Bird R.B., Stewart W.E., Lightfoot E.N. Transport Phenomena

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Generalization of Newton's Law of Viscosity

any instant of time we can slice the volume element in such a way as to remove half the

fluid within it. As shown in the figure, we can cut the volume perpendicular to each of

the three coordinate directions in turn. We can then ask what force has to be applied on

the free (shaded) surface in order to replace the force that had been exerted on that sur-

face by the fluid that was removed. There will be two contributions to the force: that as-

sociated with the pressure, and that associated with the viscous forces.

The pressure force will always be perpendicular to the exposed surface. Hence in (a)

the force per unit area on the shaded surface will be a vector p6,-that is, the pressure (a

scalar) multiplied by the unit vector 6, in the x direction. Similarly, the force on the

shaded surface in

(b)

will be p6,, and in

(c)

the force will be p6,. The pressure forces will

be exerted when the fluid is stationary as well as when it is in motion.

The viscous forces come into play only when there are velocity gradients within the

fluid.

general they are neither perpendicular to the surface element nor parallel to it,

but rather at some angle to the surface (see Fig. 1.2-1). In (a) we see a force per unit area

exerted on the shaded area, and in

(b)

and

(c)

we see forces per unit area

and

7,.

Each of these forces (which are vectors) has components (scalars); for example,

has

components

T,,,

T,~,

and

T,,.

Hence we can now summarize the forces acting on the three

shaded areas in Fig.

1.2-1

in Table

1.2-1.

This tabulation is a summary of the forces per

unit area (stresses) exerted within a fluid, both by the thermodynamic pressure and the

viscous stresses. Sometimes we will find it convenient to have a symbol that includes both

types of stresses, and so we define the molecular stresses as follows:

paii

rii

where

and

may be x,

(1.2-2)

Here

Sij

is the Kronecker delta, which is

and zero if

Just as in the previous section, the r,j (and also the

'rr$

may be interpreted in two ways:

rZi

pa,,

ril

force in the

direction on a unit area perpendicular to the

direction,

where it is understood that the fluid in the region of lesser xi is exerting

the force on the fluid of greater xi

'rrY

paij

rij

flux of j-momentum in the positive

direction-that is, from the region

of lesser

to that of greater xi

Both interpretations are used in this book; the first one is particularly useful in describ-

ing the forces exerted by the fluid on solid surfaces. The stresses

.rr,,

T,,,

ryy,

'rrzz

T,,

are called normal stresses, whereas the remaining quantities,

IT,,

T,,,

T~~,

. .

are called shear stresses. These quantities, which have two subscripts associ-

ated with the coordinate directions, are referred to as "tensors," just as quantities (such

as velocity) that have one subscript associated with the coordinate directions are called

Table

1.2-1

Summary of the Components of the Molecular Stress Tensor (or Molecular

Momentum-Flux Tensor)"

Direction

normal

to the

shaded

Vector force

per unit area on the

shaded face (momentum

face flux through shaded face)

Tr,

pajx

p6,

Components of the forces (per unit area)

acting on the shaded face (components of the

momentum flux through the shaded face)

x-component y-component z-component

These are referred to as components of the "molecular momentum flux tensor" because they are

associated with the molecular motions, as discussed in

g1.4

and Appendix

The additional "convective

momentum flux tensor" components, associated with bulk movement of the fluid, are discussed in

51.7.

Chapter

Viscosity and the Mechanisms of Momentum Transport

vectors." Therefore we will refer to

as the z~iscous stress tensor (with components

T~)

and

.rr

as the molecular stress tensor (with components

qj).

When there is no chance for

confusion, the modifiers "viscous" and "molecular" may be omitted.

discussion of

vectors and tensors can be found

Appendix A.

The question now is: How are these stresses rij related to the velocity gradients in

the fluid? In generalizing Eq. 1.1-2, we put several restrictions on the stresses, as follows:

The viscous stresses may be linear combinations of all the velocity gradients:

dvk

7..

-CkClp..

where

and

may be 1,2,3 (1.2-3)

vk'

dx1

Here the 81 quantities

pijkl

are "viscosity coefficients." The quantities x,, x,, x3 in

the derivatives denote the Cartesian coordinates x,

z, and v,, v,,

are the same

as v,, v,, v,.

We assert that time derivatives or time integrals should not appear in the expres-

sion. (For viscoelastic fluids, as discussed in Chapter 8, time derivatives or time in-

tegrals are needed to describe the elastic responses.)

We do not expect any viscous forces to be present, if the fluid is in a state of pure

rotation. This requirement leads to the necessity that

ri,

be a symmetric combina-

tion of the velocity gradients. By this we mean that if

and

are interchanged, the

combination of velocity gradients remains unchanged. It can be shown that the

only symmetric linear combinations of velocity gradients are

If the fluid is isotropic-that is, it has no preferred direction-then the coefficients

in front of the two expressions in

Eq.

1.2-4 must be scalars so that

We have thus reduced the number of "viscosity coefficients" from 81 to 2!

Of course, we want Eq. 1.2-5 to simplify to Eq. 1.1-2 for the flow situation in Fig.

1.1-1. For that elementary flow Eq. 1.2-5 simplifies to

T,,

dv,/dy, and hence the

scalar constant

must be the same as the negative of the viscosity

Finally, by common agreement among most fluid dynamicists the scalar constant

is set equal to

where

is called the dilatational viscosity. The reason for

writing

in this way is that it is known from kinetic theory that

is identically

zero for monatomic gases at low density.

Thus the required generalization for Newton's law of viscosity in Eq. 1.1-2 is then

the set of nine relations (six being independent):

Here

T~~

T,~,

and

can take on the values 1,2,3. These relations for the stresses in a

Newtonian fluid are associated with the names of Navier, Poisson, and ~tokes.~ If de-

C.-L.-M.-H.

Navier,

Ann.

Chimie,

19,244-260 (1821);

S.-D.

Poisson,

~cole Polytech.,

13,

Cahier 20,l-174

(1831);

G. G.

Stokes,

Trans.

Camb.

Phil. Soc.,

8,287-305 (1845).

Claude-Louis-Marie-Henri

Navier

(1785-1836)

(pronounced "Nah-vyay," with the second syllable accented) was a civil engineer whose specialty was road

and bridge building;

George Gabriel Stokes

(1819-1903) taught at Cambridge University and was president

of the Royal Society. Navier

and

Stokes are well known because of the Navier-Stokes equations (see Chapter

3).

See

also

Acheson,

Elemazta

Fluid Mechanics,

Oxford University Press (1990), pp. 209-212,218.

1.2 Generalization of Newton's Law of Viscosity

sired, this set of relations can be written more concisely in the vector-tensor notation of

Appendix A as

in which

is the unit tensor with components

SV,

Vv is the velocity gradient tensor with

components (d/dxi)vj,

(Vv)'

is the "transposer' of the velocity gradient tensor with com-

ponents (d/dxj)vi, and

v) is the divergence of the velocity vector.

The important conclusion is that we have a generalization of Eq. 1.1-2, and this gen-

eralization involves not one but two coefficients3 characterizing the fluid: the viscosity

and the dilatational viscosity

Usually, in solving fluid dynamics problems, it is not

necessary to know

If the fluid is a gas, we often assume it to act as an ideal

monoatomic gas, for which

is identically zero. If the fluid is a liquid, we often assume

that it is incompressible, and in Chapter

we show that for incompressible liquids

and therefore the term containing

is discarded anyway. The dilational vis-

cosity is important

describing sound absorption in polyatomic gases4 and in describ-

ing the fluid dynamics of liquids containing gas

bubble^.^

Equation 1.2-7 (or 1.2-6) is an important equation and one that we shall use often.

Therefore it is written out in full in Cartesian (x,

z), cylindrical (r,

z), and spherical

(r,

Of+)

coordinates in Table

B.1.

The entries in this table for curvilinear coordinates are

obtained by the methods outlined

55A.6 and

A.7.

It is suggested that beginning stu-

dents not concern themselves with the details of such derivations, but rather concen-

trate on using the tabulated results. Chapters

and

will give ample practice in doing

this.

In curvilinear coordinates the stress components have the same meaning as in Carte-

sian coordinates. For example,

r,,

in cylindrical coordinates, which will be encountered

in Chapter

can be interpreted as:

(i)

the viscous force in the z direction on a unit area

perpendicular to the r direction, or (ii) the viscous flux of z-momentum in the positive r

direction. Figure 1.2-2 illustrates some typical surface elements and stress-tensor compo-

nents that arise in fluid dynamics.

The shear stresses are usually easy to visualize, but the normal stresses may cause

conceptual problems. For example,

T,,

is a force per unit area in the z direction on a

plane perpendicular to the z direction. For the flow of an incompressible fluid in the

convergent channel of Fig. 1.2-3, we know intuitively that v, increases with decreas-

ing z; hence, according to Eq. 1.2-6, there is a nonzero stress

r,,

-2p(dv,/dz) acting

in the fluid.

Note on the Sign Convention for the Stress Tensor We have emphasized in connection

with Eq. 1.1-2 (and in the generalization in this section) that

T~~

is the force in the posi-

tive x direction on a plane perpendicular to the

direction, and that this is the force ex-

erted by the fluid in the region of the lesser

on the fluid of greater

In most fluid

dynamics and elasticity books, the words "lesser" and "greater" are interchanged and

Eq. 1.1-2 is written as

r,,

+p(dv,/dy). The advantages of the sign convention used in

this book are: (a) the sign convention used in Newton's law of viscosity is consistent

with that used in Fourier's law of heat conduction and Fick's law of diffusion; (b) the

sign convention for

rij

is the same as that for the convective momentum flux

(see

Some writers refer to

as the "shear viscosity," but this is inappropriate nomenclature inasmuch

can arise

nonshearing flows as well as shearing flows. The term "dynamic viscosity" is also

occasionally seen, but this term has a very specific meaning in the field

viscoelasticity and is an

inappropriate term

for

Landau and

Lifshitz,

op. cit.,

Ch.

VIII.

Batchelor,

An Introduction to Fluid Dynamics,

Cambridge University Press

(1963,

pp.

253-255.

Chapter

Viscosity and the Mechanisms of Momentum Transport

of radius

Force

fluid in

direction on

surface element

(RdB)(dz)

-~~o(~=RRdedz

surface element

(RdNdz)

-7,.,(,.=~Rd~d~

Solid notched

Force

fluid

direction on

surface element

(dr)(dz)

+To2

(,/zI

dr dz

Yak

Solid sphere

of radius

Force

fluid in

direction on

surface element

(RdB)(R

sin

8 d4)

-rrolr=R~2sin0d8d4

Solid sphere

of radius

Force

fluid in

direction on

surface element

(RdMR

sin

8 d4)

-T~+~~=~R~

sin

8 d8d4

Force

fluid in

direction on

surface element

Solid cone

(dr)(r

sin

d+)

with half

ror(o=ar

sin

drd4

Fig.

1.2-2

(a)

Some typical surface elements and shear stresses in the cylindrical coordinate system.

(b)

Some typical surface elements and shear stresses in the spherical coordinate system.

51.7

and Table 19.2-2); (c) in Eq.

1.2-2,

the terms

paij

and

T~~

have the same sign affixed,

and the terms

and

T~~

are both positive in compression (in accordance with common

usage in thermodynamics); (d) all terms in the entropy production in

Eq.

24.1-5 have

the same sign. Clearly the sign convention in Eqs.

1.1-2

and 1.2-6 is arbitrary, and either

sign convention

can

be used, provided that the physical meaning of the sign convention

is clearly understood.

3 Pressure and Temperature Dependence of Viscosity

Fig.

1.2-3

The flow in a converging duct is an example of a situation

in which the normal stresses are not zero. Since

is a function of

and

the normal-stress component

T,,

-2p(dv,/dz)

is nonzero.

Also, since

depends on

and

the normal-stress component

T,,

-2p(dvr/dr)

is not equal to zero. At the wall, however, the

'z")

normal stresses all vanish far fluids described by Eq.

1.2-7

provided

that the density is constant (see Example 3.1-1 and Problem

3C.2).

1.3

PRESSURE AND TEMPERATURE DEPENDENCE

OF VISCOSITY

Extensive data on viscosities of pure gases and liquids are available in various science

and engineering handbooks.' When experimental data are lacking and there is not time

to obtain them, the viscosity can be estimated by empirical methods, making use of other

data on the given substance. We present here a corresponding-states correlation, which fa-

cilitates such estimates and illustrates general trends of viscosity with temperature and

pressure for ordinary fluids. The principle of corresponding states, which has a sound

scientific basis: is widely used for correlating equation-of-state and thermodynamic

data. Discussions of this principle can be found in textbooks on physical chemistry and

thermodynamics.

The plot in Fig. 1.3-1 gives a global view of the pressure and temperature dependence

of viscosity. The reduced viscosity

p/p,

is plotted versus the reduced temperature

T/T,

for various values of the reduced pressure

p/p,.

"reduced quantity is one

that has been made dimensionless by dividing by the corresponding quantity at the criti-

cal point. The chart shows that the viscosity of a gas approaches a limit (the low-density

limit) as the pressure becomes smaller; for most gases, this limit is nearly attained at

atm

pressure. The viscosity of a gas at low density increases with increasing temperature,

whereas the viscosity of

liquid decreases with increasing temperature.

Experimental values of the critical viscosity

are seldom available. However,

may be estimated in one of the following ways: (i) if a value of viscosity is known at a

given reduced pressure and temperature, preferably at conditions near to those of

Schetz and

Fuhs (eds.),

Handbook of Fluid Dynamics and Fluid Machinery,

Wiley-

Interscience, New York (1996), Vol. 1, Chapter

W. M. Rohsenow,

Hartnett, and Y.

Cho,

Handbook

of Heat Transfer,

McGraw-Hill, New York, 3rd edition (19981, Chapter

Other sources are mentioned in

fn.

of 91.1.

Millat,

Dymond, and C.

Nieto de Castro (eds.),

Transport Properties of Fluids,

Cambridge

University Press (1996), Chapter

11,

Mason and

Uribe, and Chapter

12,

Huber

and

Hanley.

Chapter 1 Viscosity and the Mechanisms of Momentum Transport

Fig.

1.3-1 Reduced vis-

cosity

p/p,

as a

function of reduced

temperature for several

values of the reduced

pressure.

10.

Uye-

hara and

Watson,

Nat. Petroleum News,

Tech. Section, 36,764

(Oct. 4,1944); revised

M. Watson (1960).

large-scale version of

this graph is available

Hougen,

Watson, and

Ragatz,

P. P.

Charts, Wiley, New

York, 2nd edition

(1960)l.

Reduced temperature

T/T,

interest, then

can be calculated from

p/pY; or (ii) if critical p-V-T data are avail-

able, then p, may be estimated from these empirical relations:

Here

is in micropoises, p, in atm, Tc in

and

in cm3/g-mole. A tabulation of critical

viscosities3 computed by method (i) is given in Appendix

Figure 1.3-1 can also be used for rough estimation of viscosities of mixtures. For

N-component fluids with mole fractions x,, the "pseudocritical" properties4 are:

That is, one uses the chart exactly as for pure fluids, but with the pseudocritical proper-

ties instead of the critical properties. This empirical procedure works reasonably well

Hougen and

Watson,

Chemical Process Principles,

Part

111,

Wiley, New York (1947),

p. 873. Olaf Andreas Hougen (pronounced "How-gen") (1893-1986) was a leader in the development of

chemical engineering for four decades; together with

Watson and R.

Ragatz, he wrote

influential books on thermodynamics and kinetics.

Hougen and

Watson,

Chemical Process Principles,

Part

11,

Wiley, New York (1947), p.

604.

1.4 Molecular Theory of the Viscosity of Gases at Low Density

unless there are chemically dissimilar substances in the mixture or the critical properties

of the components differ greatly.

There are many variants on the above method, as well as a number of other empiri-

cism~. These can be found in the extensive compilation of Reid, Prausnitz, and Poling.'

EXAMPLE

1.3-1

Estimate the viscosity of

at 50°C and 854 atm, given

28.0 g/g-mole,

33.5 atm, and

126.2

Estimation of Viscosity

from Critica

Properties

SOLUTION

Using

Eq.

1.3-1 b, we get

7.70(28.0)"~(33.5)~'~(126.2)'/~

189 micropoises

189

poise (1.3-3)

The reduced temperature and pressure are

From Fig. 1.3-1, we obtain

p/pc

2.39. Hence, the predicted value of the viscosity is

p,(p/p,)

(189

1OP6)(2.39)

452

poise

(1.3-5)

The measured value6 is 455

lop6

poise. This is unusually good agreement.

1.4

MOLECULAR

THEORY

THE

VISCOSITY

GASES

LOW

DENSITY

To get

better appreciation of the concept of molecular momentum transport, we exam-

ine this transport mechanism from the point of view of an elementary kinetic theory of

gases.

We consider a pure gas composed of rigid, nonattracting spherical molecules of di-

ameter

and mass

and the number density (number of molecules per unit volume) is

taken to be

The concentration of gas molecules is presumed to be sufficiently small

that the average distance between molecules is many times their diameter

In such

gas it is known1 that, at equilibrium, the molecular velocities are randomly directed and

have an average magnitude given

(see Problem

1C.1)

in which

is the Boltzmann constant (see Appendix

F).

The frequency of molecular

bombardment per unit area on one side of any stationary surface exposed to the gas is

C. Reid,

Prausnitz, and

Poling,

The Properties of Gases and Liquids,

McGraw-Hill, New

York, 4th edition (19871, Chapter 9.

Michels and R.

Gibson,

Proc. Roy. Soc.

(London), A134,288-307 (1931).

The first four equations in this section are given without proof. Detailed justifications are given in

books on kinetic theory-for example,

H. Kennard,

Kinetic Theory of Gases,

McGraw-Hill, New York

(1938), Chapters

and

111.

Also

A. Guggenheim,

Elements of the Kinetic Theory of Gases,

Pergamon

Press, New York (1960), Chapter

has given a short account of the elementary theory of viscosity. For

readable summaries of the kinetic theory of gases, see R.

Silbey and R. A. Alberty,

Physical Chemistry,

Wiley, New York, 3rd edition (2001), Chapter 17, or R.

Berry,

A. Rice, and

Ross,

Physical Chemistry,

Oxford University Press, 2nd edition (2000), Chapter 28.

Chapter 1 Viscosity and the Mechanisms of Momentum Transport

The average distance traveled by a molecule between successive collisions is the

mean

free

path

given by

(1.4-3)

V?ird2n

On the average, the molecules reaching a plane will have experienced their last collision

at a distance

from the plane, where

is given very roughly by

The concept of the mean free path is intuitively appealing, but it is meaningful only

when

is large compared to the range of intermolecular forces. The concept is appropri-

ate for the rigid-sphere molecular model considered here.

To determine the viscosity of a gas in terms of the molecular model parameters, we

consider the behavior of the gas when it flows parallel to the m-plane with a velocity

gradient

dvx/dy

(see Fig. 1.4-1). We assume that Eqs. 1.4-1 to

remain valid in this non-

equilibrium situation, provided that all molecular velocities are calculated relative to the

average velocity

in the region in which the given molecule had its last collision. The

flux of x-momentum across any plane of constant

is found by summing the x-momenta

of the molecules that cross in the positive

direction and subtracting the x-momenta of

those that cross in the opposite direction, as follows:

In writing this equation, we have assumed that all molecules have velocities representa-

tive of the region in which they last collided and that the velocity profile

vx(y)

is essen-

tially linear for a distance of several mean free paths. In view of the latter assumption,

we may further write

By combining Eqs. 1.4-2,5, and

we get for the net flux of x-momentum in the positive

direction

This has the same form as Newton's law of viscosity given in Eq. 1.1-2. Comparing the

two equations gives an equation for the viscosity

nmiiA

ipiiA (1.4-8)

Velocity

profile

vx(y)

Fig.

1.4-1

Molecular transport

of x-momentum from the plane at

to the plane at

Molecular Theory of the Viscosity of Gases at Low Density

or, by combining Eqs.

1.4-1,3,

and

This expression for the viscosity was obtained by Maxwell2

1860.

The quantity

called the

collision cross section

(see Fig.

1.4-2).

The above derivation, which gives a qualitatively correct picture of momentum

transfer in a gas at low density, makes it clear why we wished to introduce the term

"momentum flux" for

§1

.l.

The prediction of Eq. 1.4-9 that

is independent of pressure agrees with experimen-

tal data up to about

atm at temperatures above the critical temperature (see Fig.

1.3-1).

The predicted temperature dependence is less satisfactory; data for various gases indi-

cate that

increases more rapidly than

1/T.

To better describe the temperature depen-

dence of

it is necessary to replace the rigid-sphere model by one that portrays the

attractive and repulsive forces more accurately. It is also necessary to abandon the mean

free path theories and use the Boltzmann equation to obtain the molecular velocity dis-

tribution in nonequilibrium systems more accurately. Relegating the details to Appendix

we present here the main

result^.^'^^

of area

/--

Fig.

1.4-2

When two rigid spheres of diameter

approach

each other, the center of one sphere (at

0')

"sees" a circle of

area

md2

about the center of the other sphere (at

O),

which a collision can occur. The area

&I2

referred to as the

.---/

"collision cross section."

James Clerk Maxwell (1831-1879) was one of the greatest physicists of all time; he is particularly

famous for his development of the field of electromagnetism and his contributions to the kinetic theory

of gases. In connection with the latter, see

C. Maxwell,

Phil. Mag.,

19,19, Prop.

XI11

(1860); S.

Brush,

Am.

Phys,

30,269-281 (1962). There is some controversy concerning Eqs.

1.4-4

and 1.4-9 (see

Chapman

and

Cowling,

The Mathematical Theory of Non-Uniform Gases,

Cambridge University Press, 3rd

edition 1970), p. 98;

E. Cunningham and

Williams,

Diffusion in Gases and Porous Media,

Plenum

Press, New York (1980), s6.4.

Sydney Chapman (1888-1970) taught at Imperial College in London, and thereafter was at the

High Altitude Observatory in Boulder, Colorado; in addition to hs seminal work on gas kinetic theory,

he contributed to kinetic theory

plasmas and the theory of flames and detonations. David Enskog

(1884-1947) (pronounced, roughly, "Ayn-skohg") is famous for his work on kinetic theories of low- and

highdensity gases. The standard reference on the Chapman-Enskog kinetic theory of dilute gases is

Chapman and

Cowling,

The Mathematical Theory of Non-Uniform Gases,

Cambridge University

Press, 3rd edition (1970); pp. 407409 give a historical summary of the kinetic theory. See also

Enskog,

Inaugural Dissertation,

Uppsala (1917).

addition

Ferziger and

Kaper,

Mathematical Theory of

Transport Processes in Gases,

North-Holland, Amsterdam (1972), is a very readable account of molecular

theory.

The Curtiss-Hirschfelder5 extension of the Chapman-Enskog theory to multicomponent gas

mixtures, as well as the development of useful tables for computation, can be found in

Hirschfelder,

Curtiss, and

Bird,

Molecular Theory of Gases and Liquids,

Wiley, New York, 2nd corrected printing

(1964). See also C.

Curtiss,

Chem. Phys.,

49,2917-2919 (19681, as well as references given in Appendix

Joseph Oakland Hirschfelder (1911-1990), founding director of the Theoretical Chemistry Institute at

the University of Wisconsin, specialized in intermolecular forces and applications of kinetic theory.

Curtiss and

Hirschfelder,

Chem. Phys.,

17,550-555 (1949).

Chapter

Viscosity and the Mechanisms of Momentum Transport

rigorous kinetic theory of monatomic gases at low density was developed early in

the twentieth century by Chapman in England and independently by Enskog in Sweden.

The Chapman-Enskog theory gives expressions for the transport properties in terms of

the intermolecular potential energy

&),

where r is the distance between a pair of molecules

undergoing a collision. The intermolecular force is then given by F(r)

-dp/dr. The

exact functional form of p(r) is not known; however, for nonpolar molecules a satisfac-

tory empirical expression is the Lennard-Jones

(6-12)

potential6 given by

in which

is a characteristic diameter of the molecules, often called the collision diameter

and

is a characteristic energy, actually the maximum energy of attraction between a

pair of molecules. This function, shown in Fig. 1.4-3, exhibits the characteristic features

of intermolecular forces: weak attractions at large separations and strong repulsions at

small separations. Values of the parameters

and

are known for many substances; a

partial list is given in Table E.l, and a more extensive list is available el~ewhere.~ When

and

are not known, they may be estimated from properties of the fluid at the critical

point (c), the liquid at the normal boiling point

(b),

or the solid at the melting point (m),

by means of the following empirical relations:"

Here

E/K

and

are in

is in &tgstriim units (1

lop1'

m), ?is in ~rn~/~-rnole, and

is in atmospheres.

The viscosity of a pure monatomic gas of molecular weight

may be written in

terms of the Lennard-Jones parameters as

Molecules repel Molecules attract

one another at

separations

r,,

separations

Whenr

-3u,

llcp(

has dropped off

to less than 0.01

Fig.

1.4-3

Potential energy function

p(r) describing the interaction of two

spherical, nonpolar molecules. The

Lennard-Jones

(6-12)

potential, given

Eq.

1.4-10,

is one of the many em-

pirical equations proposed for fitting

this curve. For r

the molecules

repel one another, whereas for r

the molecules attract one another.

(Lennard-)Jones,

Proc. Roy. Soc.,

A106,441462,463477 (1924). See also

Silbey and

Alberty,

Physical

Chemistry,

Wiley, 2nd edition (2001), §§11.10,16.14, and 17.9;

and

S. Berry,

Rice,

and

Ross,

Physical Chemistry,

Oxford University Press, 2nd edition (2000), g10.2.