Graham’s Law of Effusion & Intermolecular Collisions | Chemistry Optional Notes for UPSC PDF Download

Introduction

• An important consequence of the kinetic molecular theory is what it predicts in terms of effusion and diffusion effects. Effusion is defined as a loss of material across a boundary. A common example of effusion is the loss of gas inside of a balloon over time.
• The rate at which gases will effuse from a balloon is affected by a number of factors. But one of the most important is the frequency with which molecules collide with the interior surface of the balloon. Since this is a function of the average molecular speed, it has an inverse dependence on the square root of the molecular weight.
  Rate of effusion ∝ = 1/√MW
  This can be used to compare the relative rates of effusion for gases of different molar masses.

The Knudsen Cell Experiment

• A Knudsen cell is a chamber in which a thermalized sample of gas is kept, but allowed to effuse through a small orifice in the wall. The gas sample can be modeled using the Kinetic Molecular Theory model as a collection of particles traveling throughout the cell, colliding with one another and also with the wall. If a small orifice is present, any molecules that would collide with that portion of the wall will be lost through the orifice.

Figure 2.5.1: Effusion of gas particles through an orifice. (CC BY-SA 3.0; Astrang13).

This makes a convenient arrangement to measure the vapor pressure of the material inside the cell, as the total mass lost by effusion through the orifice will be proportional to the vapor pressure of the substance. The vapor pressure can be related to the mass lost by the expression

where g is the mass lost, A is the area of the orifice,  Δt is the time the effusion is allowed to proceed, T is the temperature and  MW  is the molar mass of the compound in the vapor phase. The pressure is then given by  p. A schematic of what a Knudsen cell might look like is given below.

Intermolecular Collisions

• A major concern in the design of many experiments is collisions of gas molecules with other molecules in the gas phase. For example, molecular beam experiments are often dependent on a lack of molecular collisions in the beam that could degrade the nature of the molecules in the beam through chemical reactions or simply being knocked out of the beam.
• In order to predict the frequency of molecular collisions, it is useful to first define the conditions under which collisions will occur. For convenience, consider all of the molecules to be spherical and in fixed in position except for one which is allowed to move through a “sea” of other molecules. A molecular collision will occur every time the center of the moving molecule comes within one molecular diameter of the center of another molecule.

One can easily determine the number of molecules the moving molecule will “hit” by determining the number of molecules that lie within the “collision cylinder”. Because we fixed the positions of all but one of the molecules, we must use the relative speed of the moving molecule, which will be given by
v_rel = √2 – v
The volume of the collision cylinder is given by
V_col = √2 – v Δt A
The collisional cross section, which determined by the size of the molecule is given by
σ = πd²
Some values of σ are given in the table below:

Table 2.6.1: Collisional cross-section of Select Species

Since the number of molecules within the collision cylinder is given by

and since the number density ( N/V) is given by

the number of collisions is given by

The frequency of collisions (number of collisions per unit time) is then given by

Perhaps a more useful value is the mean free path (λ), which is the distance a molecule can travel on average before it collides with another molecule. This is easily derived from the collision frequency. How far something can travel between collisions is given by the ratio of how fast it is traveling and how often it hits other molecules:
λ = ⟨v⟩/Z
Thus, the mean free path is given by

The mere fact that molecules undergo collisions represents a deviation from the kinetic molecular theory. For example, if molecules were infinitesimally small (σ≈) then the mean free path would be infinitely long! The finite size of molecules represents one significant deviation from ideality. Another important deviation stems from the fact that molecules do exhibit attractive and repulsive forces between one another. These forces depend on a number of parameters, such as the distance between molecules and the temperature (or average kinetic energy of the molecules.)