Prof. Yo-Sep Min Physical Chemistry II, Fall 2013 Lecture 2-1

• The molecular motion in gases

1. The kinetic model of gases

b) The collision frequency

c) The mean free path

2. Collisions with walls and surfaces

3. The rate of effusion

4. Transport properties of a perfect gas

• Ch. 21 Molecules in motion

Molecular motion in gases

Molecular motion in liquids

Diffusion: migration of matter down a concentration gradient

Prof. Yo-Sep Min Physical Chemistry II, Fall 2013 Lecture 2-2

• The kinetic model enables us to calculate the frequency with

which molecular collisions occur (called collision frequency) and

the average distance a molecule travels between collisions

(called mean free path).

• We count a hit whenever the centers of

two molecules come within a distance d

of each other.

• The distance d is the collision diameter*.

* The collision diameter is of the order of the molecular diameters.

Assuming the molecules are hard spheres, the d is the diameter of the

molecules.

d

d

Prof. Yo-Sep Min Physical Chemistry II, Fall 2013 Lecture 2-3

• The collision frequency (z) is the # of

collisions made by one molecules

divided by a time interval t.

• When there are N molecules in a volume V, (number density N)

Nrelcz

where the is called the collision

cross-section of the molecules. = d2

*

• In terms of pressure, using NkTTnkNnRTpV A V

N

kT

p

kT

pcz rel

* See Justification 21.3.

V

Ncz rel

Prof. Yo-Sep Min Physical Chemistry II, Fall 2013 Lecture 2-4

• At constant V, the collision frequency increases with

increasing T due to the faster relative mean speed.

• At constant T, the collision frequency is proportional to

pressure due to the greater the number density.

Ex) For an N2 molecule at 1 atm and 25 oC, z ~ 5109/s

kT

pcz rel

V

Ncz rel

Prof. Yo-Sep Min Physical Chemistry II, Fall 2013 Lecture 2-5

Substituting ,

• The mean free path (), which is the average distance

traveled by a molecule between collisions, can be calculated

from the collision frequency.

• If a molecule collides with a frequency z, it spends a time 1/z

in free flight between collisions.

• Therefore, the mean free path is: z

crel

kT

pcz rel

p

kT

pc

kTc

z

c

rel

relrel

p

kT

Prof. Yo-Sep Min Physical Chemistry II, Fall 2013 Lecture 2-6

N

V

p

kT

• Doubling the p reduces the by half (if V is variable).

• However, at constant V, the distance between collisions is

determined by the number of molecules (N) present in the

given V, not by the speed at which they travel.

NkTkTnNnRTpV A N

1

N

V

p

kT

Prof. Yo-Sep Min Physical Chemistry II, Fall 2013 Lecture 2-7

• A typical gas N2 at 1 atm and 25 oC travels with a mean

speed of ~ 500 m/s.

• Each molecule makes a collision within ~ 1 ns.

• Each molecule travels ~ 1000 molecular diameters between

collisions.

At these conditions, N2 gas behaves nearly perfectly.

Prof. Yo-Sep Min Physical Chemistry II, Fall 2013 Lecture 2-8

• For accounting for transport in the gas phase, the rate at

which molecules strike an area (called collision flux) should

be obtained.

• The collision flux (Zw) is the number of collisions with an area

in a given time interval.

timearea

collision of #

WZ

mkT

pcZW

24

1 N

*

* See Justification 21.4. Correct the printing typos.

• Ex) At 1 atm and T = 300 K, a container receives

~31023/cm2·s.

Prof. Yo-Sep Min Physical Chemistry II, Fall 2013 Lecture 2-9

• Effusion is the process where individual molecules flow

through a hole without collisions between molecules.

• Effusion occurs if the diameter of the hole is considerably

smaller than the mean free path of the molecules.

Prof. Yo-Sep Min Physical Chemistry II, Fall 2013 Lecture 2-10

• When a gas (p and T) is separated from a vacuum by a small

hole, the rate of escape of its molecules is equal to the rate at

which they strike the area of the hole.

• For a hole of area Ao,

mkT

pAAZr o

oWAeffusion2

,

• Using R = kNA and M = mNA,

MRT

NpAr Ao

Aeffusion2

,

• The rate of effusion is inversely proportional to . M

Prof. Yo-Sep Min Physical Chemistry II, Fall 2013 Lecture 2-11

•The Graham’s law: For two gases, their effusion rates are

related to their masses (mA and mB) as:

M

M

A

B

Beffusion

Aeffusion

M

M

r

r

,

,

MRT

NpAr Ao

Aeffusion2

,

• The Graham’s law of effusion reveals that the rate of effusion

is inversely proportional to because the rate at which they

strike the area of the hole is also inversely proportional to .

Prof. Yo-Sep Min Physical Chemistry II, Fall 2013 Lecture 2-12

• The equation is utilized to determine the vapor pressures of

liquids and solids, particularly of substances with very low vapor

pressures. called Knudsen method

Knudsen cell: a cavity with a

small hole

• When particles (mass m) are escaped

through a hole by the effusion, the mass loss

m is monitored to determine p.

mkT

pAAZr o

oWAeffusion2

,

tmAZm oW

tA

m

M

RTp

o

2• Substituting the equation for ,

* See Example 21.2.

*

mkT

pZW

2

Prof. Yo-Sep Min Physical Chemistry II, Fall 2013 Lecture 2-13

• Transport properties of a substance are its ability to transfer

matter, energy, or some other property from one place to

another.

• Diffusion: the migration of matter down a

concentration gradient.

• Thermal conduction: the migration of energy down a

temperature gradient.

• Viscosity: the migration of linear momentum down a

velocity gradient.

• Electrical conduction: the migration of electric charge

along an electrical potential gradient.

Gas

Prof. Yo-Sep Min Physical Chemistry II, Fall 2013 Lecture 2-14

• Transport properties are commonly expressed by

phenomenological or empirical equations.

• The rate of migration of a property is measured by its flux (J).

• The flux is the quantity of a property passing through a given

area in a given time interval (quantity per area per time).

ex) matter flux for diffusion, energy flux for thermal conduction

• Experimental observations on transport properties show that

the flux of a property (J) is usually proportional to the 1st

derivative of some other related property.

Ex) Diffusion: concentration gradient.

Thermal conduction: temperature gradient.

Viscosity: velocity gradient.

Prof. Yo-Sep Min Physical Chemistry II, Fall 2013 Lecture 2-15

• For the diffusion, the matter flux diffusing

parallel to the z-axis of a container is

proportional to the 1st derivative of the

concentration.

dz

dJ

N(matter)

where N is the number density of particles.

• The SI units of the matter flux is [number/m2·s]

• If the concentration varies steeply with position, the diffusion

will be fast.

• If the concentration is uniform (dN/dz = 0), there is no net flux.

Prof. Yo-Sep Min Physical Chemistry II, Fall 2013 Lecture 2-16

• Because matter flows down a concentration

gradient from high to low concentration, J is positive if dN/dZ is negative.

• Therefore, the proportionality coefficient

must be negative. We write it –D.

dz

dDJN

(matter)

• The constant D is called the diffusion coefficient .

• The SI units of D are [m2/s]

Fick’s 1st law of diffusion

Prof. Yo-Sep Min Physical Chemistry II, Fall 2013 Lecture 2-17

• For the thermal conduction, the energy flux associated with

thermal motion is proportional to the temperature gradient.

dz

dTJ (energy)

• The SI units of the energy flux are [J/m2·s]

• The proportionality coefficient is –.

dz

dTJ (energy)

where the is called the coefficient of thermal conductivity.

• The SI units of is are [J/K·m·s]

Prof. Yo-Sep Min Physical Chemistry II, Fall 2013 Lecture 2-18

• To see the connection between the momentum flux and the

viscosity, consider a fluid in a state of Newtonian flow.

• The Newtonian flow can be imagined as

occurring by a series of layers moving past

one another.

• The velocity of successive layers varies

linearly with distance z from the wall.

• The particles bring their initial momentum

when they enter a new layer which is

accelerated or retarded.

• The net retarding effect is interpreted as the fluid’s viscosity.

Prof. Yo-Sep Min Physical Chemistry II, Fall 2013 Lecture 2-19

• Because the retarding effect depends on the transfer of the x-

component of momentum, the viscosity depends on this

momentum flux in the z-direction.

• The flux of x-component of momentum is

proportional to dvx/dz.

• As a proportionality coefficient, we use –.

dz

dvJ xmomentum) ofcomponent -(x

• The is called the coefficient of viscosity

or simply the viscosity.

• The SI units of are [kg/m·s].

• Viscosities are sometimes reported in poise (P).

1P = 0.1kg/m·s

Prof. Yo-Sep Min Physical Chemistry II, Fall 2013 Lecture 2-20

• Next Reading:

8th Ed: p.758 ~ 766

9th Ed: p.757 ~ 762