Ambedkar Nagar

Prime Classes for IIT-JEE/PMT, Ambedkar nagar

Collision frequency and mean free path

MoleculeWall Collisions:

Lets call Zw the rate of collisions of gas molecules with a section of wall of area A.

I. Zw should be proportional to the area A II. Zw should be proportional to the average molecular speed, vavg

III. Zw should be proportional to the number of mole, n/V

The proportionality constant can be calculated from a complete analysis of the directions from which molecules impinge on the wall; it turns out to have the value 1/4 . So the wall collision rate is

= 14 Collision rate per unit area =

= 1

4

= 14 8 = 2

Diffusion

Diffusion is the process by which the molecules of different substances mingle with each other. The atoms of two solids diffuse into each other when the two solids are in contact, but the process is very slow. The diffusion of a solid through a liquid solvent is much faster but mixing normally needs to be encouraged by stirring or shaking the solid in the liquid (the process is then no longer pure diffusion). Gaseous diffusion is much faster.

Effusion

Effusion is the process in which a gas escapes through a small hole into vacuum. This occurs if the diameter of the hole is considerably smaller than the mean free path of the molecules.

Rate of Effusion

= =

=

Grahams law of effusion

The rate of effusion of a gas through a small hole into a vacuum is inversely proportional to the square root of its molar mass. Assuming that different gases are studied at the same temperature and pressure, their number density, N/V, is the same, and the rate of effusion of each gas depends only on the factor 1/ , exactly as observed by Graham.

Ambedkar Nagar

Prime Classes for IIT-JEE/PMT, Ambedkar nagar

Molecule Molecule Collisions

Length of collision tube in time = The Volume Of Collision Tube = = = If N/V is the number of molecules per unit volume in the gas (the number density of the gas), then the number of collisions per second experienced by the moving molecule is

. = =

() = . = = It can be proved that = so = Number of collision in Unit volume () Now imagine that all of the gaseous molecules in the cylinder are moving . When you count all of the collisions for every gaseous molecule moving within the cylinder in a sec, you get Zii. The relation was found to be:

Collision Cross Section: d2 A molecule will hit another molecule if the centre of the former lies within a circle of radius d. The collision cross-section is the target area, d2

An average molecule sweeps out a cylinder of volume d2 in 1 second. It will collide with any molecules whose centers lie within the cylinder. Using this construction, we can calculate the rate of collisions with other molecules.

= = (The accounts for double counting of collisions).Mean Free Path

The mean free path is the average distance travelled by a moving gaseous molecule, between successive impacts (collisions).

Z1 is the rate at which a particular molecule collides with other molecules. Its inverse, Z1-1 , therefore measures the average time between collisions. During this interval, a molecule travels an average distance vavg /Z1which is called the mean free path, .

=

=

= =

Because 1/p, we see that the mean free path decreases as the pressure increases.