the properties of dense gases, because they include the volume of the molecules. The necessary assumptions are the absence of quantum effects, molecular chaos and small gradients in bulk properties. Expansions to higher orders in the density are known as virial expansions. The definitive work is the book by Chapman and Enskog but there have been many modern developments and there is an alternative approach developed by Grad based on moment expansions.^{[citation needed]} In the other limit, for extremely rarefied gases, the gradients in bulk properties are not small compared to the mean free paths. This is known as the Knudsen regime and expansions can be performed in the Knudsen number.

Properties

Pressure and Kinetic Energy

Pressure is explained by kinetic theory as arising from the force exerted by molecules or atoms impacting on the walls of a container. Consider a gas of N molecules, each of mass m, enclosed in a cuboidal container of volume V=L³. When a gas molecule collides with the wall of the container perpendicular to the x coordinate axis and bounces off in the opposite direction with the same speed (an elastic collision), then the momentum lost by the particle and gained by the wall is:

$\Delta p = p_{i,x} - p_{f,x} = 2 m v_x\,$

where v_x is the x-component of the initial velocity of the particle.
The particle impacts one specific side wall once every

$\Delta t = \frac{2L}{v_x}$

(where L is the distance between opposite walls).
The force due to this particle is:

$F = \frac{\Delta p}{\Delta t} = \frac{m v_x^2}{L}.$

The total force on the wall is

$F = \frac{Nm \overline{v_x^2}}{L}$

where the bar denotes an average over the N particles. Since the assumption of molecular chaos imposes $\overline{v_x^2} = \overline{v^2}/3$ , we can rewrite the force as

$F = \frac{Nm\overline{v^2}}{3L}.$

This force is exerted on an area L². Therefore the pressure of the gas is

$P = \frac{F}{L^2} = \frac{Nm\overline{v^2}}{3V}$

where V=L³ is the volume of the box. The fraction n=N/V is the number density of the gas (the mass density ρ=nm is less convenient for theoretical derivations on atomic level). Using n, we can rewrite the pressure as

$P = \frac{n m \overline{v^2}}{3}.$

This is a first non-trivial result of the kinetic theory because it relates pressure, a macroscopic property, to the average (translational) kinetic energy per molecule ${1 \over 2} m\overline{v^2}$ which is a microscopic property.

Temperature and kinetic energy

From the ideal gas law

$\displaystyle PV = N k_B T$

(1)

where $\displaystyle k_B$ is the Boltzmann constant, and $\displaystyle T$ the absolute temperature,
and from the above result $P = {Nmv^2\over 3V}$ we have $PV = {Nmv_{rms}^2 \over 3}$
we have $\displaystyle N k_B T = \frac {N m v_{rms}^2} {3}$
then the temperature $\displaystyle T$ takes the form

$\displaystyle T = \frac {m v_{rms}^2} {3 k_B}$

(2)

which leads to the expression of the kinetic energy of a molecule

$\displaystyle \frac {1} {2} mv_{rms}^2 = \frac {3} {2} k_B T.$

The kinetic energy of the system is N times that of a molecule $K= \frac {1} {2} N m v_{rms}^2$
The temperature becomes

$\displaystyle T = \frac {2} {3} \frac {K} {N k_B}.$

(3)

Eq.(3)₁ is one important result of the kinetic theory: The average molecular kinetic energy is proportional to the absolute temperature. From Eq.(1) and Eq.(3)₁, we have

$\displaystyle PV = \frac {2} {3} K.$

(4)

Thus, the product of pressure and volume per mole is proportional to the average (translational) molecular kinetic energy.
Eq.(1) and Eq.(4) are called the "classical results", which could also be derived from statistical mechanics; for more details, see .^[1]
Since there are $\displaystyle 3N$ degrees of freedom in a monoatomic-gas system with $\displaystyle N$ particles, the kinetic energy per degree of freedom is

$\displaystyle \frac {K} {3 N} = \frac {k_B T} {2}.$

(5)

In the kinetic energy per degree of freedom, the constant of proportionality of temperature is 1/2 times Boltzmann constant. In addition to this, the temperature will decrease when the pressure drops to a certain point. This result is related to the equipartition theorem.
As noted in the article on heat capacity, diatomic gases should have 7 degrees of freedom, but the lighter gases act as if they have only 5.
Thus the kinetic energy per kelvin (monatomic ideal gas) is:

per mole: 12.47 J
per molecule: 20.7 yJ = 129 μeV.

At standard temperature (273.15 K), we get:

per mole: 3406 J
per molecule: 5.65 zJ = 35.2 meV.

Collisions with container

One can calculate the number of atomic or molecular collisions with a wall of a container per unit area per unit time.
Assuming an ideal gas, a derivation^[2] results in an equation for total number of collisions per unit time per area:

$A = \frac{1}{4}\frac{N}{V} v_{avg} = \frac{n}{4} \sqrt{\frac{8 k_{B} T}{\pi m}} . \,$

Speed of molecules

From the kinetic energy formula it can be shown that

$v_{rms}^2 = \frac{3RT}{\mbox{molar mass}}$

with v in m/s, T in kelvins, and R is the gas constant. The molar mass is given as kg/mol. The most probable speed is 81.6% of the rms speed, and the mean speeds 92.1% (distribution of speeds).

Wednesday 6 July 2011

Kinetic theory

Kinetic theory

History

History

Postulates

Postulates

Properties

Properties

Pressure and Kinetic Energy

Temperature and kinetic energy

Temperature and kinetic energy

Collisions with container

Collisions with container

Speed of molecules