Journal of the Institute of Petroleum, Vol. 26, No. 198

V I S C O S I T Y A N D C O N S T I T U T I O N

P . < 1 C l ) L j O

J O U R N A L O F

T H E I N S T I T U T E O F P E T R O L E U M

F O U N D E D 19 1 3 I N C O R P O R A T E D 1 9 1 4

Vol. 26 APRIL 1940 No. 198

C O N T EN T S

P A C E

Dependence of Viscosity of Liquids on Constitution.

A. H. Nissan, L. V. W. Clark and A. W. Nash .

Institute Meeting

Obituary Abstracts Book Reviews Book Received Institute Notes

212 223 183

214

215

P u b lis h e d b y T h e I n s t i t u t e o f P e tr o l e u m .

E m e r g e n c y A d d r e s s : c / o T h e U n iv e r s it y o f B ir m in g h a m , E d g b a s t o n , B ir m in g h a m , 15.

P r i n t e d in G r e a t B r it a in b y R ic h a r d C la y a n d C o m p a n y , L t d ., B u n g a y , S u f f o lk .

A l l rights o f Publication or Tram latiotr a rt P r i c e

7

6

THE INSTITUTE OF PETROLEUM

C O U N C IL , 1 9 3 9 - 4 0

P R E SID EN T : Prof. A . W . N a s h , M .Sc.

A lfre d C . A d a m s Lt.-C o l. S. J. M . A u ld ,

O .B.E., M .C ., D .Sc.

Prof. J. S. S. Bram e, C.B.E., F.I.C.

The Rt. Hon. Lord C a d m a n , C .C .M .C ., D .Sc.

P A S T -P R E S ID E N T S :

T. D ew hurst, A .R .C .S . A . E. D unstan, D .Sc., F.I.C.

Sir Thom as H. H o lla n d , K .C .S .I., K .C .I.E ., D .Sc., F.R.S.

J. K e w le y , M .A ., F.I.C.

V IC E -P R E S ID E N T S : A s h le y C a rte r, A .M .I.M e ch .E .

C . D a ile y , M .I.E.E.

F. H. G a r n e r , Ph.D ., M .S c., F.I.C.

J. M cC o n n e ll S a n d e rs, F.I.C.

F. B. Th o le , D .S c., F.I.C.

M EM BER S O F C O U N C I L : G . H. C o x o n

A . Frank D a b e ll, M .I.M ech.E.

E. A . Evans, M .I.A .E .

E. B. E vans, Ph.D ., M .S c ., F.I.C.

W . E. C o o d a y , A .R .S .M ., D .I.C . A . C . H a rtle y , O .B .E., F .C .G .I.

Prof. V . C . Illin g , M .A .

E. R. R e d g ro v e , Ph.D ., B.Sc.

C . A . P. So u th w e ll, M .C ., B.Sc.

H. C . T elt, B .Sc., D .I.C . A . B e e b y Th om p so n , O .B.E.

A . W a d e , D .S c., A .R .C .S . W . J. W ils o n , F.I.C ., A .C .C .I.

C . W . W o o d , F.I.C.

J. S. Ja ckso n , B .Sc., F.I.C.

A rth u r W . Eastlake, A .M .I.M e ch .E ., H o n o ra ry S e cre ta ry

H O N O R A R Y E D IT O R : Dr. A . E. Dunstan H O N O R A R Y A S S O C IA T E E D IT O R : D r. F. H . G a r n e r H O N O R A R Y T R E A S U R E R .• T he Rt. H on. Lord P le n d e r, G.B.E.

S E C R E T A R Y : S. J. A s tb u ry , M .A .

T H E D E P E N D E N C E OE V IS C O S IT Y O F L IQ U ID S O N C O N S T IT U T IO N .*

B y A. H.

B.Sc., A.M.Inst. Pet., L. V. W.

Ph.D., M.I.Mcch.E., F. Inst. Pet. and A. W.

M.Sc., M.I.Meeh.E.,

F. Inst. Pet.

Sy n o p s i s.

In review ing tho litoraturo tho authors considered certain points to be accepted univorsally and took these as tho basis of their work. A n ultim ate flowing un it of liquid w as assum ed to m ove in “ jum p y ” fashion and energy to bo consumed in giving tho ultim ate un it tho necessary increment to enable it to “ ju m p .” This energy was represented by Q in 77 = A e(^ RT. The con­

nection betw een Q and tho boiling point 'Va w as found to bo more funda­

m en tal than the relationship botween Q and tho freezing poin t.

Tho ratio of m olal latont heat of vaporization to tho corresponding heat of viscosity was found to bo a constan t, n, characteristic of tho liq uid;

“ « ” appeared to afford tho koy to tho stu dy of v iscosity as it is affocted by constitution.

Tho authors obtained a formula connecting tho m olal laton t h eat of vaporization and the m olting p oin t of m onoatom ic liquids and concluded th a t tho three sta tes of m attor, solid, liquid, and gaseous, can be defined b y a continuous series of changes w ith regard to the laws of viscous flow.

T he relationship betw een viscosity and vapour pressure w as stu died in detail and tho conclusion reached th a t both phenom ena are functions of essentially the sam e variables. Tho value of “ n ” for «-paraffin was proved to bo fractional, 4-13.

Trouton’s rulo com bined w ith Clausius Clapeyron’s equation indicated th a t log 17 p lotted against T j T B for the «-paraffins w ould yield a single curve.

This w as called tho “ specific viscosity curvo.” A n artifice w as used whereby the temperature scalo w as so changed th a t log 77 p lotted against T / T b on this constructed scale yielded a straight lino. I t w as then proved th a t any liquid which yielded a straight lin e on these charts obeyed the samo law as did tho «-paraffins w ith regard to change of 77 w ith temperature. A ll tho 137 liquids tested , shown to represent m any hundred chem ical entities, yiolded straight lines. H ence it was concluded th a t a ll liquids, associated or unassociated, m ust possess only one formula connecting 77and T .

I t w as found th a t tho h eat of activation of viscosity was a function of temporaturo oven for «-paraffins. Thus tho chief distin ction between associated and unassociatod liquids from tho p oin t of view of viscosity disappears in principle.

The liquids studied fell into three classes :—

(1) U nassociated com pounds which fell in a fan-shaped manner betw een tw o lim iting positions.

(2) A ssociated liquids, tho viscosity’ a t th e boiling poin ts of which fell in a similar manner to (1) but tho “ specific viscosity curve ” of which deviated a t a more rapid rato to tho right.

(3) Tho elem ents.

In stu dying these liquids it w as found th a t th e chief variablo which characterizes tho “ specific v iscosity curve ” was tho molecular shape.

Molecular shape has been discussed and postulated os the shape of the equipotential surfaces surrounding tho m olecule. W hen tho m olecular shape w as nearly' sym m etrically spheroidal th e “ specific v isco sity curve ” was found to lie to th e right of th e curve for a liquid of which tho shape of tho equipotential surface was less spheroidal. D ipole m om ents wore found to bo of secondary im portance. The nature of the atom s of th e moloculo has been shown to be im m aterial; once a certain shape has been attained

* Paper received 27th May, 1939, and presented a t a m eetin g of tho In stitu te in London on Thursday, 14th March, 1940.

M

1 5 6 N I S S A N , C L A H K , A N D N A S H : T H E D E I ’ E N D E N C E O F

th e “ specific viscosity curve ” assum es a position on th e chart independent of the naturo of tho m olecule. The specific v iscosity curvo for CC14 was found to lie in th e lim iting position to tho right.

Various liquids were discussed in view of this hypothesis, and were found to support it.

Pyridine and dibonzyl afforded additional and moro definite proofs of tho hypothesis, as shown from tho X -ray analysis of theso molocules.

A ssociation w as discussed on tho sam e lines. I t w as found th a t association, like 77, w as a function of ( T j T ¡ ¡ ) .

V iscosity index was stu died from this poin t of view . A remarkable conclusion was th a t a liquid composod entirely of cyclic structure would y ield high V .I. if tho rings wero a ll joined end-on, so th a t tho m oleculo was elongated in shape.

Tho olomonts afforded im portant anom alies, and a stu d y of viscosity at the boiling poin t of theso elem ents showed th a t there w as an unsuspected rise in -q w ith rise in tem perature for all liquids. Thus tho chief distinction betw een gases and liquids w ith regard to viscosity disappears.

I t has beon shown th at ono formula m ust ex ist for all flowing m atter w ith certain term s predom inating, depending on the sta te of m atter and tho nature of flo w ; for exam ple, th e term defining fall of viscosity w ith rise of temporaturo predom inates in th e order : gases a t low pressure, gases a t high pressure, helium , m etallic elem ents, non-m etallic elem ents, unassociated liquids, associated liquids, non-N ew tonian liquids, plastics, and crystals.

T

viscosity o f a pure liquid is a function o f many variables, the most important being the nature of the liquid, the degree of its association, and the temporaturo and pressure at which tho viscosity is determined. In certain liquids the stress existing at the tim e of measurement is another important factor, whilst in all liquids specific volume appears to exert some influence. A study of the dependence of viscosity on any one variable requires that all other variables must either be eliminated or reduced to a constant datum level.

To facilitate the study of viscosity all liquids in non-Newtonian state o f flow have been avoided and viscosity values at atmospheric pressure only have been used. Two variables were therefore made constant. The elimination of the effects of temperature or the maintenance of this factor at a constant value is not so easy to forecast. Thorpe and Rodger,1 in their classical research on viscosity, assert, “ It seems futile to expect that any definite stoichiometric relations should become evident by com­

paring observations taken at tho same temperature.” When comparisons were attempted at a reduced temperature of 0-6

no results worthy of publication were obtained. Similarly a recent expression for a “ reduced temperature ” which yielded excellent results in comparing static pro­

perties o f pure liquids failed with the dynamical property o f vicosity.2 On the other hand, Dunstan and Thole 3 showed that the logarithm of viscosity is additive with molecular weight and structural characteristics

Again, the majority of viscosity-tomperature equations which have proved accurate are based on the absolute-temperature scale. Nevertheless, to assume that equal temperatures should prove a fundamental basis for the comparison of viscosity o f liquids boiling at various temperatures and of various critical constants appears at least an unsatisfactory foundation for this study.

To decide the correctness or otherwise of this assumption, and to under­

stand fully the fundamental variables which were found to affect viscosity,

a critical review of the literature on the subject was made. A list of the publications which were found useful for this purpose is given in the bibliography appended at the end of the paper.1-63 It is felt that a brief

of the accepted concepts regarding the nature o f viscous forces and the dependence of these forces on temperature would be useful here, particularly to determine the correct temperature levels for comparisons.

Various authorities have testified to the fact that the equation, vj =

where

viscosity in absolute units,

T

== absolute temperature at which the viscosity measurement is affected,

A

and

constants,

holds for a large number o f liquids to a remarkable degree of accuracy.

Thus it has been shown that it holds for metals, paraffins, ethers, alcohols, acids, acid anhydrides, bromides, chlorides, iodides, aromatic hydrocarbons, ketones, and esters, with deviations of the general order of Ibss than 1 per cent. (Dunn). Again, Andrade states, “ It is doubtful if tho experimental error is less than tho very slight divergence between the calculation and experiment.” Hence this formula became the basis— and the objective—

of many theories attempting to explain the nature of viscosity phenomena.

Again it was found that if the constant

is multiplied by

or

the gas or Boltzmann constant, respectively, the dimensions of the product are that of energy. Thus

B — Q /R

where

increment of energy of activation of viscosity.

.'. y] = A e « '* 1'

Several variants of this equation have been proposed, but in view of the remarkable accuracy of the simple formula stated here, and also in view of the many statements that the improvements introduced by using the complex variants are generally negligible except in the case o f such liquids as water, this formula will be used principally in this survey.

The variously accepted ideas about viscosity o f liquids have some broad similarities, although there are differences in detail. Thus it is postulated that a liquid “ at rest ” is composed of discreet particles, each of which oscillates about a temporary point of equilibrium. Occasionally, probably due to collision with others, some particles are possessed of greater energy than the average, and when this energy is o f greater value than a certain critical value, the particles possessing them take a jump in the direction of least resistance. Thus the mean position of equilibrium is transferred from one point in space to another. Since the liquid is not subjected to any external force and the distribution of the particles is at random, the direction of the jump is similarly at random. Therefore, on an average, as many particles leave a portion o f space as those coming into it per unit time, and the liquid remains “ a t rest.”

When the liquid is subjected to a pressure differential it is seen that the

direction of least resistance has now been influenced. Instead of being

equal in all directions, it has become increasingly difficult for a particle

to jump against, and easier to jump with, the direction o f the pressure

1 5 8 N I S S A N , C L A R K , A N D N A S H : T H E D E P E N D E N C E O P

differential. A greater number of particles would jump in a unit time towards tho lower pressure than towards the higher one, and flow ensues.

Some particles would jump in a direction across that of the main line of flow, but this point is controversial. Certain authorities maintain that the number of particles jumping from one layer to another is negligibly small.

Others are equally certain that tho mam cause for viscous drag is duo to such particles jumping from layers o f slow speeds to neighbouring layers of higher speeds and

In tho first hypothesis it will be required to impart to the particles the energy increment necessary before a sudden jerk is accomplished. W ith tho latter hypothesis the loss o f energy on collision has to bo made good by an external source, and thus a viscous drag manifests itself. Of recent years the greater number of publications assume that the energy is required for particles to jump in the direction of flow.

Shearing S tress

E N E R G Y P R O F I L E O F T H E P A T H O F A F L O W IN G M O L E C U L E .

Fig. 1 is a mathematical rendering of this physical picture (Ewell).

Ordinarily a molecule, or group o f molecules if necessary, can jump from right to left or

from one mathematical minimum of energy—

i.e.,

position of equilibrium— to another, passing through an activated state represented by the maximum on tho curve. When a shearing stress is imposed on the system in the direction of the arrow, the effect is to depress the mathematical minimum by a certain quantity, , in the direction of the arrow and raise it by the same quantity in tho opposite direction.

/ = tangential force per unit area.

X2 and >.3 = dimensions of the moving particle in the plane of the shearing force.

X = distance between two points o f equilibrium.

Thus flow ensues in the direction of the arrow as the system tends to attain equilibrium again.

On this basis the action is monomolecular. An equation obtained on these assumptions by Eyring and his collaborators, however, tended to show that there might be a bimolecular form of action as well. This fact is emphasized by the Andrade equation for the viscosity of molten mono- atomic metals at the melting point which assumes entirely bimolecular phenomena. The manner in which two molecules take part in a viscous flow is not certain, and different workers postulated different modes of action.

Fig. 2 is one example of such a mechanism (Eyring

Andrade assumes another, nearer the formation of a temporary structure in the liquid.

B IM O L E C U L A R M E C H A O T SM O F F L O W .

Probably higher orders of actions exist, too, in viscous flow. There is evidence, however, towards the postulation that at the melting point the action is probably mainly bimolecular, whilst as the temperature rises monomolecular flow predominates. A t the boiling point all flow is probably in a monomolecular form, but the fact that certain organic acids evaporate in the bimolecular form show that this generalization should not be taken too strictly. Even then, however, if the two particles forming a bimolecular complex movo in the monomolecular manner, the flow is monomolecular.

Irrespective o f the type of flow, it is certain that flow in ordinary liquids requires energy. This energy is represented by

in the equation

v) =

Attem pts were made to correlate

with the latent heat of fusion and the

latent heat o f vaporization. The most recent attem pt to correlate it

with the former is due to Bernal, who points out that the quantity

in

the simple exponential formula varies irregularly, but that

or the

“ viscosity entropy chango of the substance ” at freezing point, is, for each chemical typo of liquid, relatively constant. This is substantiated by the following table.

ICO N ISSA N , CLAKK, AND NASH : THE DEPEN D EN C E OF

B . B / T f x 10’. ^{B j T j ,} x 103.

NaCl . . . . 9 1 0 8-44 5-40

N aB r . . . . 8 - 0 0 7-79 4-81 T .

K ... 7-40 7-09 4-39

K B r . . . . 7-96 7-93 4-82

o ä ... 0-406 7-41 4-49 A ... 0-524 6-24 6-03

n2 ... 0-468 7-39 6-35 •Simple.

C O ... 0-463 7-00 6-46 c h4... 0-740 8-31 6-61 H g ... 0-598 2-56 0-94 N a ... 0-96 2-59 0-95 K . ... 1-15 3-43 1-16

P b ... 2-32 3-S6 1-33 1 M etallic.

C d ... 1-59 2 - 6 8 1-57 A g ... 4-87 3-95 2-25 Z n ... 2-92 4-22 2-47

For ionic and simple liquids the value o f

is G-2-8-4, whilst for metallic liquids the value is 2-6-4-2.

The authors have added the column for

or the viscosity entropy change at the boiling point, and it is seen that the agreements are as good as, if not better than those for freezing p o in t: 4-4—5-4 for ionic (7-1-8-4 for

6-0-6-6 for ordinary liquids (C-2-8-4 for

and 0-94-2-5 for metallic liquids (2-G-4-2 for

). This can be- explained on the assumption (Eyring) that while the metals flow without their valence electrons

in ionic form), the ordinary liquids flow as molecular units.

I t is significant that the fused salts have values intermediate between fused metals and ordinary liquids. (On this basis the. regularity of

becomes due simply to the fact that the ratio

is usually around 0-5).

The division into ionic, simple, and metallic types is more marked with

than with

This fact suggests a closer relationship between viscosity and boiling point (or vapour pressure) than with viscosity and freezing point.

Previously Dunn found that the ratio o f the heat o f vaporization to that of viscosity was a constant characteristic of the liquid and compiled the following list.

Compound. n = L I Q .

Paraffins . . . . . 4- 0

A cetone . . . . . 4-45

E thers . . . . . 4-11

Iodides . . . . . 4-39

Benzene . . . . . 2-75

Form ic acid . . . . . 2-62

M ethyl alcohol . . . . 3-62

W ater . . . . . . /"2-72 a t 0° C.

\ 3-3 at 100° C. approx.

Dunn attributed the variation of

with temperature for water to the association of this liquid.

Later Eyring and Ewell again found that the equation for viscosity could bo given by substituting — for

in the exponential term and

71

calculated “

” for varipus compounds. These workers state that “

” has a value of 4 (to nearest 0-5) for liquids composed of “ polar or elongated molecules ” and a value of 3 for liquids composed of “ spherically sym ­ metrical molecules.” A sharp separation of liquids into

= 3-liquids and

= 4-liquids is assumed as follows :—

n

= 3 CCI4, C0H 0, cycZoHexane, CH4, N 2, CO, A.

n =

4 n-CsH 12, n-CsH 14, »-C7H 16, CS2, CHCI3, C6H 5-C H 3, C2H 5-OC2H 5, C,H50 2C2H3l

J > 0 0 ’ C2H 6I, C2H 5Br, CH3T, C2H 4

No explanation is given for the fact that C2H 4CL2 and C2H 4Br2 have

= 3 |, although the fact that 0 2 also possesses

34 is “ attributed to its paramagnetic property.”

The

3-liquids CH4, CC14, and A have spherical fields o f force about the molecules (approximate, probably, in the case of CH4 and CC14), whilst N 2 and CO have probably the same shape of

potential energy shells,”

and the same is apparently true in the case of benzene and cycZohexane.

The concept is put forward that viscous flow is a form of vaporization in one degree of freedom for the

= 3-liquids. To explain the meaning of the last statement, reference should be made to the “ theory of holes.”

This theory is only an elaboration of the postulate put forward in the previous pages that a molecule or a combination of molecules require an increment of energy to move from one position of equilibrium to another.

It is stated in the theory of holes that the activation energy is to be used for providing a hole into which the molecule may flow; but this m ay not necessarily be equal to the volume of the molecule. The energy required to transfer a molecule from liquid to vapour without leaving a hole in the liquid is A

whilst 2 A

is required to move the molecule and leave the hole em pty in the liquid. Thus for a fraction of a hole — p- is needed,

n

where

denotes the ratio of the volume of the molecule to that o f the hole A

— i.e.,

A

= = = 2 5 \

I t appears that for spherically shaped molecules an activation energy of flow is the energy required to make a hole one-third the size o f a m ole­

cule—

A /ivnp /3— whilst for polar or elongated molecules, where certain preferred orientations are possible, a smaller fraction o f the energy of

\

i.e., n

3 in

Af?Tis_ = -— A2Er j.

When

assumes greater values than 4—as in the metals—the phenomenon

is well explained by maintaining that only a small fraction of the molecular

volume is being affected in each elementary process— i.e., the metals are

flowing without their valence electrons as ions (Ewell). This explanation

receives emphasis from the fact that the ratio

for m etals is a lower

1 6 2 NISSA N , CLARK, AND N ASH : THE D EPEN D EN C E OE

constant than

for ordinary liquids, whilst

for fused salts, which should consist of ionized metallic and acidic radicals, occupies an intermediate position.

A point o f paramount importance arises here. In view o f the close relationship of viscosity and vaporization phenomena—to be further illustrated below—should the liquid state bo treated as a continuous set of changes in series with the gaseous state ? Andrade believes th at in studying viscosity of liquids success is possible only if the investigator starts with the solid state. In fact, the equation this worker obtained for monoatomic metals predicting the viscosity at the melting point, assuming solid vibration frequency to ex ist in the liquid, is remarkably accurate.

v, p.p. = 5 -1 0 X 1 0 - 1

' A

where 7) p.p. = viscosity at the melting point,

atomic weight o f liquid,

melting point of liquid,

V A —

volume of 1 gram atom of liquid at

The authors succeeded in correlating this equation with that o f Eyring, who, unlike Andrade, did not assume that liquids are only connected with solids, but regarded viscosity as a manifestation similar to vaporization—

and believe that these strict divisions are, indeed, only apparent. Eyring’s equation is :—

Aim

■n — 1 -0 9 X IQ-3 _{f] l} A IV p m - 2 /3 — i — e~^>w* E C

* A ^ J ^ v a p .

Thus for viscosity at the m elting point

7]p.p. = 5-10

10“1-

_ i - 0 9 x 10-3 T m

- 1 0 9 X 1 0 { y Ay m AE^

Eyring has shown that this expression gives a value for

twice as large as that given by experimental data in the case o f anisotropic, and three times as large in the case of spherically symmetrical, molecules. Thus :—

T

Agrap.

V) F -P . = ■ 'iF .P . X 3 *

’¿ J.”' ' e"HTm taking a spherical monoatomic molecule—

argon.

Hence, if these assumptions are correct

T A£Vnp.

-*■ ttl--- ---

2.

3 A-k’yap.

X

should be equal to 1.

For argon

83-9 A

1505-7 and A

Hence (log -§ — (- — vSty ) = 1-89, or the value for the expression

\ v ap . ^ * AW c/t./ m/

is 0-78 instead of unity.

Substance. T m . dKvap,.

V ° g * A E r w . 1 13-7 2’„ /•

Argon . . . . 83-9 1505 — 0 - 1 1 instead of 0 - 0

H elium . . . . 0- 9 23-8 0-34 „

K ryp ton . . . . 104-1 2240 0-05 „

X en on . . . . 133-1 3200 0-18 „

The fact that the value of

in Erying’s equation is not exactly 3, his further correction of J is not definite, and that Andrade is uncertain about his constants, should be considered. Further, the value o f A2?vap. should be taken at

Thus the rule for monoatomic spherical liquids,

log ( £ ■ ■

-|- AA’top- = 0 0g \ J A

1 13-7

seems to bridge the vapour, liquid, and solid states, since phenomena characteristic o f each have been employed to deduce it. Since this rule was obtained from viscosity consideration, the authors believe that the three states o f matter are not as sharply divided in their manner o f flow as they have been maintained to be (cf. Andrade and Bernal for extremes).

This fact will be of greater significance later in this study when the laws of viscosity arc investigated. To summarize these studies :—

A t the melting point the elementary process appears to be m ainly a bimolecular phenomenon. The exact nature of the mechanism by which momentum is transferred from one layer to another or from one molecule to another is not certain. A knowledge of such a mechanism is not essential in deriving an expression for viscosity at the melting point. All that is required is to assume a bimolecular phenomenon, and that the chief energy possessed by the molecule is duo to vibration at a frequency almost identical

—if not exactly so— with that existing in the solid state.

As the temperature is raised, the molecules acquire sufficient excess energy to enable them to pass over a potential energy barrier— i.e., passing through an activated state— from one mean position o f equilibrium to another. This denotes a monomolecular form of motion. The bimole­

cular system does not disappear, and evidence points to the presence of such system s, only to a smaller extent than the viscosity at the melting point indicates. Thus with increase in temperature the monomolecular system supersedes the bimolecular one. It appears reasonable to assume, therefore, that at the boiling point the monomolecular elementary process is either the sole method or at least a major one. This is corroborated by the fact that m ost associated liquids distil over in the monomolecular form. Acetic acid, on the other hand, boils m ostly in the bimolecular form—but for the elementary process the two molecules would constitute one unit, and therefore the flow may still be considered monomolecular.

Nearly all the quantitative theories agree that loss o f momentum on collision is negligible (if present)—

gaseous type o f viscous drag is non-existent in liquids. The authors will deal with this point later in this study.53

The quantitative analysis o f the variation of

with temperature may be given in the general equation

y) = A eBIT

1 6 4 N ISSA N , CLARK, AND NASH : THE D EPEN D EN C E OF

Variants of this equation containing the specific volume add slightly to its accuracy.

Different workers have had an alm ost equal degree of success or failure in the approximate evaluation of the factor A. The chief factors affecting it appear to be the molecular weight and the temperature.

The value o f

appears to be connected with the latent heat of vaporiza­

tion and to a lesser degree of certainty with that o f fusion. Possibly the latter fact is due to the inter-relationship o f the heats of vaporization and fusion.

B

is the quotient o f a latent heat of viscosity,

divided by the gas constant,

m ay bo found by taking a fraction o f AEvap- the latent heat of vaporization thus :—

n

The constant “

is characteristic of the “ size and shape ” of the molecule. These terms should, however, bo defined more rigorously.

Thus to study the relationship of viscosity with constitution

appears to provide the key. Since the elementary process changes from the freezing to the boiling points from a predominantly bimolecular action to an almost entirely monomolecular action, the ideal method is to study the liquid over its entire temperature range. I f this is not possible, then the boiling point should provide tho next best comparative basis, since (1) the elemen­

tary process is probably of one type, (2) the value of A2?Tap is most ac­

curately known at

tho boiling point, (3) there is a minimum o f dis­

turbance due to association. I t is clear that comparing liquids at equal temperatures for this purpose is futile.

An assumption has been permeating these notes that

is invariant with temperature. Ward stated that where

is invariant with tempera­

ture the mean co-ordination is also invariant; whilst the mean co-ordina­

tion will change with temperature if

is found to change. Change of

with temperature can be easily seen by plotting log

a curve result­

ing if

changes, whilst a straight line results otherwise. This point is to be verified, as it is considered o f importance, although it is agreed that, in view of the accuracy of the simple exponential formula, any change of

with temperature appears to be of secondary importance for interpolating values of viscosity from a given set of experimental data for ordinary liquids.

Another point which has been implied throughout is tho characteristic decreaso o f viscosity with temperature— (at least the net result is a decrease).

Liquid helium was found by Keesom

to increase with temperature.

Th e Pr e s e n t Wo r k.

MATERIAL.

With the conclusions summarized above as the guiding principles, the

following list of substances were investigated with the object o f elucidating

the relationship of viscosity with constitution. Tho data for viscosity

will be found in the references indicated.“ -68

1. N orm al Paraffins,1-

- 57.

« -B u tan e to «-dodecane inclusive; » -te tra d c c a n c ; n-hexadecane an d w-octadccano.

2

. isoParaffins

. . . . 15 m em bers.

3. Acyclic Olefines

. . . 7 m embers.

4. Cyclic d e fin e

. . . . . cyc/oHexene.

5. Alkyl H alides

. 25 m embers.

6

. Oxygen-containing Compounds

25 m embers.

Alcohols . . . . .

m embers.

Aldehyde . . . . . Acetaldehyde.

K etones . . . . . . 4 m em bers.

A nhydrides . . . . .

m em bers.

E th e r . . . . . . . D iethyl ether.

Acids . . . . . . . 5 members.

W ater.

7. Cyclic Compounds

30 m embers.

8

. E lem ents 61> °5.

m embers.

9. Inorganic Compounds

m embers.

(The boiling points of these com pounds were found from th e litera tu re as cited in referen ces

and

tak in g th e figures given by th e investigators who gave th e viscosity m easurem ents where possible.)

A lthough only 137 liquids have been studied, it is th o u g h t th a t th e y represent th e behaviour of a far g reater num ber of pure liquids. More im p o rta n t th a n th is is th e fact th a t th e y include (

) unassociated liquids an d liquids associated to various degrees, (

) ionic liquids— i.e., fused salts—

an d m etallic liquids, (3) liquids of ideal spherical sym m etry such as argon an d liquids of various degrees of anisotropy, (4) th e principal representative organic stru ctu res and radicals, an d (5) elem ents and com pounds. T hus by com parison and co n trast th e effects o f such factors are expected to present them selves, especially as viscosity is one of th e m ore sensitive properties of such factors.

Exploratory and Confirmatory W ork.—A t th e o u tset of th is project th e degree of connection between th e vapour pressure (or vaporization phenomenon) and viscosity was investigated. F o r th is prelim inary work th e norm al paraffins were chosen, since th e y appeared to be least affected by secondary factors.

Thus th e viscosity o f th e norm al paraffins from n-pentane to n-decane was p lo tte d against th e corresponding vapour pressure as illu strated in Fig. 3 (For vapour-prossure d a ta cf. 61- °9-

) \

C ertain facts became e v id e n t:—

(1) I f tem peratures o f equal v apour pressure be accepted as correspond­

ing tem peratures, then, for norm al paraffins, tem p eratu res of equal viscosities are corresponding tem peratures.

(2) Since a single curve connects v apour pressure and viscosity irre ­ spective o f m olecular weight, th e factors responsible for, and th e laws governing, vapour pressure are identical w ith those affecting viscosity.

(3) I n a tte m p tin g to stu d y th e effects o f molecular stru c tu re and m ole­

cular forces i t is possible to elim inate th e disturbing effect of m olecular

woiglit. This fa c t is im p o rtan t, since b u t few of th e liquids to be studied have an ex act co u n terp a rt in m olecular w eight w ith th e rem ainder.

1 6 6 N I S S A N , C L A R K , A N D N A S H : T H E D E P E N D E N C E O F

Fia. 3.

E ach of th e above conclusions has previously been shown indirectly, b u t th e norm al paraffins show strikingly th e close connection betw een th e tw o phenom ena discussed. I t will bo rem em bered th a t plo ttin g log.

vapour pressure against log. viscosity yields a stra ig h t line for m any

R E C I P R O C A L O f V A P O U R P R E S S U R E O f P A R A F F I N S ( L O G . S C A L E )

F i g . 4.

substances, th e slope of which represents th e ratio betw een th e tw o h eats o f vaporization an d viscosity. I t is now evident th a t for ?i-paraffins above m-C

H

th ere will be only one line an d one “ n ” = The

A-&Yts.

au th o rs prefer to com pare log ■/] vs log

jP instead o f log

vs log P .

Fig. 4 shows th e stra ig h t line for th e norm al paraffins («C

"«C10). The value for n for norm al paraffins, or — .,vap', is 4-13. H ero it m ay be sta ted th a t th e au th o rs cannot agreo th a t th e sam e error was m ade in all these d a ta , so th a t 4-13 results in stead o f E y rin g ’s and E w ell’s integral num ber 4-0.

To decide definitely w hether exact integers or fractional num bers exist—

th e im portance of this p o in t will be discussed later— th e equation of th e stra ig h t line was determ ined and found to be :—

log y) (cp.) = 0-242 log (l/'P mm) or

)cp. = P ^ n

A t th e boiling p oint all th e «-paraffins have th e sam e viscosity. This value of viscosity is 0-194 cp. for a value of « = 4-0 and 0-201 for n = 4-13.

K uenen and W isser gave 0-208. Thorpe and R odger gave 0-198 for n- octane, 0-199 for «-heptane, 0-200 for n-pentane, an d 0-204 for «-hexane.

B ingham et al. gave 0-217 for «-hexane a t 65° C. (b. p t. 69° C.). T h u s, although th e difference is small, w hatever evidence th ere is points to th e correctness of th e fractional value of “ « .” (L ater riu for th e «-paraffins is found to be 0-2145 ± 0-005.)

A m ore definite course would be th e calculation of viscosity from the C lausius-Clapeyron equation for v apour pressure— i.e. :—

lQg © = - T - I' k ^ 1 - assum ing 1 = co n stan t’

where M — m olecular w eight

I = la te n t h e a t in gm .-cal./gm .

T t a , log ( I I ) _ _ 0-242 log ( A ) - 0-242 - ¿ ) where th e suffix B denotes boiling point.

0-242 M l 1 0-242 M l 1 or log y)! -

x

. T 2

x 1>gg . T

= A +

0-697

The ex te n t of th e error in th e calculated viscosity o f «-hexane justifies

th e conclusion th a t a non-integral value for « is quite accurate. As the

te m p eratu re deviates to a greater e x te n t from th e boiling p o in t th e error

in th e calculated value o f

) increases («-heptane and «-octane). T h a t this

is due to th e v ariatio n of I w ith tem p eratu re, an d n o t to an inaccurate

evaluation o f «, is proved by th e fact th a t th e m ore accurate form ula for

vapour pressure, in which th e la te n t h e a t is assum ed to v a ry w ith te m ­

perature, gives b e tte r results for these com pounds. T hroughout this

discussion it should be borne in m ind th a t H atschek has shown th a t Thorpe

and R odger’s figures for viscosity (m ostly used here) are n o t reliable in

th e value of th e th ird significant figure.

1 6 8 N I S S A N , C L A R K , A N D N A S H : T H E D E P E N D E N C E O P

Ta r le I ,

«-Paraffin. M. I. Tb. A. B.

V iscosity in cp. a t

0° C. 20° C. 50° C. 80° C. 100° C.

n-H exane 86 80-11 341-8 - 1-8853 406-16

Observed.

0-398 1 0-319 1 0-2415 1 0 1915 1 (0 1675) Calculated.

0-400 1 0-317 1 0-235 | 0-185 | 0-160 Difference.

0-002 | 0-002 | 0-007 j 0 006 | 0-0075

« -H eptane 100 76-3 371-5 -1 -7 8 5 5 404-4 Observed.

0-520 1 0-411 | 0-305 1 0-2335 1 0-1975 Calculated.

0-496 i 0-393 10-293 1 0-229 1 0-199 Difference.

0-024 | 0-018 10-012 | 0 004 | 0 001

n-O ctanc 114 70-95 398-7 -1 -7 7 2 0 428-67 Observed.

0-7015 1 0-5395 | 0-3845 1 0-28S5 I 0-243 Calculated.

0-627 1 0-491 10-359 1 0-277 1 0-238 Difference.

0-075 10-049 10-026 10-011 | 0-005

Tlius assum ing M lr = L r = L 0 aT

an d since j d loge P = d T

log

) = — 0-242 log P — ^ — B ' log T + C’

where A ', B ’, and C‘ m ay be calculated from fundam ental constants and th e values for l / n — 0-242 an d for t\D = 0-201. This equation for th e norm al paraffin is

log

] = 0-053 ^ a log T + p i p a log T

+ log 0-201 - 0-053 To calculate the values of L 0 and a in L T = L 0 -f- aT , th e d a ta supplied by Schultz’s graph for th e la te n t heats of norm al hydrocarbons a t various te m p eratu res were em ployed in th e assum ed form ula for th e change of this h ea t w ith te m p eratu re71. Table I I gives th e values for calculated viscosities obtained from th is equation.

T able I I shows th a t where th e m agnitude of the error in L T is n o t serious th e to ta l error due to “ n ” and other sources is of th e order

cp. m ax i­

m um , and h a lf th is am ount for the average. T hus a fractional value o f n appears to satisfy th e requirem ents w ithout an y im posed restrictions, bearing in m ind th e following sources of error :—

(

) A pproxim ate nature of the equation when applied over a wide range o f tem perature.

(2) A ssum ption th a t a is in v arian t w ith tem perature.

(3) E rrors in reading and estim ating th e la te n t h eats from Schultz graphs.

(3) E rrors (in th ird place) in rep o rted values o f

.

(5) A ssum ption th a t the com pounds are pure in b o th m easurem ents.

T a b i æ

II.

«-Paraffin. 7-e. a’ Tb. ^{00 5 3}1/0 A'.

0-242a (Const.)

V iscosities in cp. a t B \It _{0 ° C .} 20° C. 50° C. 80° C. 100° C.

n-IIcx an c 1*2,502 - 1 6 1 341-8 06 2 0 -1 -0 0 4 - 7-0122 Observed.

0-398 1 0-319 1 0-2415 1 0-1915 1(0-1675) Calculated.

0-398 10-312 10-233 10-186 10-104 Difference.

0-000 | 0-007 | 0-009 | 0-000 | 0-004

n-H cp tan c 14,077 -1 7 - 5 371-5 746-1 -2 -1 3 8 0 -8 -2 0 4 4 Observed.

0-520 1 0-411 | 0-305 | 0-2335 | 0-1975 Calculated.

0-548 10-415 | 0-297 | 0-228 | 0-198 Difference.

0-028 | 0-004 | 0-008 | 0-006 10-000

«-O ctane 16,060 - 2 0 308 851-2 2-4444 -0 -1 0 2 9 Observed.

0-7015 1 0-5395 1 0-3845 I 0-2885 | 0-243 Calculated.

0-757 1 0-551 i 0-376 10-279 10-237 Difference.

0-055 | 0-011 | 0-009 | 0-010

j

I t is obvious th a t no g reat value can be atta c h e d to calculations applied to higher m em bers o f th e series, since u n certain ties ab o u t L T grow p ro ­ p o rtio n ately . Two facts are th e n established :—

(1) I n th e case o f «-paraffins w hatever is tru e w ith regard to v apour pressure (or its reciprocal) is au tom atically tru e w ith regard to viscosity when tem p era tu re is th e fundam ental v arian t.

(

) N o necessity exists to ro u n d off th e value of “ n ” to a n integer.

E xam ining th e expression

iog ( h ) J _ * L ( . J _ _ ± ) b \ P j 2 . 3 B \ T 1 T j W hen P

= 760 rum. H g T 2 = T B — boiling point.

( m -

\760/ 2 . 3 B T „ \ T J

Since = const, by T ro u to n ’s ru le

B

B u t tak in g I a t T , log ( ^ ) = - - l )

or log == K " - l ) if = M S S = C onst

w here §S — en tro p y change on boiling a t T .

170 N I S S A N , C L A R K , A N D N A S H : T H E D E P E N D E N C E O F

T hus log (1/P) of th e norm al paraffins was p lo tted against values of T j T u a n d one curve was obtained for th e m em bers m-C

H

to «-C

H

inclusive in accordance w ith T ro u to n ’s rule.

Fio. 5.

T ro u to n ’s rule was applied to viscosity for th e following reasons :—

(1) A p lo t of log (•»]) against T nj T will yield a curve representing all th e m em bers o f «-paraffins for which T ro u to n ’s rule is operative and th e value o f n — 4-13.

(2) N o m em ber of th e paraffins has been studied over th e entire range of

th e liquid state. I f one curve represents all th e m em bers o f th e norm al

paraffins— excepting th e first few as indicated by E w e ll40—th e n this

curve will represent th e viscosity characteristic of these liquids from th e

freezing to th e boiling p o in ts ; since th e higher mem bers are studied near

th e first p oint, w hilst th e lower are studied nearer th e second point. The

m easurem ent tem peratures usually rep o rted in th e literatu re are from

0° to 100° C. The boiling p o in t o f n-b u tan e is near 0°, w hilst th e freezing

p oint o f «-octadecano is n o t m uch above it. T hus only «-C

to n-C

are necessary and sufficient to elucidate th e behaviour of th e norm al paraffins.

(3) Log v) vs 1 /T has been ta k e n to bo a stra ig h t line for unassociated liquids indicating th e invariancy of B w ith tem p eratu re in th e expression

v, = A e'»T

V IS C O S IT Y IN C E N T IP 0 1 5 E S

Fig.

. T ro u to n ’s rule applied here gives

° W 2-3n R T B \ T >

or log -r\ vs T jj/T should yield a straig h t line if B is in v a ria n t w ith te m p e ra ­ tu re . Fig. 0 is a p lo t o f log -q ag ain st TDj T for th e m em bers of «-paraffins ta b u la te d above.

Thus i t is clear th a t th e characteristic curve, while n o t curving exces­

sively, is definitely n o t a stra ig h t line. T h a t various authorities obtained

172 N I S S A N , C L A B K , A N D N A S H : T H E D E P E N D E N C E O P

stra ig h t lines can only be due to th e fact th a t th e tem p eratu re range studied was n o t sufficiently wide to exhibit th e v ariatio n of B w ith te m ­ p eratu re. This v a riatio n of B w ith tem p eratu re for oven such substances as th e paraffins clearly indicates :—

(1) A ll theories which attempt to explain the mechanism o f liquid viscosity which assume a constant B fo r unassociated liquids, and all formulae con- necting the variation o f viscosity o f liquids with temperature on the assumption that B is constant cannot represent the complete truth.

(2) T he strict distinction between unassociated arid associated liquids, based on the assumption that with the former B is constant, whilst with the latter it is variable with temperature, disappears. I n its place stands a relative rule only.

(3) Nevertheless the variations of B wth temperature fo r non-associated liquids are small, and m ay be ignored over narrow lim its of temperature fo r general purposes.

T he recognition of th e variatio n o f B w ith tem p eratu re is o f im portance in so far as it renders th e laws o f viscosity m ore general th a n th e y have been presented hith erto . This p o in t will be elaborated la te r on, in view of m ore evidence.

I t is clear th a t T b /T , or its m ore convenient reciprocal T / T B, is a fundam ental basis for com paring viscosities from th e point of view of stru ctu re,#since all th e paraffins fall on one line. Again th e function t is seen to be another characteristic function from T ro u to n ’s rule. Thus th e function (^¿/,) was p lo tted against (T jT n ) for th e liquids listed above, in Fig. 7.

B y th is m ethod th e liquids which were studied fell in to th ree divisions.

The shaded area represents all “ unassociated ” liquids and those “ a s­

sociated ” liquids which do n o t change th e degree of th eir association w ith tem p eratu re. (The «-paraffins represent th e first type, a n d acetic acid—which boils m o stly in th e bim olecular form —-represents th e second.) I n th e u pper portion fall th e characteristic curves for th e associated liquids.

W a ter does n o t appear to change its degree of association w ith te m ­ p era tu re to a g reat ex te n t, w'hilst trim eth y l carbinol does so. The lower p ortion o f th e graph is characteristic of liquids which are grouped under

“ m etallic,” in th a t th e y appear to flow w ith o u t th eir valence electrons.

F o r clarity, only certain representative liquids are shown.

T he e x act in te rp re ta tio n o f th e graph will be m ade clearer la te r; th e position of th e lino is controlled by th e value o f “ n ” a n d th e change of B w ith tem p eratu re. This fact will bo b e tte r understood w hen fu rth er details are studied, b u t it m ay be seen directly from th e form ula

assum ing th a t M lTj T = constant in accordance w ith th e generalized form

o f T ro u to n ’s rule. S trictly speaking, th e pressure should have been

varied from ono atm osphere w ith th e different m easurem ents, b u t change

o f viscosity w ith pressure is negligible a t low pressures an d for sm all pressure changes.

F

. 7.

B

P

.

A t this stage i t was clear th a t T { T B represented a rational, if n o t th e only basis, for com parison, for th e following reason. All th e norm al paraffins listed above fell on one characteristic c u rv e ; sim ilarly th ere were curves representing alkyl iodides (excepting th e m ethyl iodide), alkyl brom ides a n d chlorides, probably th e ketones and th e acids above form ic acid, alk y lated benzene above th e eth y l benzene m em ber, and pro b ab ly th e alkali halides. Since these groups are m ainly connected by stru ctu re, T j T B should be th e basis for investigating this phenom enon.

Sim ilarly it appeared th a t i t was th e logarithm ic function o f viscosity which was o f m ajor im portance— as predicted by D u n stan an d Thole.

T hus plots of log

) vs T j T B were m ade for all th e substances u nder stu d y . T j T B was preferred to T njT , since w ith th e la tte r th e degree of cu rv a­

tu re is sm all and m a y be obliterated by inaccuracy o f experim ent an d

plotting, w hilst in th e form er th e curve was m ore definite. Straight-line

AQ

/

functions have m an y advantages, and therefore th e following artifice was em p lo y e d :—

The greater proportion o f tho viscosity figures rep o rted in th e literatu re

Fio.

.

can only be relied upon to th e th ird decim al figure. Sim ilarly, only th e

th ird decim al in th e fractional value of T \ T B can represent significant

values. T hus all th e graphs used below have been chosen so th a t 500 m m .

represent values of T ¡T 2i from 0-500 to 1-000. Sim ilarly th e log -q values have been plo tted to a n accuracy corresponding to th e th ird decim al place.

I n Pig.

th e p lot of log

) vs T j T n for th e «-paraffins from n-C

to «-C

is shown to fall on one curve.

A stra ig h t line joins tho points T /7 '^ = 0-500 a n d T \ T B = 1-000. Since th e curve represents tho viscosity characteristic of a species of substances, tho curve will bo called “ Specific Viscosity Curve ” for «-paraffins. The stra ig h t line is th e n considered to represent th e viscosity characteristic of th e «-paraffins as p lotted, log r, vs a certain scale (on th e rig h t of th e plot) which is <f>(T/Tn). I n other words, a n artificial scale o f T ¡T 2, is to bo con­

stru cted so as to yield a stra ig h t line for tho «-paraffin “ specific viscosity curve ” when p lo tte d on it. This stra ig h t line is designated «-paraffin line.

The d o tted lines show tho construction of th e scale. A value of T j T 0 on th e n a tu ra l scale is connected by a horizontal line to th e specific viscosity curve. T he p o in t on tho «-paraffin line an d v ertically above th e in te r­

section o f th e horizontal line an d th e “ specific viscosity curve ” lias th e sam e value of log vj as th e point o f intersection. T hus a vertical line is erected a t th e p o in t o f intersection, and tho horizontal th rough th e p oint on th e «-paraffin lino represents tho value o f T j T ^ on th e artificial scale.

To read th e values of T ¡T B of a p o in t on th e artificial scale th e reverse is em ployed— i.e., draw a horizontal lino th ro u g h the p o in t to m eet th e «- paraffin line, drop a vertical line through tho intersection to m eet th e

“ specific viscosity curve,” draw a horizontal line through th e la tte r in te r­

section point to read T /T # on th e n atu ra l scale. Log

is com m on for b o th scales. Thus th e accuracy of plo ttin g points on th e artificial scale is governed by tho accuracy of reading tho n a tu ra l scale. All in terp o lated values are to be p lo tte d in detail in th e m anner described, for m ere in te r­

polation on tho scale is only guess-work, as th e scale varies in a com plex m anner.

I t is to be p roved th a t all curves which yield s tra ig h t lines on this artificial scale are governed by th e same equation as th e one governing th e basic curve— i.e., th e “ specific viscosity curve ” for norm al paraffins.*

T B — H eight from base o f graph representing T j T B on n a tu ra l scale

T r ’ = „ „ „ „ „ artificial scale

Vv

)

> % correspond to T R ', T St', a n d T K,' Since th e «-paraffin line is a stra ig h t lino,

m > m r m / r p /

-* Rj ~~ J- R2 __ -L R% R, log

!

log

j

" log

log

B y cross m ultiplication and cancellation

I o - t, - T .. ' ( loS Yn ~ l0g M T *' l0g

~ T r '

°

* \ T n ,' - T n; ) ’ T n; - T n,' loggia

* This work has been dono independently from th a t of Iran y/ 1 and tho tw o m ay bo looked upon as confirm atory, affording proof of each other’s conclusions.

The results Irany obtained w ith T I T C (T c = critical tem perature), and for which no explanation w as offered (i.e., for th e closeness of som o lines and n ot others), can all bo understood from this work, since T s / T , is generally In other words, T / Tb is tho real “ reduced tem perature ” and not V an dor W aal’s T ¡Tc in as far as viscosity is concerned (cf. Thorpe and Kodgers) .1