A detailed presentation on Pressure of dilute gases, effusion and wall collision from the topic of kinetic theory of gases and chemical kinetics. References from physical chemistry - Atkins

1. CHEM324 : KINETIC THEORY OF GASES AND
CHEMICAL KINETICS
DONE BY : R. BARANI DHARAN
21380206
INTEGRATED M.SC CHEMISTRY
III YEAR

2. PRESSURE OF A DILUTE GAS
• Let us assume that the box confining our model system is rectangular with walls that
are perpendicular to the coordinate axes and that the walls are smooth, slick, flat, and
impenetrable.
• A collision of a molecule with such a wall is called a specular collision,
which means: (1) It is elastic. i.e., the kinetic energy of the molecule is the same
before and after the collision. (2) The only force exerted on the particle is perpendicular
to the wall.

3. • Consider a particle strikes the wall at the right end of the box ( perpendicular to the y-axis).
• The values of vx and vz do not change because no force is exerted in these directions. Because the
kinetic energy does not change, v2y does not change, although the sign of vy does change. Let v(i) be
the initial velocity and v(f) be the final velocity:
• As the particle strikes the wall, the force exerted on the wall rises suddenly and drops to zero just as
suddenly. Consider the force exerted on the particle by the wall, which by newton’s third law is the
negative of the force exerted on the wall by the particle.

4. • The time average of the y component of the force on the particle over a period of time from 0 to τ is
given by
• From Newton’s second law, F = ma
• Since we already concluded that vy(f) = - vy(i),
• By Newton’s third law, the force on the wall Fwall = - Fparticle

5. • The total time-average force on the area A of the wall is the sum of the contributions from all particles
that strike this area in time τ.
• Consider first, the particles whose velocities lie in the range dvxdvydvz, abbreviated by d3v. The fraction
of all particles whose velocities lie in this range is
where v is a velocity in the range d3v. All of the particles with velocities in this range
are moving in the same direction with the same speed

6. • The particles that will strike the area A on the wall are contained within a
prism with base area A and sides parallel to the direction of motion of the
particles
• The altitude of the prism is vyt
• The volume of the prism is equal to the area of its base times its altitude:
Vprism = Avyt
• We assume that our system is uniform. The number density N = N/V,
where N is the number of particles and V is the volume of the system.
• Number density in prism, N =
• Number density in the prism with velocities in the specified range:

7. • Let the total time average force on A be denoted by FA. Thus, the total contribution to FA due to particles
in the specified range
= X
=
• We now add up the contributions for molecules with different velocities by integrating over the velocity
components. Total time average force on A is:
• The integrand is an even function of vy, which means that if vy is replaced by −vy, the integrand is
unchanged. The value of the integral over vy from 0 to ∞ equals half of that of the integral from −∞ to
∞, so we have replaced the lower limit by −∞ and divided by 2.

8. • This integral is the same as the integral that gives the mean value of vy
2 which equals v2/3
• The pressure is force per unit area

9. EFFUSION AND WALL COLLISIONS
• Effusion is the process by which molecules of gas pass through a small hole into a vacuum. The hole
must be small enough such that gas doesn’t flow through the hole as a fluid but as individual molecules.
• According to Graham’s law of effusion, “ at a given temperature and a given pressure the rate of
effusion of a gas is inversely proportional to the square root of its density.
• We will try to calculate the number of molecules striking the area A instead of the force exerted on this
area. If the area is a hole in the wall, we obtain the rate of effusion. If the area is a section of the wall,
then we obtain the rate of wall collisions.

10. DIFFUSION AND RATE OF DIFFUSION
• Diffusion, process resulting from random motion of molecules by which there
is a net flow of matter from a region of high concentration to a region of low
concentration without bulk motion.
• According to Fick's laws, the diffusion flux is proportional to the
negative gradient of concentrations.
• Diffusion occurs more quickly in gases than it does in liquids.
• The concentration gradient, membrane permeability, temperature, and
pressure all influence the rate at which diffusion occurs.

11. • Number of molecules with velocities in the velocity interval dvxdvydvz that will strike the area A in time τ
is given by
• Total number of molecules striking the area A in the time interval τ is
• The triple integral can be factored:

12. • The integration over vx and vz gives factors of
• Total number of molecules striking the area A in time τ =
• Let w = vy
2, then the integral becomes,
• Total number of molecules striking the area A in time τ =

13. • The number of particles striking unit area per unit time is denoted by v
• This equation for the rate of wall collisions also gives the rate of effusion per unit area in the case of a
small hole in the wall.

14. THANK YOU!