Kinetic theory Totally Explained

Everything about Kinetic Theory totally explained

Kinetic theory (or kinetic theory of gases) attempts to explain 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. Kinetic theory is also known as the kinetic-molecular theory or the collision theory.

History

In 1740 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 wasn't immediately accepted, in part because conservation of energy hadn't yet been established, and it wasn't 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 Dani Bernhardt (1747), Georges-Louis Le Sage (ca. 1780, published 1818), John Herapath (1816)
   and John James Waterston (1843),
   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.
   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.
   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. In his 1875 thirteen page article 'Molecules', published in the September issue of Nature, 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.”
   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)
   and Marian Smoluchowski's (1906)
   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, each of which has a mass or weight in SI units, kilograms.
The number of molecules is large such that statistical treatment can be applied.
These molecules are in constant, random motion. The rapidly moving particles constantly collide with each other and with the walls of the container.
The collisions of gas particles with the walls of the container holding them are perfectly elastic.
The interactions among molecules are negligible. They exert no forces on one another except during collisions.
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 relatively 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 can be treated as classical objects.
The time during collision of molecule with the container's wall is negligible as comparable to the time between successive collisions.
The equations of motion of the molecules are time-reversible. In addition, if the gas is in a container, the collisions with the walls are assumed to be instantaneous and elastic.

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's an alternative approach developed by Grad based on moment expansions. 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.
The kinetic theory has also been extended to include inelastic collisions in granular matter by Jenkins and others.

Pressure

Pressure is explained by kinetic theory as arising from the force exerted by gas molecules impacting on the walls of the container. Consider a gas of N molecules, each of mass m, enclosed in a cuboidal container of volume V. 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:
ť

Delta p_x = p_i - p_f = 2 m v_x,

where v_x is the x-component of the initial velocity of the particle.
The particle impacts the wall once every 2l/v_x time units (where l is the length of the container). Although the particle impacts a side wall once every 1l/v_x time units, only the momentum change on one wall is considered so that the particle produces a momentum change on a particular wall once every 2l/v_x time units. ť

Delta t = frac

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).

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