1. Derivation of PV = nRT
The ideal gas equation is an empirical law - i.e., deduced from the results of various experiments.
Several great scientific minds (incl. Boltzmann, Maxwell) attempted to lay a theoretical basis for this law -
i.e., to deduce a mathematical model for an ideal gas.
// This is the very essence of science - an attempt to explain and understand measurements and observations of the natural world within a self-consistent, predictive model. //

2. The Postulates of Kinetic Theory:
1 The volume occupied by a gas is very large compared with the volume of the molecules themselves.
2 Gas molecules are considered as point-like hard spheres.
3 Molecules exert no forces on one another except during collisions which are perfectly elastic (no energy loss).
4 Molecules move in random directions.

3. Ramifications:
Consider a cubical box of gas side L containing N molecules: vx vy vz L A
Consider one molecule with velocities vx, vy, vz in x, y, z directions such that its total speed v is given by
Consider the x direction only:
Upon collision with side A the molecule undergoes a momentum change = -2mvx. vx
-vx
The molecule hits the wall every 2L/vx seconds.
Hence the rate of change of momentum =

4. Recall Newton’s Law: Force = rate of change of momentum
Hence the force on wall A arising from one molecule is mvx2/L.
The total force exerted on the wall is the sum of all the forces exerted by each molecule of which there are N.
But pressure = force per unit area and so
Because motion is random, vx2 = vy2 = vz2 = v2/3,
so we arrive finally at:

5. Kinetic Energy The KE of a single molecule is ½ mv2
The average KE per molecule ½ mv2
Hence for N molecules:
And yet we know
So PV = 2/3 KEN
We also know from the ideal gas equation that PV = nRT and so if N = N0, we can extract a molar kinetic energy:

6. Profound Implications:
All gases at the same temperature have the same molar kinetic energy.
Kinetic energy  T, in fact they can be used interchangeably
The average kinetic energy per molecule is given by:
k = R/N0 = x J K-1 is called the Boltzmann constant (effectively the universal gas constant per molecule).

7. The Maxwell-Boltzmann Speed Distribution
The shape of the speed distribution curve is similar for all gases and is called the Maxwell-Boltzmann speed distribution.
Number of molecules
Speed / m s-1 T1
T3 > T2 > T1 T2 T3

8. Different Average Speeds
We can identify three important speeds on the M-B distribution
Number of molecules
Speed v / m s-1
vMP the most probable speed
v the mean, or average, speed
vrms the root mean square speed =

9. Any one of which is enough to characterize the distribution:
So for 298 K N2: vMP = 420 ms-1, v = 474 ms-1, vrms = 515 ms-1 _
(n.b., take care to use SI units in calculations)

10. Collision Rate and Mean Free Path
Imagine following a particular molecule moving through a gas for one second:
hit
miss d
crel x 1s = c (m)
How many collisions will our molecule make per second?

11. The volume of the cylinder is:
Our molecule will collide with every other molecule in the “cylinder” it sweeps out.
Which is how many?
The volume of the cylinder is:
p d2 crel
The ideal gas equation gives us the number of molecules per unit volume: PV = nRT and n = N / N0
Hence the number density N / V = P N0 /RT = P/kT
So the number of collisions per second is: P
where s = pd2 is called the collision cross-section. kT x c d
density) (#
(Volume) z
rel 2 σ = p

12. So for example the number of collisions made per second by an average N2 molecule in air at STP is:
z ≈ 2 x 109 s-1
Put another way the molecule typically travels for an average of only 5 x s between collisions.
At an average speed of 475 ms-1, the mean distance traveled between collisions – the, mean free path, l, can be calculated to be
l = 475 m s-1 x 5 x s = 2.4 x m
(which is about 103 molecular diameters)

13. Discuss How Cross Section Depends on Probe and Velocity
Molecular beam attenuation experiments

14. Attenuation data for the scattering of a thermal beam (1100 K) of CsCl by Ar atoms and by the polar CH2F2 molecules in a 44 mm cell. The log of the transmission decreases linearly with the pressure of the target gas. [Adapted from H. Schumacher, R. B. Bernstein, and E.W. Rothe, J. Chem. Phys. 33, 584 (1960).]

15. The operational definition of a collision is that one molecule collides with another when a force acts between them. We need to make it quantitative, but, loosely speaking, the longer the range of the force, the more likely is a collision. The very polar molecule CsCl has a longer range of attraction to another polar molecule than to a spherical atom.
The implication that the cross-section depends on the range and strength of the intermolecular force carries with it an obvious corollary: the magnitude of the collision cross-section is a property of the two molecules that are colliding. Strictly speaking, a molecule does not have 'a size' whose value is independent of how we probe it.