Materials Science and Engineering/Derivations/Models of Micro and Nanoscale Processing/Kinetic Theory of Gases

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 kinetic-molecular theory or collision theory.

 $f(v){\frac {dC/dv}{C}}=4\pi v^{2}\left({\frac {M}{2\pi RT}}\right)^{3/2}\exp \left({\frac {-{\frac {1}{2}}Mv^{2}}{RT}}\right)$

• $f(v)$= fraction, $dC/C$, of the number of molecules between velocity $v$ and $dv+v$
• $M$ = mass per mole
• $N_{av}$ = Avogadro's number
• $R$ = Gas constant
• $k_{B}$ = Boltzman constant

The theory for ideal gases makes the following assumptions:

• The gas consists of very small particles, each of which has a mass.
• 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.

The Langmuir Equation

The Langmuir equation or Langmuir isotherm or Langmuir adsorption equation relates the coverage or adsorption of molecules on a solid surface to gas pressure or concentration of a medium above the solid surface at a fixed temperature. The equation was developed by Irving Langmuir in 1916.

 $\Phi ={\frac {1}{2}}C{\bar {v}}_{x}$
 $\Phi ={\frac {1}{2}}C\left({\frac {\int _{0}^{\infty }v_{x}f(v_{x})dv_{x}}{\int _{0}^{\infty }f(v_{x})dv_{x}}}\right)$
 $\Phi ={\frac {1}{2}}C\left({\frac {2RT}{\pi M}}\right)^{1/2}$
 $\Phi ={\frac {1}{4}}c{\bar {v}}$