KINETIC MOLECULAR THEORY OF GASES

It is based on the following assumption;

The gas consists of a vast number of small particles; it is known as a molecule. Molecules are tiny, so the actual volume is small to the total space occupied by the gas.
Molecules are in constant fast motion in all possible directions, and colliding with the vessel’s walls is also random.
The kinetic energy is transferred from one molecule to other but not converted to the other form of energy like heat.
No attractive force between molecules and molecules of the walls of the vessels; Therefore, molecules move independently from each other.
Newton’s second law of motion applies to gaseous molecules’ motion.

The above assumption is explained for ideal gas only and is approximately valid for real gas.

PRESSURE OF AN IDEAL GAS

Above postulate, by applying the laws of classical mechanics to derive the pressure of a gas,

Here, N molecule of gas has m mass and closed in a cubic box with V volume, and each side of cube length l. Molecule motion is totally random.

One molecule of the gas has velocity C. This velocity is divided between the three compartments, u, v, and w, along with the axis x, y, and z.

The velocity of components is perpendicular to the walls of the container. It is explained by,

\[\displaystyle {{C}^{2}}={{u}^{2}}+{{v}^{2}}+{{w}^{2}}---(1)\]

The motion of one molecule along the x-axis is a collision with the wall, which is perpendicular to its motion. Sine collision is elastic, and the wall remains constant. The only sign of the velocity component changes. Resulting change of momentum in the x-direction (∆P_x)

\[\displaystyle \begin{array}{l}\Delta {{P}_{x}}=m\left\{ {u-\left( {-u} \right)} \right\}\\=2mu--------(2)\end{array}\]

After the collision, the molecule takes times = l/u to collide with the opposite wall. The frequency of collisions o two opposite walls by u/l and change momentum per unit is,

\[\displaystyle \begin{array}{l}\frac{{\Delta {{P}_{x}}}}{{\Delta t}}=2mu\times \frac{u}{l}\\=\frac{{2m{{u}^{2}}}}{l}-------(3)\end{array}\]

Total change in momentum of single molecule per unit from a collision on all the six walls,

\[\displaystyle \begin{array}{l}\frac{{\Delta {{P}_{x}}}}{{\Delta t}}=\frac{{2m{{u}^{2}}}}{l}+\frac{{2m{{v}^{2}}}}{l}+\frac{{2m{{w}^{2}}}}{l}\\=\left( {\frac{{2m}}{l}} \right)\left( {{{u}^{2}}+{{v}^{2}}+{{w}^{2}}} \right)\\=\frac{{2m{{c}^{2}}}}{l}-------(4)\end{array}\]

Total change in momentum per unit time for all N molecules was obtained by summing all molecule contributions.

\[\displaystyle \begin{array}{l}\frac{{\Delta {{P}_{{\text{total}}}}}}{{\Delta t}}=\sum\limits_{{i=1}}^{N}{{\frac{{2m{{c}^{2}}}}{l}}}\\=\frac{{2m}}{l}\sum\limits_{{i=1}}^{N}{{c_{i}^{2}}}----(5)\end{array}\]

Mean square velocity as

\[\displaystyle \left\langle {{{c}^{2}}} \right\rangle =\sum{{\frac{{c_{i}^{2}}}{N}}}----(6)\]

\[\displaystyle \frac{{\Delta {{P}_{{\text{total}}}}}}{{\Delta t}}=\frac{{2mN}}{l}\left\langle {{{c}^{2}}} \right\rangle ---(7)\]

Newton’s second law of motion, the rate of change of momentum is force (f) and per unit area pressure (P).
Face area (A) of a cubical vessel is 6l²and volume is v, pressure exerted by N molecules of gas on the walls of vessel is,

\[\displaystyle \begin{array}{l}P=\frac{f}{A}=\frac{{2mN\left\langle {{{c}^{2}}} \right\rangle }}{{l\left( {6{{l}^{2}}} \right)}}\\=\frac{1}{{3v}}mN\left\langle {{{c}^{2}}} \right\rangle -----(8)\end{array}\]

= root mean square velocity of a gaseous
molecule is .
Root mean square velocity denoted by C.
Equation (8) for the pressure of a gas,