Dr. Joseph N. Grima, Department of Chemistry
University of Malta, Msida, MSD 06, MALTA
http://staff.um.edu.mt/jgri1/teaching/ch237

(1-0) Aside - The kinetic theory of gasses

(1-0-0) Introduction
(1-0-1) Molecular speeds
(1-0-2) The collision frequency and mean free path  

(1-0) Aside - The kinetic theory of gasses

(1-0-0) Introduction

The kinetic model of gases is based on three assumptions:

1. The gas consists of molecules of mass m in ceaseless random motion.
2. The size of the molecules is negligible, in the sense that their diameters are mich less than the average distance travelled between collisions.
3. The molecules do not interact, except that they make perfectly elastic collisions when they are in contact. 

In a time interval Dt we have many molecules colliding with a particular wall. Thus the change in momentum in a Dt time interval is given by:

Total momentum change = D momentum for 1 molecule x no. of colliding molecules

where:

1. No. of Colliding molecules in Dt is as follows:
• Distance travelled by a molecule in Dt in the x-direction is v_x Dt.
• Thus all molecules in a 'v_x Dt' vicinity that are moving towards wall will collide in this time Dt.
• Thus if wall has area A, all molecules in volume 'A v_xDt' will collide in the time period Dt (see fig. 1)
• But if there are n moles of molecules in the total volume V, then the number of molecules that will collide in Dt is given by:
Fig. 1: A molecule will reach the wall on the right within an interval Dt if it is within a distance v_xDt of the wall and travelling to the right.
2. Assuming that on average, half of the molecules move the right and the other half to the left, then average number of collisions is given by:
3. The momentum change for each collision is given by (see fig. 2):
Dmomentum for 1 cossision = 2mv_x

 
Fig. 2: In an elastic collision of a molecule with a wall perpendicular to the x-axis, the x-component of velocity is reversed. (The y- and z-components are unchanged)

Thus the total change in momentum is given by:

This means that the force exerted by the molecules on wall of area A perpendicular to the x-direction is give by:

i.e.:

1. The speeds of an individual molecule spans a wide range. 
2. Collisions continuously re-distribute speeds, i.e. the speed of any individual molecules may change with every collision.

Fig. 3: The distribution of molecular speeds with temperature and molar mass. Note that the most probable speed (corresponding to the peak of the distribution) increases with temperature and with decreasing molar mass and, simultaneously, the distribution becomes broader. 

Given this expression, let us now calculate the mean speed of molecules (not the root mean square speed). For a discrete system, a mean value can be calculated by:

1. Multiplying each speed by the fraction of the molecules that have that speed;
2. Adding the products obtained from (1) together.

where m is the reduced mass of the molecules that relates to the individual masses of two dissimilar molecules (m_A and m_B) though:

Let us now calculate the frequency with which molecular collisions occur and the distance a molecule travels on average between collisions for a single-species gas.

We count a 'hit' whenever the centres of two molecules come within a distance d of each other, where d, the collision diameter, is of the order of the actual diameters of the molecules (for impenetrable hard spheres d is the diameter) (see fig. 4). 

Let us assume that the positions of all the molecules except one to be frozen. When one mobile molecule travels through the gas with a mean relative speed for a time Dt, it sweeps out a 'collision tube' of cross-sectional area s = p d² and length  (see fig). The number of stationary molecules with centres inside the collision tube is given by:

Fig. 4: In interval Dt a molecule of diameter d sweeps out a tube of diameter 2d and length 'c'_relDt . As it does so it encounters other molecules with centres that lie within the tube, and each such encounter counts as one collision. 

Given this expression for the collision frequency, z, it may be deduced that the mean time between collisions is 1/z, i.e. the average distance traveled between collisions, or, the mean free path, l, is given by:

Thus to summarise, a typical gas (nitrogen or oxygen) at 1 atm and 25 °C can be thought of as a collection of molecules traveling with a mean speed of about 350 ms^-1. Each molecule makes a collision within about 1 ms, and between collisions it travels about 102 to 103 molecular diameters. The kinetic model of gases is valid (and the gas behaves nearly perfectly) if the diameter of the molecules is much smaller than the mean free path, for then the molecules spend most of their time far from one another and do not interact.