chapter 6 irreversible thermodynamics 6.0...

Chapter 6

IRREVERSIBLE THERMODYNAMICS

6.0 INTRODUCTION

We begin here the study of thermodynamics in the proper sense

of the word, by exploring a variety of physical situations in

a system where one or more intensive variables are rendered

nonuniform. So long as the variations in T, P, ~ or other

intensive quantities are 'small' relative to their average

values, one can still apply the machinery of equilibrium

thermodynamics in a manner discussed later. It will be seen

that the identification of conjugate forces and fluxes, the

Onsager reciprocity conditions, and the rate of entropy

production play a central role in the analysis provided later

in the chapter.

6. i SHOCK PHENOMENA

As our first illustration of nonequilibrium effects we

consider the case of shock effects in conjunction with Fig.

6.1.1.

(a) i: Let a piston be suddenly accelerated to a velocity

u, traveling to the right in the shock tube depicted in Fig.

6.1.i. Assuming steady-state conditions, the material to the

right of the piston moves along in the sam direction as the

524

525

p is ton

u

u

F' o .~p

ium moving at speed u

shock f r o n t

SHOCK PHENOMENA

u:O

P o

Co ~ medium at r e s t

FIGURE 6.1.1 lllustration of the motion of a piston and gas in a shock tube.

piston and with the same velocity. The compressed material at

pressure Po + Ap is preceded by a "shock front," ahead of which

is the undisturbed material at pressure Po and at rest. The

shock front moves with sound velocity c o and extends over the

region where P changes from Po to Po + AP. The density of the

undisturbed material is Po and that of the region behind the

shock front is Po + Ap.

(ll): Let an observer ride along the shock front down the

tube. He would see material 'entering' on the right with

velocity c o at a density Po, and ~leaving' on the left with

velocity c o - u at a density Po + Ap. The mass of material

processed in this manner must be conserved; per unit time, the

mass crossing unit area into the shock front, and the mass

leaving unit area at the back of the shock wave is given by

m t - poCo- (po + A p ) ( C o - u ) . (6.1.1)

This relation may be solved for

u - Co Ap/(po + Ap). (6.1.2)

(ill) " Next, we invoke Newton's Second Law of Motion" We

note that per unit time and cross section a mass m t of material

526 6. IRREVERS,BLETHERMODYNAMICS

changes in momentum from mt.O to mtu; according to Newton's Law,

this rate of change in momentum must be accounted for by a

force per unit area which changes from Po to Po + AP. Hence,

A P - PoCo u. (6.1.3)

We eliminate u from (6.1.2) and (6.1.3) and solve for

Co 2 == (Po + Ap) AP/PoA p. (6.1.4)

(iv)" Under the further assumption that all disturbances

are small, we set Ap << Po and AP/Ap- dP/dp, so that

dP/dp - Co 2. (6.1.5)

Assume next that the compression occurs so rapidly that the

material has no time to respond before it is transformed from

the undisturbed state to the steady state behind the shock

front. In this event the transformation occurs adiabatically.

(v): We now specialize to the case where the material

under study is an ideal gas. For the small disturbances

envisaged in (iv) the temperature is assumed to remain

constant, and under adiabatic conditions, PV v = constant, where

V" Cp/Cv. Since V- pol - - p-i the adiabatic condition may be

reformulated to read P- ApTo -- Po. Then

dP/dp - "fAPTo -i = ~fPo/Po, ( 6 . 1 . 6 )

and in view of (6.1.5),

Co - V'TPo/Po. ( 6 . 1 . 7 )

For an ideal gas, Po/Po = RTo/M, where T o is the temperature of

the undisturbed medium and M is the gram molecular mass.

Accordingly,

c o - 4VRTo/M. (6. i. 8)

SHOCK PHENOMENA ~ 2 7

Note how (6.1.8) may be inverted to determine ~ from a

m e a s u r e m e n t o f t h e v e l o c i t y o f s o u n d . I n a s m u c h a s V - (Cv +

R)/Cv, CV I R/(7- i) and Cp l 7R/(7- i) are directly available

f r o m s u c h m e a s u r e m e n t s .

(b) So far, we have assumed only infinitesimal departures

from equilibrium. We generalize considerably by allowing for

steady state conditions extensively removed from equilibrium;

this forces us to take into account severe excursions of T, P,

p, or v i p-1 from the equilibrium properties To, Po, Po, and v o

-i The situation may be visualized with the diagram shown PO "

in Fig. 6.1.2.

As before, we invoke the conservation law for matter, m t

being the mass of material that is being overtaken by unit area

of the shock front per unit time. Then, in analogy to (6.1.1),

mt l p o c l ( C - - U) p | (6.1.9)

where c is the velocity of propagation of the shock front. We

will later relate this quantity to co; the two differ because

with rising temperatures the propagation velocities increase.

As was done in conjunction with (6.1.3) we can set up an

equation based on Newton's Second Law of Motion:

P- Po- Po cu - m t u . (6.1.10)

u ~ C ~

To, Po,~Oo = Vo-

T ,p ,o : v -1 u=O, eo

u,e

Medium movinq at speed u Medium at rest Piston Shock f ront

FIGURE 6.1.2 Shock tube conditions under severe departures from equilibrium

S ~8 6. IRREVERSIBLE THERMODYNAMICS

Finally, we introduce the First Law of Thermodynamics as

follows: Let e o and e be the energies of the material per unit

mass of material being overtaken in unit time by unit area of

the shock front (for which m t - i); the difference e- e o in

energy, before and after the shock wave has hit, must reflect

any chemical reactions initiated by the shock. The change in

kinetic energy acquired by this material is u2/2; thus, we write

A e - e - e o + u2/2. (6.1.11)

The work performed by the piston on the material per unit

time and transmitted across units cross-section is Pu.

Assumln~ adiabatic shock conditions, the First Law of w

Thermodynamics then states that

Pu- = t , ( e - e o + u2/2). (6.1.12)

We have at hand now all the laws needed in our further

development; the rest is algebra.

First, solve (6.1.9) for c

c - [ p / ( p - p o ) ] U - [ V o / ( V o - v ) ] u . (6.1.13)

Second, divide (6.1.12) by (6.1.I0),

I 2 Pu/(P- Po) - (e- e o +~u )/u (6.1.14)

and solve for

1 e - e o -[[(P + Po)/(P - Po)] u2 (6.1.15a)

u2/2 for P >> Po. (6.1.15b)

Third, eliminate c in (6.1.9) by use of (6.1.13) and

simplify. This yields

SHOCK PHENOMENA 5 2 9

m t - [Vo/(V o - v) - l]up - vpu/(v o - v) - u/(v o - v). (6.1.16a)

for

Fourth, eliminate m via (6.1.I0) and solve the resultant

u 2- (P- Po)(Vo- v). (6.1.16b)

Finally, use (6.1.16b) in (6.1.15a) to obtain

e- e o - (P + Po)(Vo- v)/2, (6.1.17)

which is known as Hugoniot's equation. If we set h - e + Pv,

h e - e o + PVo, we may write

h - ho - (P - Po)(V + Vo)/2. (6.1.18)

(c) We now specialize considerably by dealing with the

perfect gas as a working substance. Then

PV - nRT - (m/M)RT, (6.1.19)

or

Pv - RT/M (6.1.20)

and

w.

e - cvT + constant - (Cv/M)T + constant, (6.1.21)

where M is the molecular weight, and Cv the molar heat capacity

at constant volume.

Use (6.1.21) on the left and (6.1.20) on the right of

(6.1.17)"

- IR (Cv/M) (T - To) - ~. ~ (P + Po)[ (To/Po) - (T /P ) ] . (6.1.22)

3 0 6. IRREVERSIBLE THERMODYNAMICS

Note the manner in which e has been eliminated in favor of C v.

We rewrite the above by defining a shock strength by ~ -

P/Po, in terms of which Eq. (6.1.22) becomes

Cv(T - To) - R(n + I) (T O - T/n)/2. (6.1.23)

Now collect terms in T and in T O �9

T(2C v + R + R/n) - (2~ + R + IIR) To, (6.1.24)

or

T 2Cv + R + fIR

To 2C v + R + R/If (6.1.25)

For II >> R this relation reduces to

T/To ~ [R/(2Cv + R)Ill. (6.1.26)

The factor on the right appears so frequently that we introduce

for it a new symbol, ~, - R/(2C v + R) - R/(Cp + Cv). We then

obtain

T/T o - (i + p,H)/(l + ~,/H) (6.1.27a)

~,H for II >> I. (6.1.27b)

At high T, Cv ~ 3R/2 for a monatomic gas, and Cv ~ 5R/2 for a

diatomic gas. Hence, T/To ~ n/4 or H/6 for monatomic or

diatomic gases respectively. Note the route we took to obtain

information on the rise in temperature when an ideal gas is

shocked and note that the asymptotic limits for T/To differ for

monatomlc and diatomic gases.

(d) On the basis of the above we can now establish a

considerable number of interrelations using various algebraic

SHOCK PHENOMENA 531

manipulations. For instance"

(1) We can find the ratio P/Po from

P/Po" (P/Po)(To/T) - 9(1 + #,/~)/(I + ~,ff)

- (n + ,.)/(z + ,.n) (6.1.28a)

i/#, for n >> i. (6.1.28b)

Thus, there exists a distinct upper limit on P/Po, of 4 and of

6 for monatomic and diatomic gases, for very large shock

strengths.

(li) Information on the mass flow velocity is obtained by

first using (6.1.21) to determine

e- e o - (Cv/M)(T - To) , (6 .1 .29)

and then using this result in (6.1.15a), eliminating T through

Eq. (6.1.27), and reintroducing H - P/Po. This yields

u 2 2~MC-~v]To[ (H- I)2] mm ~S'

H+~, (6.1.30)

which shows that there exists a connection between shock

strength and mass flow velocity in a perfect gas.

(ill) We may eliminate To for the undisturbed medium from

Eq. (6.1.30) by recalling (6.1.8) and noting that ~s - R/Cv(I +

~); on carrying out the indicated operation and taking square

roots of the resultant we find"

u___I 2 (n - z)~l ~2 c o ~(~ + i) ~ + ~, (6.1.31a)

(6.1.31b)

0.716 4~ monatomic gas, H >> I (6.1.31c)

5 3 2 6. IRREVERSIBLE THERMODYNAMICS

0.890 ~ diatomic gas, H >> I. (6.1.31d)

For 11 sufficiently large, u/c o > I; i.e. , the mass flow velocity

becomes supersonic.

(iv) Let us examine the ratio c/c o next. We begin with

(6.1.13)

C U m i

Co ( l - po /p)Co ' +oi t- [(t + ~,,n)l(n + ~,,)]

II+#a [-.~j - (n - i) (i - ~,,1

(6.1.32)

where we had used (6.1.28a). Eliminate u/c o in (6.1.32) from

(6.1.32). This yields

c I I + p ~ - - ~ ,

Co (n - 1)(z - ~,) 2 ( n - 1) z] z/2

~(~ + 1) (n + # , )]

2(II + #s) , , . . . .

-y(~ + 1 ) ( 1 - #, )z i12

(6.1.33)

This relation may be simplified by noting from the definition

of ~, and V that ~, - (V - l)/(V + I) and i - Ps - 2/(V + I).

Then

C / C o - r + 1 ) / 2 v ] ( n + ~,) (6. I. 34a)

for H >> I (6.1.34b)

0.895 ~, monatomic gas, H >> i ( 6. i. 34c)

0.926 ~, diatomic gas, II >> I. (6.1.34d)

A comparison of (6.1.34) with (6.1.31) establishes that c > u;

the shock wave will always outrun the mass velocity of the gas.

c/c o - M, is called the MaGh number.

FIRST AND SECOND LAW IN LOCAL FORM 533

(d) We can write a shock equation of state by defining P/Po

-vo/v" I'/. Then (6.1.28c) may be rearranged to read

n - (,7 - .,)/(I - ~,.). (6.1.35)

Compare this to the case of the reversible, adiabatic equation

of state ~ -77 and to the reduced isothermal equation of state

EXERCISES

6.1.1 Calculate the fractional rise in temperature for an ideal monatomic and dlatomic gas subject to adiabatic shock strengths H- I0, i00, i000. Compare with the fractional rise obtained under similar conditions for reversible adiabatic compressions.

6.1.2 The velocity of sound in water at 30 ~ is 1.528 m/sec. Find the compressibility K- (l/p) (dp/dP) at that temperature.

6.1.3 (a) Assuming adiabatic conditions to apply, derive an appropriate equation for the sound velocity c in terms of T for a gas at relatively low pressure. (b) Taking V- 1.41, and an average molecular weight M- 28.9 g/tool, calculate the sound velocity in air at room temperature and the change in sound velocity in air with temperature at 273 K.

6.1.4 Prove the following relations involving shocked materials :

m2- PPo (P- Po)/(P - Po) u2 - (P - Po)(P0 - P)/PPo ~,- (v- i)/(~ + z).

6.2 IRREVERSIBLE THERMODYNAMICS : INTRODUCTORY COMMENTS - THE

FIRST AND SECOND LAWS IN LOCAL FORM

(a) In the concluding part of our study we deal with the

phenomena of flow of matter and energy, thus initiating an

examination of thermodynamics in its literal sense. Any such

flow necessarily involves nonequilibrium states in which

different portions of a given system generally display


different physical properties. To deal with this situation we

subdivide the system into many subunits; in the limit when the

volume of each subunlt tends to zero, every intensive and

extensive property will have been specified as a function of

position. Each of the n thermodynamic properties 41 of a

uniform system will have been replaced by an instantaneous

field 4s(r) defined everywhere inside the boundaries of the

total system.

We render such a system subject to the scrutiny of

thermodynamics by establishing the Principle of Local State,

which makes two assertions: (i) The instantaneous values of

all thermodynamic quantities 4i at any given point satisfy the

same general thermodynamic principles and relations as the

corresponding quantities for a large copy of that small region

at that instant of time. This will permit us to extend the

thermodynamics of equilibrium configurations to the present

case. (il) The local, instantaneous gradients in 41, and their

rates of change, do not enter the description of the states of

each local system. This point addresses the fact that at any

point r all relevant parameters are likely to be characterized

by different values in contiguous regions, and hence, by a

gradient. Nevertheless, as long as the variation of ~i from one

region to the next is 'sufficiently small' (what sufficiently

small means must be decided by experiment), it may be left out

of account. This represents an assertion that is verifiable

only by appeal to experiment.

In general ~i will also depend on time; in such a case one

must specify not only 41(r,t), but also a corresponding velocity

function v(r,t) to describe a process. There do exist cases

where one wishes to treat the evolution of a system in a

restricted interval of time. If it so happens that the

properties of the subsystems remain unaltered, and only the

characteristics of the surroundings change, then the various ~i

remain independent of t in the time interval under study and

the system is said to have reached steady state conditions. A

more precise specification of this state is furnished in

Section 6.4.


(b) In discussing properties of inhomogeneous systems it

is conventional to represent extensive variables in terms of

specific quantities- that is, quantities per unit mass rather

than per unit volume. When such quantities are multiplied by

the density and integrated over the volume of the system one

obtains the total extensive variable. Let 4(r,t) represent a

quantity per unit mass whose distribution over the volume

element dSr is governed by the density function p(r,t). The

extensive variable for the entire system is then given by

@(t) - ~v P(r't)~(r't)d3r' (6.2.1) ( t ) - - "

where V is the volume enclosed by the boundary. One should

note that V in general may be a time variable quantity.

To determine the rate of change of @ (d@/dt -@) we must

take into account that not only the integrand but also the

integration limits change with time. Reference to Fig. 6.2.1

shows that the system whose initial boundary is schematically

indicated by the solid curve passes in time dt to a system

whose boundaries are schematized by the jagged curve; in

general this involves a deformation of the system due to a

center-of-mass flow with velocity v. The evaluation of d@/dt

may proceed in three steps. First, there is a contribution

from the central region V" encompassing the two overlapping

(v... t ;.

FIGURE 6.2.1 Change in volume of a system subjected to a barycentric flow. A" and A"" are the original and new boundaries adjacent to V" and V'".

5 3 6 6. IRREVERSIBLE T H E R M O D Y N A M I C S

volumes, drawn as a shaded region in Fig. 6.2.1. This

contributes a quantity dt~v, (8~p/at)dar to ~dt. In the limit as

dt ~ 0, and when keeping only first order terms, one may ignore

the difference between V " and V. Next, there is a contribution

due to a volume V" that is newly occupied in the deformation

process. This is composed of the elements- p~(v.n)d2rdt, where

n i s t h e O u t e r u n i t n o r m a l t o t h e s u r f a c e e l e m e n t d2r a t t h e

original boundary, and v is in the direction of the velocity of

m o t i o n o f t h e p a r t i c l e s as t h e y c r o s s f rom V" i n t o V ' . I t i s A

clear that p~(v.n)d2r represents the rate of transfer of

m a t e r i a l (more p r e c i s e l y , o f p~) a c r o s s t h e b o u n d a r y d2r i n a

direction normal to the element of area. When multiplied by dt

one obtains a volume element containing the material

transferred in time dt across the boundary along the direction

of flow. The minus sign arises because n is the outer unit

normal, whereas ~dt represents the _increase in ~ in the system.

(Thus, when v is precisely oppositely directed to n then a

positive contribution to ~ results in the amount dtp~vd2r. ) The

overall contribution to ~dt due to the transfer just described

L ^ is- dt - p~v.nd2r, where A" is the bounding surface separating

L ^ V" from V'. A similar contribution, - dt .p~v.nd2r, arises

from the volume V" relinquished in the transfer. The latter

two terms may now be combined into a single integral,

- dt~A(p~)v.nd2r, for which n differ in sign as well as in

magnitude as one passes from region V" to V" . It follows that

~A A ~V "' - - (p~)v-nd2r + 8(p~)d3 r (6.2.2) ( t ) - - - ( t ) at - '

and when Gauss' theorem is applied (see Table 1.4.2, line (j))

one obtains

I ~V {[a(p~)/at] - v.(~pv)} d3r. (6.2.3a) ( t ) - - - -

Equation (6.2.3a) is often referred to as the Reynolds

transport equation.

FIRST AND SECOND LAW IN LOCAL FORM 5 3 7

$ Note that if we write I Jv(t)(dp~/dt)dSr then Eq. (6.2.3a)

can be cast in the local form

d(p~) I a(p~)_ v.(p,~v) (6 2 3b) dt at - " "

for any specific ~ which is an extensive quantity per unit mass

of the system. It should be evident that if ~ is chosen as an

extensive quantity per unit volume rather than per unit mass

then the density factor p may be dropped from Eq. (6.2.3b).

Equation (6.2.3b) is typlcal of the form encountered

whenever conservation laws apply. The overall rate of change

of the quantity p~ i R, i.e., dR/dr, is governed by two terms:

(1) The rate of production (or disappearance) of R locally,

i.e., 8R/at; this term can be traced to the occurrence of

processes totally within the system, without referring to flows

across boundaries. (il) The balance between influx from or

outflow to the surroundings, as expressed in the divergence of

the flux vector _V'~, with J~R i Rv,_ which was earlier related to

flows or transport across boundaries, as specified by Stokes'

Law. One should note that (6.2.3b) or its equivalent

d /dt - ( a R / a t ) - v . ( 6 . 2 . 3 c )

is much more restrictive than (6.2.3a), in that one demands a

balance between influx, outflow, and rate of generation or

dissipation, not only for the system as a whole, but on a

polnt-by-polnt, local basis. A relation of this form is

designated as an equation of continuity.

In certain physical situations the quantity R is

indestructible or uncreatable: It can neither be generated nor

destroyed locally. In such circumstances 8R/at, the rate of

local generation or annihilation of R, must necessarily vanish.

For this special case

d R / d t - - V-JR, ( 6 . 2 . 3 d )

5 3 ~ 6. IRREVERSIBLE THERMODYNAMICS

which implies that local changes in R can be brought about

solely by a net change in the balance between influx and

outflow, as expressed in divergence term - ['JR_. Eq. (6.2.3d)

is a conservation equatlon..

(c) We next turn to the establishment of the First and

Second Laws in local form. This will be done under the

important restriction that there be no motion of the center of

mass of the local system" v- O. In these circumstances no

volume deformations need be considered" dV/dt = O, thereby

greatly simplifying the analysis, while not unduly restricting

its applicability. Reference to a generalized treatment of the

problem with v ~ 01 shows that the present restriction involves

the dropping of a P(dV/dt) term from the local formulation of

the First Law, and of a tensorial pressure-volume term

appropriate to anisotropic media from the local formulation of

the First and Second Laws. However, since we will continue to

confine our studies to isotropic media, and since in the steady

state dV/dt vanishes, all results cited later are attainable by

the present, more restricted approach. We divide the

subsequent discussion into several subunits.

(i) Consider a system in which n different chemical species

k (k- I, 2,...,n) are subject to r distinct chemical reactions

(~ - 1,2,...,r). In an extension of the procedure of Section

2.9 we now represent the ~th chemical reaction by X(k)VktAk -- O,

where the Vke are the appropriate stoichiometry coefficients

matched to the various chemical species A k in reaction ~. These

quantities may have either sign, as discussed in Section 2.9.

Then, if AI represents a unit advancement of reaction ~ we

designate the rate of advancement of this reaction by ~i -

dAi/dt. The product PkVki~i then represents the mass production

or depletion rate per unit volume of species k in the ~th

chemical reaction. Furthermore, we allow for the fact that

each species may be subject to any conservative external

IS. R. de Groot and P. Mazur, Nonequilibrium Thermodynamics (North-Holland, Amsterdam, 1962).

FIRST AND SECOND LAW IN LOCAL FORM S 3 9

force per unit mass fk --- _V~k, where ~k is an appropriate

time-independent potential per unit mass. It should be noted

that while v- 0, individual particle flows do not vanish; in

fact, we write the mass flux vector for species k as Jk m PkVk,

where Pk is the density, and v k the average drift velocity per

unit mass, of the kth species. Note that in the present case

ECk)PkV_k/P -- I - - O.

( i t ) We now t u r n t o an a n a l y s i s o f t h e l a w o f c o n s e r v a t i o n

of mass. The time rate of change of mass for a given species

within a fixed volume is equal to the net rate of influx across

the bounding surfaces plus the net rate of generation of k in

chemical reactions. Thus,

r

(dPk/dt)d3r --- Pk[k "nd2r + X PkVkeWt d3r, ~--I

(6.2.4)

where the minus sign occurs because n is the unit normal vector

directed toward the outside. Observe that for some species k

the last term in (6.2.4) is positive and for others it is

negative; this balances out in such a manner that there is no

net formation or disappearance of total mass in the reactions

that occur locally. On introducing Gauss' Law (line (j), Table

1.4.2), Eq. (6.2.4) reduces to

r

d P k / d t 1 - V . J k + ~ PkVk,~,, s

(k- i, 2, ..., n). (6.2.5)

(ill) We next construct expressions for energy balance.

Since by assumption v- 0, the kinetic energy term for motion

of the center of mass vanishes. The rate of change of the

total potential energy density p~ m Z(k)Pk~, so long as one

considers only time-independent potentials ~k, may be determined

from (6.2.5) by multiplication of both sides with ~k and

subsequently summing over k"

S 4 0 6. IRREVERSIBLE THERMODYNAMICS

dp~ dt

n

--z-( + x k-I k-I

r n (n + I X Pk~"k,' , - - V. I ~_Jk

i-i k-i k-i

n

- I Jk'fk k-I

(6.2.6)

Here we have written V.(~Jk) -- ~V.Jk + Jk'V~ and we have

noted the fact that Z(k)Pk~Vkl--0 because the total potential

energy of the system is conserved in every chemical reaction.

In the preceding discussion and in what follows below we

have introduced to the bar symbol to emphasize that we refer to

a specific (i.e. per unit mass) quantity. Note that Jk is a

mass flux vector; although ~Jk has a strange appearance this

quantity is dimensionally correct" Jk- PkVk, whence ~Jk -

#k~/k iS the rate of transport of potential energy density.

(d) As a further step we introduce the First Law of

Thermodynamics which requires that the total energy of the

local system be strictly conserved. This means that any change

of total energy U can occur solely through influx or outflow of

energy across boundaries. This flux will be characterized by

the flux vector Ju, which corresponds to the energy density pu; m

Ju - puv. The total energy balance equation thus reads

..Iv{ (dpu)/dt}d3r - - JAJu-{~dZr; with the aid of Gauss' Theorem we

then obtain the strictly local equation for conservation of

total energy density:

d(p~)/dt - - V.J u. (6.2.7)

(iv) We now introduce the distinction between total

specific energy u and the internal specific energy e by the

relation u- e + @; if the center of mass of the system were in

motion then an additional kinetic energy term would have to be ~ . a m

added. The distinction between u and e may perhaps be most

readily grasped by an example. Consider a system placed in

a gravitational potential" Changes in its energy are then


specified by d(Mgz) - gzdM + Mgdz. The quantity on the left

refers to the overall change of energy du. The first term on

the right corresponds to a case where additional mass is

transported from infinity and accreted to the system at the

location z; the internal, thermodynamic coordinates of the

system have now been altered, leading to a change in internal

energy, de. The second term, Mgdz, corresponds to a

displacement of the entire system in the gravitational

potential #s; thus the potential energy of the entire system is

uniformly altered, but the thermodynamic coordinates of the

system remain unchanged. This latter alteration is symbolized

by the quantity d~s. In thls elementary example d(Mgz) - gzdM

+ Mgdz corresponds to d(pu) - d(pe) + d(p~s).

(v) The First Law of Thermodynamics is frequently written

out in terms of the quantity pe rather than pu, and this is the

path we shall follow here. We specify the total energy flux as

n

k-I (6.2.8)

The reason for the JQ nomenclature is that we have consistently

regarded the net energy flux density V-J u as arising from the

performance of work, the effects of potential energy changes,

and heat flow. The first two contributions are contained in

the term V-~Ck)~Jk; the remaining energy flux must thus be

identified as a net heat transfer _ V.JQ, involving the

corresponding flux vector JQ.

Let us subtract (6.2.6) from (6.2.7); this yields an

expression for the rate of change of internal energy density"

n

d(p~) -_ V-JQ + ~ Jk-fk. (6 2 9) dt - - " "

k=l

Equation (6.2.9) represents the local formulation of the First

Law of Thermodynamics when there is no motion of the center of

mass. Note again that Eq. (6.2.7) and (6.2.9) are not averaged

542, 6. IRREVERSIBLE THERMODYNAMICS

over a finite volume but must be obeyed locally. Also, Eq.

(6.2.9) does not satisfy the conservation law except when fk m

0 for all k. This situation arises because e is not the total

energy density unless the system is free from all external

forces.

(e) The Second Law is handled by recourse to the Gibbs

equation (1.18.34) in the form (for dV-0) TdS-dE + ~(k)~kdnk.

Set dS - d(sV) - Vd(p~), dE - Vd(p~), and write

~k - ~kMk, nk - n~/Mk - pkVlMk (6.2.10)

Then

T d (p~) d (p~) dPk - dt + ~ ~k dt " (6.2.11) dt

k

Finally, introduce (6.2.5) and (6.2.9) into (6.2.11); this

yields

n r

- + z + x + z k k-i ~=i

where A e -- ~(k)PkVki~k is called the chemical affinity. Note

that 3 k and ~e refer to specific quantities. However, the above

quantities occur exclusively in combinations such as 3.fk, ~kJk,

and ~tNe; this permits the following alternative interpretation"

3 k is the flux in moles of k past unit cross section in unit

time, ~k is the usual partial molal Gibbs free energy, fk =

--V~ is the negative of the potential energy gradient per mole,

Vke~ ~ is the rate of molar concentration change of species k in

reaction ~, and A t the corresponding affinity. These quantities

will be used interchangeably.

We now rewrite (6.2.12) as

FIRST AND SECOND L~W IN LOCAL FORM 543

d(pg)_dt V-[(JQ- n ) j n

~kJk /T - (I/T 2) JQ.VT + (I/T 2) ~. ~kJk .VT k-I k-I

n r

- ( I / T ) ~. JT k.(~T~k - fk) + ( l / T ) ~. ~,A,. k-i ~-i

(6 .2 .13)

This relation can be split into two types of contributions: The

first term in (6.2.13) involves the divergence of the flux

T-I(_JQ - ~.(k)~kJk). In the context of Eq. (6.2.13) it therefore

clearly makes sense to define an entropy flux vector by the

following relation:

n

Js " (I/T){Jo - ~- ~kJk} �9 k-I

(6.2.14)

The remaining terms on the right of (6.2.13) must represent

source terms if Eq. (6.2.13) is to be interpreted as an entropy

balance equation d(p~)/dt-- V.J s + 0. Having thus identified

- V.J s we can express 0 as the rate of entropy density

generation locally as follows"

n r

- - (l/T) Js'_VT - (l/T) ~. Jk'_V~k + (l/T) ~. wtA, _> 0,

k-i ~=i (6.2.15)

in which we have written _Vk~k -- _fk -- V(~k + ~k) -- V~k, thus

recognizing that the specific chemical potential P-k and the

external molar potential ~ can be combined into a specific

ge.neralized chemical potential ~'k" Equation (6.2.15) is of

great fundamental interest, as will be demonstrated later.

Note that we have succeeded in setting up a continuity

equation for entropy density, the Second Law, in local form,

[d(ps)/dt] - - V-J s + 0, (6.2.16)

which should be contrasted with (6.2.7). We have also achieved

S 4 4 6. IRREVERSIBLE THERMODYNAMICS

an important separation. The term- V-J s specifies the net

transfer of entropy density across the boundaries of the local

system, whereas 8 refers to the rate of entropy density

generation due to (irreversible) processes occurring totally

within the local volume element. In subsequent sections this

latter quantity will play a cardinal role.

(f) A reformulation of (6.2.15) may be achieved by first

eliminating JQ between (6.2.8) and (6.2.14); one obtains

n

Ju - - I rk_J - k-1

(6.2.17)

In the case of a single species (k- I) the ratio Ju/J1 may be considered as the total energy transported per unit mass of

species i, UI, likewise, Js/J1 is the total entropy carried per

unit mass of species I, S I. Thus Eq. (6.2.17) specializes to

- -*- TS~- ~i which is an U I-* -TS~ [i, or to its equivalent, E l ,

analog of H- TS - Pl. Now substitute (6.2.17) into (6.2.15);

then a slight rearrangement yields

n n

- - (I/T 2) Ju'VT + Z (rk/T2)Jk'VT -- (I/T) Z Jk'V[k k-I k-i

r + ~ (I/T)~,A,

~-I n r

-Ju'V(I/T) - 7. Jk'V(rk/T) + ~ (I/T)~,A, _> 0. (6.2.18) k-i ~-i

The form of (6.2.15) and (6.2.18) is highly significant.

In each case the rate of local entropy density generation, due

to irreversible processes occurring totally within a local

volume element, may be written as a sum of terms of the general

form 8 - E(j)Jj-X_~ >_ O, wherein the Jj represent either general

fluxes or reaction velocities, and the X_~ represent generalized

forces. As already explained in Section 2.2, this nomenclature

arises because 8 - 0 can only occur when equilibrium prevails,

LINEAR PHENOMENOLOGICAL EQUATIONS 545

m

at which point both X j and Jj go to zero for all J. In this

sense Jj is a response to the imposition on the system of an

external force X_j. Of great importance to our future

development of irreversible thermodynamics are the particular

forces and fluxes that occur pairwise in the expression 8 -

~.(j)Jj-X_j. Such pairs are said to constitute conjugate

variables. In Eq. (6.2.15) these pairs are respectively (J,/T,

-VT), (Jk/T,-~k), (wt, Ae/T); in Eq. (6.2.18)they are (Ju,

-VT-I), (Jk/T, ~r~k/T), and (wl, A,/T). It is therefore obvious

that no unique set of such pairs may be set up; the selection

of a set as a starting point for further development is then

simply a matter of convenience.

Since a flux may be considered to be ~driven' by a

corresponding force, no flux can occur without a force field,

in which case all irreversible phenomena cease; 8 now vanishes,

and Eq. (6.2.16) becomes a conservation condition for entropy.

EXERCISES

6.2.1 Derive the equation of continuity for a system of constant mass in the form dp/dt + V.pv- O.

6.2.2 Explain why for reactions occurring totally within a system ~ does not contain terms in reaction velocities or affinities [see Eq. (6.2.9)], whereas ~ does: see Eq. (6.2.13). Hint: Consider the possible sources for reaction energies, and how such reaction energies would be dissipated.

6.2.3 Derive the equation r

dpNk/dt - - l'_Jk + Z PkUk~k k-1

and explain its relation to the equation of continuity. 6.2.4 In Eqs. (6.2.8) and (6.2.9) JQ was identified as a

heat flux vector, yet this quantity corresponds to e, the internal energy density. Consult Sections 1.8 and 1.16 and explain again why this particular designation is appropriate.

6.3 THE LINEAR PHENOMENOLOGICAL EQUATIONS, AND THE ONSAGER

RECIPROCITY CONDITIONS

(a) We had earlier derived an equation for the local rate of

entropy density production when no volume changes occur:

546 6. IRREVERSIBLETHERMODYNAMICS

n r

- (Sps"/ST) -- T-IJs.VT - T -I ~ Jk.V[k + ~ ~,(A,/T) _> O.

k-I 2-1 (6.3.1)

In circumstances where no chemical reactions take place and

where no particle fluxes occur 8 - - T-IJs.VT _> 0. One can have

- 0 only if equilibrium prevails, in which case Js and VT both

attain a value of zero. In the absence of any particle flux

the quantity T-IJs represents the heat flux JQ/T 2, as is evident

from (6.2.14). Equation (6.3.1) now reads 8 - - T-2JQ-VT -

JQ.V(I/T). According to our standard interpretation JQ is

'driven' by a gradient in I/T. In the simplest approach to the

problem one postulates that JQ is linear in V(1/T) ; moreover,

the linear relationship must be homogeneous, so that no

additive constants prevent JQ from vanishing simultaneously with

V(1/T). The postulated relation

JQ- I..V(1/T) (6.3.2a)

should hold under conditions not too far removed from

equilibrium. Here L- L(T,p) is a scalar function of the

temperature and density; one cannot introduce a tensorial

quantity, as this would generate a set of preferred directions.

Equation (6.3.2a) may be rewritten as

JQ - - (L/T 2)VT - - ~VT, (6.3.2b)

which represents Fourier's Law of Heat Conduction (1818). We

have thus recovered a well-known law, which attests to the

correctness of the procedural methods adopted here. The

quantity ~ is known as the thermal conductivity of the medium.

The same argument may now be repeated for the case where

again no reactions occur, the temperature is held constant, but

now a particle flux of one type is permitted. Equation (6.3.1)

reduces to

= - _> o. (6.3.3)

LINEAR PHENOMENOLOGICAL EQUATIONS 54" /

We now revert to the use of molar quantities for which Ji.V~i -

(ml/V)v.V(Gi/m i) - (nl/V)v.V(Gi/n i) -Ji.V~i. We shall omit the

tilde for ease of notation. The form of (6.3.3) suggests that

V~i is a driving force to which T-iJi is the responding

conjugate molar flux for species i. Again, a linear relation

is invoked, of the form

Ji - L" (T,p)V~i , (6.3.4a)

which connects the force and flux. The extra factor T -I has

been absorbed in the definition of L'. Equation (6.3.4a) is in

the form of Fickes Law (1856) for diffusion, usually written in

the form

J - - D Vc, (6.3.4b)

where D is the diffusion coefficient and c, the concentration

of particles. The Eqs. (6.3.4a) and (6.3.4b) are equivalent;

for in the absence of electric field, or when only uncharged

particle flows occur, ~ - ~ - ~ + RT ~n c. Thus, V~ -

(RT/c)Vc, so that D - - (L'RT/c).

On the other hand, if the particles are charged, then one

may recast Eq. (6.3.4a) in the form Je = L"(T,p)V(~/e), where

-e is the electronic charge, and Je is the charge flux; Je-

- eJ. Now write V(~/e) - V(~/e) - V~, where ~ is the chemical

potential and ~ the prevailing electrostatic potential; such a

shift back from molar to atomic quantities is considered in

Exercise 6.3.2. Suppose further that the charge carrier

concentration in the system remains uniform. In that event one

finds

_fl'e - L " ( T , p ) ( - V~) - L " ( T , p ) E - a E , (6.3.4c)

where E is the electrostatic field. Equation (6.3.4c) is one

formulation of Ohm' s Law (1826) ; a is the electrical

conductivity.

~48 6. IRREVERSIBLE THERMODYNAMICS

(b) These examples suffice to illustrate the more general

concept of a set of processes for which the rate of local

entropy density production has the form 8 - ~(J)JJ'X_0, where the

Jj are all the relevant fluxes and the Xj, the corresponding

forces. Pairs of variables Jj, X_j, satisfying this particular

form for 8 are said to be conjugate. For every such pair one

postulates a direct proportionality of the form J• - LIjX_0

between the various conjugate Ji, Xj pairs. The validity of the

llnearlty principle ultimately rests on a comparison with

experimental data and in no way invokes a new principle of

thermodynamics. However, in lowest order of approximation, a

microscopic theory of heat and of mass flow does lead precisely

to such linear relationships.

When more than one force at a time acts on a local system,

a corresponding number of flows must occur simultaneously. In

this event one enlarges on the original postulate by writing

down simultaneous equations of the form

Jl- LIIXI + LIzX2 + " ' " + Lln_Xn

Jn- + 2x2 + . . . + (6.3.5)

These relations are known as phenomenological or macroscopic

equations, in which the various Jj, Xj satisfy the relationship

- E(j)Jj-X_0 for i <_ j <_ n. E-very flux is accorded a

phenomenological equation of its own, which involves additively

every force acting on the system as a whole. The result is a

linear superposition in which each force influences each of the

fluxes. The class of coefficients Lij inclusive of L• are

known as phenomenological coefficients or macroscopic

coefficients. Those for which i- j connect the conjugate

flux-force pairs; they are termed proper coefficients. The

remainder provide cross-coupling effects between forces of one

type and fluxes of another type and are known as interference

coefficients. The ultimate validation of Eq. (6.3.5) again

LINEAR PHENOMENOLOGICAL EQUATIONS 549

rests on comparing predictions based on Eq. (6.3.5) with

experiment.

(c) In conjunction with Eq. (6.3.5) we introduce a set of

interrelations, known as the Onsager Reciprocity Conditions

(1931)"

Lij - Ljl. (6.3.6)

These reciprocity relations are derived in Section 6.5. It

should be clearly recognized that Eq. (6.3.6) holds only if the

phenomenological relations involve conjugate fluxes and forces.

If nonconj ugate quantities are used, Lio and Lol are

functionally related but no longer equal. In the presence of

magnetic fields H or angular rotations ~ this principle must be

enlarged to the following form"

LIj(H,~) - + Lji(- H,- ~), (6.3.7)

where the sign is determined by whether the phenomenological

forces X i or Xj do or do not change sign when all microscopic

velocities of particles are reversed; reversals of X with

velocities call for use of the minus sign. Equation (6.3.7),

known as the ~Casimir-Onsa~er relation (1945), is here taken to w

be justified on the basis of empirical verification, but may be

derived from statistical theories.

EXERCISES

6.3.1 Consider a gas at pressure P separated by a frictionless piston from a second gas at pressure Pa. The piston is now released. Determine the rate of entropy production and identify the force and the flux.

6.3.2 Show that the molar flux of charged particles obeys the relation J1- LIIV(#I + F4), where F is the Faraday. Define current flux and show that it is indeed possible to write J1 as proportional to V(~/e + 4).

6.3.3 Verify that the equation for charge flux can be formulated on an atomic or a specific or a molar basis.

~ 6. IRREVERSIBLE THERMODYNAMICS

6.4 STEADY-STATE CONDITIONS AND PRIGOGINE'S THEOREM

Steady-state conditions are characterized by the requirement

that the fluxes and forces characterizing irreversible

phenomena in a system be independent of time. Some of these

forces and/or fluxes may vanish; in the extreme case where all

Ji and X i are zero the system is necessarily in an equilibrium

state.

The above, intuitive concepts may be made much more precise

with the aid of Pri~ogine's Theorem which characterizes steady-

state conditions as follows" Let irreversible processes in a

system be characterized by n independent forces X I, X 2, ...,

and corresponding fluxes J1, J2, ..., Jn. Let the first k

forces be kept at fixed values X_ -~ ~0, ..., ~. Then a state of

minimum entropy production 0 is r-cached w-hen the particular

fluxes Jk+1, Jk+2, -.., Jk+n, all vanish.

We first prove the assertion and then discuss its relevance

to steady-state conditions. As usual, we write 0 - l(•177

from which one sets up the phenomenological equations

Ji - ~(o)LIjXj, Lij - Ljl, (6.4. i)

and the relation

n n

"- ~ ~ LIjX_-I'X_0 _> O. 1-i j-1 - -

(6.4.2)

An extremum is found by differentiating 0 with respect to the

nonfixed forces 2

n

-- - 0 - ~ (Lij + Ljl)X_0 (i - k+l, k+2, ..., n). (8 0/8X i)~i j-I (6.4.3)

This extremum is a minimum since 0 i s nonnega t ive . On a p p l y i n g

2For the remainder of the proof we dispense with vectorial notation; the reader may readily generalize the argument outlined here in a manner appropriate to the vectorial notation.

STEADY-STATE CONDITIONS ~ I

Eq. (6.4.1) one flnds that

2 Z LIjXj - 2Ji - 0 (i - k+l, k+2, ..., n), (6.4.4) j-I

w h i c h p r o v e s t h e t h e o r e m .

To d e t e r m i n e t h e i m p l i c a t i o n s o f t h e t h e o r e m one n e e d s t o

study the structure of the resultant phenomenological

equations. By hypothesis, forces I through k, namely, X i -X_ -~

(i- I, 2, ..., k) are fixed; by (6.4.4) the last n- k

phenomenologlcal equations ~(O<_k)el3~ + Y~(j_~k§ - 0 vanish

for k+l _< i _< n. Hence, the X_0 with j _> k+l may be solved in

terms of the known ~ with j _< k. Thus, all X_0 are now fixed;

hence, by (6.4.1), all fluxes Ji are likewise time-invariant.

In short, steady-state conditlons have been achieved. By

Prlgoglne's Theorem, this steady-state is characterized by a

minimum rate of irreversible entropy generation; this must

henceforth be included as a criterion for the establishment of

stationary states.

Moreover, stationary states tend to be inherently stable,

as may be seen by the following argument. Suppose all forces

are maintained at their steady-state values save one, ~, for

which k+l < m _< n. We apply a perturbation 6~ to this

particular force; this provokes a nonzero flux among the set of

J's that had previously ceased to exist" Now Jm- Lm6~" No

other flux is altered because all other forces are maintained

in their original state. I~= _> 0 because the biquadratic form

Z(1)~(~)eljxix_j _> 0 is nonnegative. Hence, Lmm(~) 2 _> 0, which

implies that Jm6~ >_ O.

Now all spontaneous fluxes Jm of a given sign naturally

bring about a change of opposite sign in their associated,

conjugate forces [i.e., in 6~ in this case; see also Exercise

6.4.1]. This, in combination with the result just cited, means

that Jm cannot be sustained; the system ultimately returns to

the initial, quiescent, stationary state. We thus deal here

with an extension of Le Ch~telier's principle to steady-state

S S'2 6. IRREVERSIBLE THERMODYNAMICS

conditions: any system initially in a stationary state that is

perturbed in a singular manner will tend to react so as to

return to the initial steady-state.

The extended principle of Le Ch~telier and Prigogine's

theorem thus leads to the conclusion that if k out of n forces

Xi, X2 , ..., X~ are maintained at fixed values by means of

external constraints the system will ultimately reach a state

of minimum entropy production that is truly stationary; this

will be termed a steady-state condition of order k.

EXERCISES

6.4.1 Discuss the correctness of the statement that a flux always spontaneously occurs in such a manner as to reduce the conjugate force which maintains it, using the following illustrations: (a) the flow of current originated by a battery whose leads are maintained originally at an electrochemical potential difference ~; (b) the passage of matter from one phase to a second phase originally at a different chemical potential; (c) the injection of matter into a device maintained in a steady-state at a steady difference in temperature.

6.4.2 Provide a proof of Prigogine's Theorem in which you maintain the vectorial notation throughout.

6.4.3 How does the factor 2 arise in Eq. (6.5.4)? 6.4.4 Specify carefully under what conditions all fluxes

vanish. Does this always mean that equilibrium has been achieved? This matter needs proper elaboration. [Hint: there may be counterfluxes] .

6.5 THE ONSAGER RECIPROCITY RELATION

We provide here a simplified version of the Onsager reciprocity

relation; as derived in this section, the result holds only

under steady state conditions. It is actually far more widely

applicable, but a rigorous derivation of the reciprocity

relations would require use of the machinery of statistical

mechanics, which would lead us too far afield. Since we are

generally interested only in steady state effects the

restrictions to which the derivation is subject does not

adversely affect its utility. We follow the procedure adopted

ONSAGER RECIPROCITY RELATIONS 553

by Tykodi s. The reader is advised to write out the steps of

the derivations in full, using the specific phenomenologlcal

equations for the special case of three fluxes and three

forces.

Consider the set of phenomenologlcal equations

r

21- Z LIjX_J �9 J-I

(6.5.1a)

r

Ji- Z LIjX_J �9 j-1

(6.5.1i)

r

Jk- Z �9 j-i

(6.5.1k)

r

J= " Z LrjX_j, j-1

(6.5.1r)

and solve Eq. (6.5.1k) for

~k -- Jk/l~k -- ~ (Lkj/L~k)~- j~k

(6.5.2)

Substitute this result in Eq. (6.5.1i) so as to eliminate Xk;

this yields

Ji- Z LijX_j - LikJk/l~e -- Z (Liklej/~)X_-j" j~k j~k

(6.5.3)

Notice that all the Ji have now been written as a function of

the ~ and of Jk.

We next consider the dissipation function

3R. Tykodi, Thermodynamics of Steady States (MacMillan, New York, 1967), pp. 31-33.


r

i-i (6.5.4)

and its partial derivative

- a J, [L,k]_

i~k i~k (6.5.5)

We further rewrite Eq. (6.5.1k) in the form Jk- le~_Xk + Z leiXi

and use t h i s e x p r e s s i o n to e l i m i n a t e X_k from Eq. ( 6 . 5 . 5 ) .~t~This

leads to the final result

a~ Jk aJk L~ i~k L~ "

(6.5.6)

Reference to the preceding section, and to Eq. (6.4.4),

shows that if steady state conditions are applied and if there

are no constraints placed on the various forces all currents

vanish; thus, the first term drops out. This is also the

condition for which the rate of entropy conduction is a

minimum. Thus, consldering 0 to be an implicit function of the

fluxes, the left hand side must then vanish. The Onsager

reciprocity condition is thereby established: Lik- le• One

should note that if the reciprocity condition were to fail the

various X i would be interrelated, contrary to the assumption

that they are independent. Furthermore, the assumption of the

Onsager relations leads directly to the minimization of 8 under

steady state conditions.

6.6 THERMOMOLECULAR MECHANICAL EFFECTS

By now we have set up the basic machinery which permits the

principles of irreversible thermodynamics to be applied to

problems of interest. We next illustrate the method of

procedure by an elementary example. The same approach will be

used in later sections, with appropriate variations on the

basic theme.

THERMOMOLECULAR MECHANICAL EFFECTS 5 5 5

(a) The system under study consists of two vessels at

constant volume filled with a single type of fluid and

connected by a small opening; the vessels are maintained at two

different, uniform pressures and temperatures. We wish to

examine the heat and mass flows between the two portions of the

system.

Attention must first be focused on the quantity

representing the rate of change of entropy density due to

processes occurring totally within the system. This permits

identification of pairs of conjugate fluxes and forces. We can

use one of two formulations, namely Eqs. (6.2.15) or (6.2.18);

other formulations have also been specified in the literature.

Select Eq. (6.2.18) as the basis of further operations and

switch to molar rather than specific quantities: The fluxes

(on a molar basis) are then taken to be Ju and Jl, and the

corresponding conjugate forces are V(I/T) and - V(~I/T )

respectively. Let us temporarily replace Ju by J0 and set

V(I/T) - X0, - V(,I/T ) - X I. The following phenomenological

relations now result, valid for fluxes and forces operating

along one dimenslon:

R m

Jo-LooX_o + (6.6.1a)

m m

Jl- LI0X0 + LilXl. (6.6.1b)

Inasmuch as conjugate force-flux pairs have been selected,

Onsager's reciprocity conditions apply: L01- Ll0.

(b) For further progress it is desirable to recast (6.6.1)

in terms of experimentally measurable driving forces: We set

X_- 0 -- (I/r2)vr and X_- I -- T-IV#I + (#I/T2)VT-- T-I[-~IVT +

~I vP] + (HI- T~I) T-2VT -- ~i T-IVP + (HI/T2)VT; thus

m

Ju-- L01(VI/T)VP + [(L01HI- L00)/T2]VT (6.6.2a)

m m

Jl-- LII(VI/T)VP + [(LIIHI- L01)/T2]VT. (6.6.2b)

.5 ~ 6 6. IRREVERSIBLE THERMODYNAMICS

In the present system nonuniformities in P and T are

encountered only at the junction between the vessels ;

accordingly, VP and VT may be replaced by the pressure and

temperature differences at the junction, AP and AT,

respectively; the thickness of the connecting unit may be

absorbed in the coefficients L.

The next step consists in imposing a variety of steady-

state conditions on (6.6.2), in attempts to endow the

coefficients with physical interpretations and to arrive at a

variety of predictions.

Consider first the special case where the temperature is

maintained at a uniform value. The sole driving force is now

the pressure difference between the vessels. Setting VT- 0

and dividing (6.6.2a) by (6.6.2b) yields a relation of the form

(Ju/J1) VT-0 -- L01/L11 - Ul. (6.6.3)

Here Ju is the rate of energy density transfer across unit

cross-sectlon in unit time arising from the flux in moles of

species I across unit-cross section in unit time. This ratio

is clearly the energy transported under isothermal conditions

per mole of species I, denoted by U I in Eq. (6.6.3). We see

then that a thermomechanlcal effect is predicted; for a fixed

pressure difference across the junction, Ap, and at constant

temperature, a particle flux J1 gives rise to a proportional

energy transport Ju- U*IJI. This is a very sensible conclusion.

(c) A second special case is now invoked, namely the

stationary state under which no mass transfer occurs, but heat

flux is permitted. We now set J1 - 0 in Eq. (6.6.2b) and solve

for the ratio

(VP/VT)jI-0 - [(HI - L01/LII)/~IT] " (HI - U~)/~IT -- QI/Vl T,

(6.6.4)

where the quantity on the right results from use of (6.6.3);

Q~ is a molar 'heat of transfer', defined by H I -U I. We thus

encounter a second physical prediction" Under conditions where

THERMOMOLECULAR MECHANICAL EFFECTS 5 5 7

mass flow is blocked, a difference in temperature between two

vessels, which are allowed to interchange energy, necessarily

results in the establishment of a pressure difference AP-

(Q~/~IT)AT between the communicating vessels. This is a

physically sensible prediction.

(d) As a third special case consider the mass flow

resulting from a pressure difference between the two vessels

maintained at a uniform temperature. According to Eq. (6.6.2b)

this yields J1 - - (LIIVI/T) VP, which is an analogue of electric

current flow arising from a difference of electrical potential.

Accordingly, it is sensible to introduce a hydraulic

perm~ttSvity, ~, for mass flow, defined as

- (JI/Vp)vT. 0 - LIIgl/T m ~; J1 - - ~. VP. (6.6.5)

Lastly, it is instructive to determine Ju under conditions

of no net mass flow. Accordingly, we set J1 - 0, solve (6.6.2b)

for VP, and use this relation to eliminate VP in (6.6.2a).

This yields

Julal. 0 - _ (I/T2LII)(L00L11 - ~I)VT. (6.6.6)

Now the right-hand side represents an energy flux occurring

from the temperature gradient, in the absence of any net

particle flow; also, at constant volume no work is performed.

The resulting Ju thus is a heat flux; the proportionality

coefficient in (6.6.6) is equivalent to the thermal

conductivity, ~. This leads to the identification

- (L00L11 - I~I)/T2L11; J0 - - mVT. (6.6.7)

(e) The analysis may now be completed by collecting

(6.6.3), (6.6.5), and (6.6.7) and solving these three equations

for the three unknowns L00 , L01 , L11 in terms of ~, ~., and U ~ or

Q'. This yields

5 5 ~ 6. IRREVERSIBLE THERMODYNAMICS

L11 - ~ - T / ~ 1 ( 6 . 6 . 8 a )

- (6.6.8b)

Loo - 2 + (UI)'I.T/%, (6.6.8c)

and when these results are introduced into (6.6.2) one obtains

a complete phenomenological description of the form

Ju - - ~-U~VP - [~ - (UIQI~/qlT) ]VT (6.6.9a)

ms ---, �9 ~" " " " T I vP + (Q I/V )VT. (6.6.9b)

Equations (6.6.9) show explicitly, in terms of phenomenological

coefficients that may be experimentally determined, precisely

how the effects of pressure and temperature gradients superpose

in the system to produce concomitant fluxes of energy and of

material. All prior information is contained in these

relations: If a difference in T is established while no net

mass flow is encountered one recovers the effect predicted by

Eq. (6.6.4), and the energy flux is given by Eqs. (6.6.6) and

(6.6.7). If uniform temperature is maintained the mass flux is

given by Eq. (6.6.9) as J1 - -~'VP and the energy flux, by Ju -

-~.U*VP. If the pressure is held uniform one encounters a

temperature-drlven particle flux J1 (~QI/VIT)VT and an energy

flux Ju- [~ ~ qlA/ 1] VT A complete analysis of the

experimental results has now been furnished.

6.7 THE SORET EFFECT

As the second application of irreversible thermodynamics we

consider the Soret effect (1893) for a two-component system: a

flow of particles under the influence of a temperature gradient

produces a gradient in concentration. We are ultimately

interested in the magnitude of this effect under steady state

conditions. Let J0, J1, J2 be the entropy and particle fluxes

THE SORET EFFECT 559

occurring under the influence of the gradients X_- 0 -- T-ZVT and

X_-I, 2 - - T-IVDI,2 [see Eq. (6.2.15)] in a closed system. The

steady state conditions to be introduced at a later stage read

J1- 0 and J2- 0; these set in after the initial flows of

species i and 2 have settled down to the quiescent condition.

We begin with the phenomenological equations in the molar

representation

u m m

Jo " LooX~o + LolXl + Lo2X._2 (6.7.1a)

m m m

d z - LzoXo + LxxXx + Lz2X2 (6.7.1b)

m m I

J2- L20X0 + e21Xl + e22x2. (6.7.1c)

Note first that even in the absence of a temperature

gradient an entropy flux can occur. For, when X 0 - 0, Ji -

LIIX 1 + LI2X2, where i - 0, I, 2. Further, at constant

temperature we define S I and S 2 as the entropy carried per mole

of species i and 2. Then at constant T the entropy flux is

given by the postulated form

J o - S~J1 + S~.J2 (T c o n s t a n t ) . (6.7.2)

On insertion of the appropriate phenomenological equations,

this yields

J o - (S~Lzz + S~L21)X~l + (S~L12 +S~L22)X_-2 �9 ( 6 . 7 . 3 )

m N

Comparison with Jo - LozXI + Lo2X2 at constant T allows one to

i d e n t i f y t h e c o e f f i c i e n t s o f X 1 and o f X...2 and to s o l v e t h e

resulting linear equations for

S~ - (LolL22 - Lo2LI2)/(LIIL22 - L22)

S~ - (LozL11 - LoILIz)/(LIIL22 - L~2). (6.7.4)

Now apply the steady state condition under which J1 = J2 = 0 and

$60 6. IRREVERSIBLE THERMODYNAMICS

u m

allow X 0 to have a fixed, nonzero value. On eliminating X z

between (6.7.1b) and (6.7.1c) one may solve for the ratio

X1/X_o - (L02L12 - L01L22 ) / (LIIL22 - L~2 ) . (6.7.5)

m , , , , , , , ,

On introducing the representations for X 0 and X I and Eq. (6.7.4)

we f i n d t h a t

V,u I - - S~VT. ( 6 . 7 . 6 )

Here V~I is the gradient of the chemical potential of species

I with respect to coordinates, which must be evaluated at

constant temperature. Under this restriction, ~I can only

depend on changes in mole fraction: (V#I)T = (~I/@XI)T;

insertion into (6.7.6) yields

(a.~/ax~)~-- si(aT/ax~)T, ( 6 . 7 . 7 )

which is the expression for the Soret effect. This is a new,

perhaps unexpected prediction based on irreversible

thermodynamics" In a closed system a heat flow arising from a

temperature gradient must produce a gradient in chemical

potential under steady state conditions. For an ideal system

Eq. (6.7.7) may be reformulated as

d~n xl = - (S*I/RT)dT, ( 6 . 7 . 8 )

which shows by means of an analytic relation how the mole

fraction for component I in a two-component system alters with

temperature across a system under the assumed steady state

condition. For the special case considered here

~n (xi/x~) - - ~T ~ [S~(T')/RT" ]aT" . ( 6 . 7 . 9 )

If the dependence of S I on T is sufficiently weak, one finds

.en (x~/x~) - - (~*/R) ~n (T/To) , ( 6 . 7 . 1 0 )

ELECTROKINETIC PHENOMENA ~6 1

with ~* the value for the entropy carried per mole of species

i, suitably averaged over the temperature interval T o to T.

EXERCISES

6.7.1 Provide a physical mechanism which explains on a microscopic level the thermodynamic result of Eq. (6.7.7).

6.7.2 Specialize the derivation of this section to a single gaseous species. Show that under steady state conditions a temperature gradient produces a pressure gradient and express the magnitude of the latter in terms of the former.

6.8 ELECTROKINETIC PHENOMENA

Here we consider the case depicted in Fig. 6.8.1 of a charged

membrane (with appropriate counter-lons in solution) separating

two identical solutions maintained at fixed temperature. An

electric field or a pressure gradient is now applied, as a

O

FIGURE 6.8.1 Schematic diagram of apparatus for carrying out electrokinetic experiments. Pressure is applied by moveable pistons P and P" on liquids in compartments R and S. Electric fields are set up by condenser plates C and C'. Solvent and positive ions can move through a membrane M separating the two compartments. Fluids can move through an inlet I and outlet O via fitted stopcocks, mounted on the pistons.

S ~ 6. IRREVERSIBLE THERMODYNAMICS

result of which both the solvent (water, designated by O) and

positive ions in solution (designated by +) move through the

membrane unit until a new steady state has again been achieved.

Under the action of the pressure differential a potential

difference is established across the membrane; alternatively,

because of the imposition of a potential gradient, a pressure

difference fs established between the two solutions. The

physical situation may be analyzed as follows:

As emphasized earlier [see Eq. (6.2.15), for example], any

flux of charged particles ~+ (on a per mole basis) arises in

response to the establishment of a gradient V~ in

electrochemical potential. For one-dimensional flow we may

write J+- LV~ - L(V/~ + ZF~) - L(V+VP + ZF~), where the

contribution- S+dT has been dropped from the last relation

because constant-temperature conditions are presumed to

prevail. Similarly, J0 - L'VoVP, as shown in Section 6.7.

Actually, the compartments R and S in Fig. 6.8.1 may be

uniform in their properties so that the changes in P and

occur essentially only across the membrane M. In this case V~

and VP may be replaced by the discontinuities A~ and Ap across

the membrane, the constant thickness of the membrane having

been absorbed into the phenomenological coefficients.

The total flux of solvent (J0) and of ions (J+) is thus

given by

J o - (LiiVo + LizV+) vP + Li2Z+F~ (6.8.1a)

m I

J+- (L2iV0 + L22V+)VP + L22Z+F~ ~ , (6.8.1b)

where we have set Lil m L', L22 m L, and where we have taken

care of the cross interactions by introducing the coupling

coefficients Li2 and L2i that link J0 and J+ to V~ and to VP,

respectively. In Exercise 6.8.1 the reader is asked to show

that Li2- Lai.

The preceding phenomenological relations may be rendered

symmetric by considering instead of J0 and J+ the total volume m

flow Jv " V0J0 + V+J+ and total current density I+ E ZVJ+"

ELECTROKINET'C PHENOMENA ~ 6

Jv- (LIIV02 + 2LI2V+V0 + L22V+2) VP + Z+F(LI2V0 + L22V+)V~ (6.8.2a)

I+ - E+F(LI2V 0 + L22V+) VP + (Z+F) 2L22V~, (6.8.2b)

which may be abbreviated to read

'] 'V- LVV(-- VP) + LVI ( - Vr (6.8.3a)

I+- LVI (- VP) + LII (- V~). <6.8.3b)

Equations (6.8.3) satisfy the Onsager reciprocity condition,

showing that (Jr,-VP) and (I+,-V~) are sets of conjugate

variables. Equations (6.8.3a) and (6.8.3b) are the

phenomenologlcal equations of interest.

In the subsequent analysis it is sometimes convenient to

generate an inverted set of phenomenological equations, by

solving Eqs. (3.8.3) for the gradients in terms of the fluxes:

- VP - RvvJv + Rvil+ (6.8.4a)

-- V~ - RVIJ v + RIII +. (6.8.4b)

In Exercise 6.8.2 the reader is asked to determine the various

R coefficients in terms of L11, L12, and I..22.

Again, these particular relations hold only for constant

temperature conditions. Suppose that, in addition, no current

flow is permitted. Then I+ - 0; according to (6.8.3b), this

imposes the constraint

(V~/VP)I+.0 -- -- LvI/LII, (6.8.5a)

whereas, if conditions are such that no pressure gradient is

allowed to develop, i.e., with VP- 0, one finds by division of

(6.8.3a) with (6.8.3b) that

(Jv/I+)vP.o - LvI/LII - ~', (6.8.5b)

564 6. IRREVERSIBLE THERMODYNAMICS

where ~" is the so-called electro-osmotlc transfer coefficient.

The quantities on the left of Eq. (6.8.5a) and (6.8.5b) are

termed streamlng potentials and electro-osmosis respectively.

It is immediately evident that

(V4/VP)z+. 0 -_ (Jv/I+)ve, o, (6.8.6)

which relationship is known as Sax4n's Law.

In Exercise 6.8.3 the reader is asked to prove that

(VP/V4) j r , o - - ( I+/Jv)vr o . (6.8.7)

Here the left-hand side is known as the electro-osmotic

pressure, and the rlght-hand side as the streaming current.

The relations developed here point up an interesting

feature" The streaming potential (V~/VP)I+. 0 cannot readily be

experimentally determined, since it forces imposition of a

change in electrostatic potential in the absence of a net

responding current. However, this quantity is also given by

the ratio - (Jv/I+)ve.o -- ,8", which can readily be determined

experimentally. Here one measures the volume flux and current

in response to the imposition of a gradient in electrostatic

potential when the pressure in the two compartments are

identical. Analogous remarks apply to the quotient in (6.8.7).

(a) The remainder of this section is devoted to the

specification of phenomenological equations (6.8.3) and (6.8.4)

by which the coefficient L or R is eliminated in favor of

experimentally measurable quantities.

As a first step, solve Eq. (6.8.3b) for- V4 and substitute

the result in (6.8.3a) ; this yields

Jv - (Lvv - Lvz2/Lzz) (- VP) + (Lvz/LzI) I+. ( 6 . 8 . 8 )

Then, for conditions under which no current flow occurs,

[Jv/(- vP)]~+-o - Lvv- Lv~2/L~ - %, ( 6 . 8 . 9 )

ELECTROKINETIC PHENOMENA 5 6 5

where l~ is the hydraulic permeability of the membrane; note

that l~ _> O. This quantity is readily determined

experimentally. With ~" - Lvl/LII Eq. (6.8.8) now reads

Jv- Lp(- VP) + ,8"I+, (6.8.10)

which is known as the ~!rst electrokinetic equation.

(b)In conjunction with Eq.

membrane conductivity as

(6.8.3b) let us now define the

a - [I+/(- V4)]Ve,0 - LII, (6.8.11)

so that with the aid of Eq. (6.8.5b),

Lvz - a~'. (6.8.12)

Introduction of Eqs. (6.8.11) and (6.8.12)

(6.8.3b) yields the .second electrokinetic equation

in to Eq.

I+ - a~'(- VP) + a(- V4), (6.8.13)

which is simply a reformulation of the second phenomenological

equation, Eq. (6.8.3b), in terms of readily measurable

quantities. In Exercise 6.8.4 it is to be shown that

Jv- (Lp + a~'2)(- VP) + a~" (- V4), (6.8.14)

which is a reformulation of the first phenomenological

equation, Eq. (6.8.3a). Note how Eqs. (6.8.9), (6.8.11), and

(6.8.12) may be used to solve for the individual L's in terms

of experimental parameters.

(c) In addition to the preceding quantities, the following

transport coefficients are in common use" The steady state

electr~ca.l resistivity

PlJv-0 " (- V4/I+)Jv=O - RII, (6.8.15)

~ 6. IRREVERSIBLE THERMODYNAMICS

where (6.8.4b) was used to arrive at the relation on the right.

To realize this condition a difference in pressure must be

established between the right- and left-hand compartments of

Fig. 6.8.1 such as to oppose the volume flux Jv normally

accompanying the ion flux I+, which itself responds to the

imposition of the potential gradient - V~. In the steady state

the electro-osmotlc flux from left to right is counterbalanced

by the hydraulic flux from right to left.

The hydraulic ~eslstance is defined by

RB -- ( - - V P / J v ) I+-0 - R W , (6.8.16)

where Eq. (6.8.4a) was used to establish the equation on the

right.

Finally, in view of (6.8.4) and (6.8.5b), the electro-

osmotic flux may be rewritten as

i~" -- ( J v l I + ) v p , = o - _ R v I / R w . (6.8.17)

On introducing Eqs. (6.8.15)-(6.8.17)into Eq. (6.8.4)one

obtains final phenomenologlcal equations of the form

- VP - RaG v - ,8"RsI + (6.8.18a)

- V ~ - - ~8"P , .~v + p I + , <6.8.18b)

which again involve a set of measurable transport coefficients.

All the necessary information relating to electrokinetic

phenomena is contained in the phenomenological equations

(6.8.13) and (6.8.14) or in the equivalent set (6.8.18a) and

(6.8.18b). The former set is especially useful if one inquires

about state conditions under which either Jv or I+ is held

fixed. The latter set is useful to characterize operating

conditions at constant pressure or constant electrostatic

potential. The preceding discussion illustrates the

flexibility of phenomenological equations that permit either

fluxes or forces to be used as dependent variables.

THERMOELECTRIC EFFECTS S6 7

EXERCISES

6.8.1 Prove that the phenomenological coefficients L12 and lel in Eq. (6.8.1) satisfy the Onsager Reciprocity Condition.

6.8.2 Express the various R in Eq. (6.8.4) in terms of the L in Eq. (6.8.3).

6.8.3 Derive Eq. (6.8.7). 6.8.4 Derive Eq. (6.8.14). 6.8.5 By examining Eqs. (6.8.4b) and (6.8.9) obtain a

relationship between Rvl and L~. 6.8.6 Discuss Eq. (6.8.10) so as to provide insight into

the meaning of the first electrokinetlc equation. 6.8.7 Characterize the steady state of the system shown

in Fig. 6.8.1 when (a) there is no net current flow; (b) there is no net volume flow; (c) pressure is uniform; (d) the electrostatic potential is uniform.

6.8.8 What is the rate of entropy dissipation 0 for the general case, and for the situation where each of the four steady-state conditions discussed in Exercise 6.8.7 is imposed?

6.8.9 Verify that (6.8.13), (6.8.14), and (6.8.18a), (6.8.18b) meet the requirements of Exercise 6.8.2.

6.9 THERMOELECTRIC EFFECTS

In this section irreversible thermodynamics will be used to

establish the interrelation between heat flow and electric

current in a conductor. The field of thermoelectric effects

has been treated elsewhere in great detail. 4

Consider a rectangular bar (Fig. 6.9.1) that is connected

to two thermal reservoirs maintained at different temperatures.

Provision is also made for adiabatic insulation of the sample,

if needed. Charge may be made to flow through the bar in the

same direction as the flow of heat (or in the opposite

direction) by charging a set of condenser plates. This

cumbersome method is used to avoid distracting complications at

junctions between the bar and the electrical leads that would

be normally employed. We are interested in the flow of charge

4T. C. Harman and J .M. Honig, Thermoelectric and Thermomagnetic Effects and Applications (McGraw-Hill, New York, 1967).


C

Re j

/ S

C I

Fig. 6.9.1. Experimental setup for analysis of thermoelectric fields. A bar is clamped between two reservoirs maintained at two different temperatures T I and T 2. Provision is also made, by means of removable strips S and S', for adiabatic insulation of the bar. Current may be caused to flow along the bar by continuous charging of two condenser plates C and C'.

and of heat along the bar and in any interference effects that

might be encountered.

According to Section 6.2 an appropriate choice of fluxes

and forces for this problem is found by examining the

dissipation function 0 -- T-IJs.VT- T-~J~-VL. Now, as the

discussion after Eq. (6.2.12) shows, the specific quantities Jk,

fk, ~--k, Nk may be converted to molecular quantities; hence we

replace the last term by- T-iJn.V[n, where now Jn is a particle

flux vector and In is the electrochemical potential per

electron. Moreover, as the inspection of Eq. (6.2.16) shows,

J, is an entropy density flux vector. Therefore, we can set

-- T-IJ,.VT- T-IJn-V~n . It is expedient to write In" [ and to

replace the flux vector Jn by the current flux J according to

the relation J- (-e)J n, where-e is the charge on an electron.

The quantlty J is known as the current density. Thus, 0 - Js.(-

T-IVT) + J-[T-iV(I/e)]. This relation identifies the fluxes and

the forces and leads immediately to the phenomenological

relations

J,-- (Lss/T)VT + (Lsn/T)V(~/e) (6.9.1a)

THERMOELECTRIC EFFECTS ~ 69

J-- (Im,/T)VT + (L~/T)V(~'/e). (6.9.1b)

In setting up these relations we have assumed that the J's and

X's are collinear and are oriented along one dimension only,

which allows us to drop the vector notation.

(a) To identify some of the phenomenological coefficients

we first consider the special case VT- 0. Then, eliminating

V([/e) between (6.9.1a) and (6,8.1b), we obtain J,/J - L,n/Lnn;

since J,/Jn-- eJ,/J is the entropy carried per particle, we

can equate

L, nlL ~ - - S.le - Im,ILm~ - J, IJivT-O, (6.9.2)

* is the entropy carried per electron (instead of per where S.

mole, as in earlier sections) at constant temperature.

A further identification of the macroscopic coefficients

is achieved by examining Eq. (6.9.1b) when VT - O. Then J -

(Lnn/T) V (~/e) . For a homogeneous sample at constant

temperature this latter relation reduces to J- (l~m/T)V(- 4) -

(I~m/T)E , where E is the electrostatic field. This special case

is clearly a manifestation of Ohm's Law (1826), J- aE, whence

Lrm/T - a, (6.9.3)

where a is the electrical conductivity of the specimen. The

generalized procedure, involving the electrochemical potential

gradient, provides a generalized version of Ohm's Law, valid

for inhomogeneous samples at constant temperature" J- aV(~/e).

(b) Next, examine the case where no current flows" Set J

- 0 in (6.9.1b) and then substitute for V([/e) in Eq. (6.9.1a).

This algebraic manipulation yields

J, - - (I/T)[L,, - ~,/Lnn]VT (J - 0). (6.9.4)

Entropy flux in the absence of a net particle flow is


equivalent to JQ/T where JQ is the heat flux. Thus, Eq. (6.9.4)

is the form of Fourier's Law for heat conduction, JQ-- ~VT,

thereby establishing the identity for thermal conductivity as

~, = L,, s - L n s Z / L ~ . (6.9.5)

In the more general case J ~ O, one may still eliminate

V~/e between (6.9.1a) and (6.9.1b) to obtain, in view of

(6.9.5) and (6.9.2),

J , , - - ( S * / e ) J - (~/T)VT- S*J n - (~/T)VT, (6.9.6)

which represents the first physical prediction. Equation

(6.9.6) shows how the total entropy flux is related to the

presence of a temperature gradient and to the particle flux

associated with the current flow; ~ contains a contribution due

to the lattice, as well as that due to the thermal flux

accompanying the current, these being considered as additive.

(c) To obtain a second prediction, return to Eq. (6.9.1)

under the special condition J- O. One thus obtains

V(CIe)/VT - d(Cle)/dT -= - L~,I~ (Z - 0). (6.9.7)

In other words, under the conditions examined here, the

imposition of a temperature difference dT necessarily results

in the establishment of a difference d[ of electrochemical

potential: d(~/e) - (Ln,/~)dT. This effect is known as the

thermoelectric effect, and the ratio d(~/e)/dT- Ln,/~, as the

Seebeck coefficient (1823), or thermoelectric power, designated

here by =. The later appellation is highly undesirable and is

gradually being eliminated in favor of the former.

Experimentally, the difference of electrochemical potential per

unit electric charge may be measured by a voltmeter operating

under open circuit conditions, and dT is measured by means of

thermocouples; ~ is thus experimentally determined. Comparison

with (6.9.2) shows that ~ - - S*/e. Then Eq. (6.9.6) becomes

THERMOELECTRIC EFFECTS S 71

j, - aJ - (s/T)VT, ( 6 . 9 . 8 )

which shows the additive effects of particle flow and heat flow

contributing to the entropy flux.

(d) Finally, we may rewrite the phenomenological equations

as follows: Since Lnn/T- a and Ln,/~-- S*/e- =,

J - - a aVT + o r ( f / e ) ; ( 6 . 9 . 9 a )

use of this in (6.9.8) yields

J , = - ( a a z + ~ : / T ) VT + a a V ( f / e ) . ( 6 . 9 . 9 b )

The phenomenologlcal equations (6.9. i) have thus been

reexpressed in (6.9.9) solely in terms of the measurable

transport coefficients a, ~, and a. The Seebeck coefficient may

be interpreted as the entropy carried per electronic charge.

Equation (6.9.9a) represents a further generalization of Ohm's

Law, showing how the current density behaves in the presence of

a temperature gradient; see also Exercise 6.9.3. Equation

(6.9.9b) specifies the entropy flux under the joint action of

a gradient in electrochemical potential and in temperature;

this represents a generalization of Fourler's Law.

EXERCISES

6.9.1 (a) In what way, if any, does the present approach require modification to render it suitable to the description of charge and entropy transport of ions in solution under the combined influence of a temperature gradient and an electric field? (b) Reformulate the theory in the present section so that it becomes applicable to the flux of positive charge. (Refer to the monograph cited earlier for assistance, if needed. )

6.9.2 (a) Provide a physical interpretation of the Seebeck effect by noting that electrons at the hot end of the bar have a higher overall kinetic energy than those at the cold end. Show that this results in a migration of electrons to the cold


end. (b) What is the physical consequence of this migration, and in what way does this lead to a steady state condition of no current flow? (c) Explain the circumstances under which a steady state condition may be reached.

6.9.3 Prove that Eq. (6.9.9a) reduces to Ohm's Law, I - A~/R, when the sample is maintained at a uniform temperature.

6.9.4 On the basls of Eq. (6.2.17) introduce E* as the energy carried per particle moving under the influence of an appropriate force. Show that if reasonable estimates of E* can be made, the Seebeck coefficient serves as a measure of chemical potential. 'What is the reference energy for this particular case? What is a reasonable value of E* for an electron in a crystal or for an ion in solution?

6.10 IRREVERSIBLE PHENOMENA IN TWO DIMENSIONS

In this section we consider effects arising in conjunction with

the geometry illustrated in Fig. 6.10.I. A rectangular slab is

aligned with the x and y axes of a Cartesian system, and a

magnetic field is directed along the z axis. Provision is made

for flux of current and of heat along x and y. One is

interested in the possible effects that may be encountered in

such a system. This leads to a consideration of what are

termed thermoelectric and thermomagnetic effects; the magnetic

field will be shown to give rise to a host of new cross

interactions between processes occurring along the x and y

directions.

To illustrate some new principles, a somewhat different

approach will be used relative to the methods introduced in the

earlier sections. As in Eqs. (6.9.1), we select (Js,VT) and

(J,V(~/e)) as the conjugate set of variables but will include

the T -I factors in the phenomenological coefficients. Three new

points are introduced at this time" (i) Since fluxes may occur

in two orthogonal directions, the conjugate flux-force pairs

now are" (J~,,Vxr), (J~,,VyT), (JX,Vx(~/e)), (Jy,Vy(~/e)). The

appropriate geometry is depicted in Fig. 6.10.1. (il) For

later convenience we shall select as independent variables from

the this particular set the quantities VxT , VyT, jx, j-y, SO

TWO-DIMENSIONAL IRREVERSIBLE PHENOMENA S ~3

ii II

^

y

FIGURE 6.10.1 Paralleleplped geometry for current and/or for entropy flux along the x- and or y-directions in a magnetic field oriented along the z-direction.

that the phenomenological equations appear in partially

inverted form

J~s-- LiiVx T - Li2Vy T + Li3 Jx + Li4 Jy (6.10. la)

J~s - Li2Vx T - LiiVyT- L i j x + Li3 Jy (6.10.1b)

Vx(~'/e ) - L13VxT + LI4VyT + L33J x + L34J y (6.10.ic)

Vy(~'/e) - - Li4VxT + Li3VyT - L34J x + L33J y, (6. lO.Id)

where the Lij are appropriate phenomenological coefficients.

(iii) For later convenience we have arbitrarily selected the

minus and plus signs in the indicated sequence in Eq.

(6.10.1a) ; the other signs in Eq. (6.10.i) are then governed by

the Casimir-Onsager reciprocity conditions, Eq. (6.3.7), as

required by the presence of a magnetic field H- kH,._

We engage in a systematic treatment of the thermodynamics

of irreversible processes in the above configurations.

Consider first the isothermal case summarized by the

constraints: (a) JY- VxT - VyT - O; isothermal conditions are

now maintained along x and y, and no current is allowed to flow

'74 6. IRREVERSIBLE THERMODYNAMICS

along y. Then Eqs. (6.10.i) reduce to

J~s - Lx3 Jx (6. I0.2a)

J~-_ Lx4j x (6.10.2b)

V x(~'/e) - L 33 Jx (6.10.2c)

vy(~'/e) -- L34J ~. (6. lO.2d)

From Eqs. (6.10.2c) and (6.10.2d) it is seen that as a result

of current flow along x, a gradient in electrochemical

potential develops along x as well as along y. The first

effect is simply a manifestation of Ohm's Law jx _ pIVx(~/e),

wherein PI m L33 is the isothermal resistivity. The second is

an example of the isothermal Hall effect, characterized by

Vy([/e) - - (L34/Hz)JXHz (6.10.3a)

- RIJXH,, (6. i0.3b)

wherein, for convenience, the magnitude of the applied magnetic

field has been introduced explicitly. As Eq. (6.10.3b) shows,

a flow of current longitudinally induces a transverse gradient

in electrochemical potential. The magnitude of this effect is

specified by the Hall coefficient, R I -- L34/H z.

We next consider the constraints (b): JY = VxT- J~- 0.

No current flow is allowed along y and no heat flow is

tolerated in this direction. Isothermal conditions are

maintained along x. This represents an (transverse) adiabatic

set of operating conditions. Equations (6.10.i) now reduce to

J~s-- Lx2Vy T + Lx3 Jx (6.10.4a)

0 -- LxxVyT- Lx4J x (6. lO.4b)

v.([/e) - L~4VyY + L33~ (6. I0.4c)

TVVO-DIMENSIONAL IRREVERSIBLE PHENOMENA 5 7 5

Vy (~'/e) - LI3VyT - L34J ~. (6.10.4d)

Equation (6.10.4b) shows that a current flow along x

produces a temperature gradient along y; this is the so-called

Ettingshausen effect, specified by

VyT-- (L14/L1111z)JXHz ( 6 .10 .5a )

- TJ~Hz, (6. i0.5b)

in which T - VyT/JXHz is the Ettingshausen coefficient. On

inserting (6.10.5a) inte (6.10.4c) one finds

Vx(~'/e) - (L33 - LI42/Lll)J ~, ( 6 . 1 0 . 6 )

which is of the form of Ohm's Law under adiabatic conditions,

with an adiabatic resistivity PA TM L33- LI42/L11 �9 When (6.10.5a)

is introduced in (6.10.4d) one obtains the expression

Vy(~ ' /e ) -- - [ (L13LI4/LII + L34)/Hz]J'XH,, ( 6 .10 .7a )

which represents the adlabatir Hall effect, with

R^ - - (I/H,) [LI3LI4/L11 + L34] �9 ( 6 .10 .7b )

We next consider conditions (c): jx_ jy_ VyT- 0. No

current flow is permitted, but a temperature gradient is

established along x, while along y isothermal conditions are

maintained. The phenomenological equations reduce to

J~s-- LIIVxT ( 6 .10 .8a )

J~ - LIzVxT (6. I0.8b)

Vx(C/e) - L13VxT ( 6 . 1 0 . 8 c )

gy(~/e) - - LI4VxT - - (L14/Hz)HzVxT. <6.10.8d)


Under the postulated conditions TJ~s and TJ~ represent heat

fluxes. Then Eq. (6.10.8a) leads directly to the definition of

an 'isothermal heat flux' (a contradiction of terms!) : TJ~s -

- LIITVxT , whence we may write

~z = TL11, (6.10.9)

where ~;z is the thermal conductivity when no transverse

temperature gradient is allowed to exist. According to Eq.

(6.10.8c), a longitudinal temperature gradient produces a

longitudinal gradient in electrochemical potential. This

represents nothing other than the 'isothermal' Seebeck effect

introduced in Section 6.9. Thus, with Vx([/e ) - LI3VxT one

finds the relation

c, z = L13 , ( 6 . 1 0 . I 0 )

where ~I is the Seebeck coefficient under conditions where no

transverse temperature is allowed to exist. Next, according to

Eq. (6.10.8d), a temperature gradient along the x direction

produces a gradient of f along the y direction; this is a

manifestation of the transverse Nern.st effect; the relation

Vy(~/e) -- -- (LI4/Hz)HzVxT suggests the definition of a

corresponding ,coefficient as

N I = _ LI4/H ,. (6.10.11)

Another set of circumstances frequently encountered is

specified by (d): jy _ jx_ j~ _ 0. Here, no currents are

allowed to flow, and adiabatic conditions are imposed along the

y-direction. Then the phenomenological relations (6. I0. I) can

be reduced to

J~s =- L11VxT- L12Vy T

0 = LI2VxT - LIIVyT

(6.10.12a)

(6.10.12b)

TVVO-DIMENSIONAL IRREVERSIBLE PHENOMENA ~ 77

V x(~/e) - LI3VxT + LI4VyT (6.10.12c)

V y ( ~ ' / e ) - - L14VxT + L13VyT. (6.10.12d)

Equation (6. I0.12b) shows that the establishment of a

longitudinal temperature gradient gives rise to a transverse

one. This interrelation is known as the Ri~hi-Leduc effect. v

It is convenient to rewrite (6.10.12b) as

VyT - (LIz/LIIH z)Hzv.T , (6.10.13a)

whence the Righi-Leduc coefficient becomes

VyT/HzVxT - M- LIz/LIIH z. (6.10.13b)

Another relation of interest is found by inserting

(6. I0.13a) into (6. i0.12a), and multiplying through by T;

yields

Eq.

this

TJ~s = - T [ L l l + L 1 2 2 / L l l ] VxT , (6.10.14a)

which gives rise to the definition for the

conductivity,

' a d i a b a t i c ' thermal

~A -- T ( L11 + LI22/L11) . (6. i0.14b)

Use of (6.10.13a) in (6.10.12c) yields

V x ( [ / e ) -- (L13 + L14L12/Lll]VxT , (6. i0.15a)

which is the Seebeck effect

maintained in the transverse

coefficient reads

when adiabatic conditions are

direction. The corresponding

~A- L13 + LI4LI2/L11- (6.10.15b)

Finally, if (6.10.13a) is combined with (6.10.12d)

magnetic field is explicitly introduced, one finds

and the


Vy(~'/e) - ( l / H , ) [ - L 1 4 + LIsLI2/LII]H,VxT, (6.10.16a)

which gives rise to the ad~abat$c transverse Nernst effect,

with a corresponding coefficient of the form

N^ - (I/H z) [- L14 + LIaLI2/L11]. (6.10.16b)

Many more effects may be treated on an analogous basis, as

is suggested by Exercises 6.10.i and 6.10.2. The physical

basis on which these effects rest is to be explored in Exercise

6.10.3.

One should note that the various coefficients listed in

this section are all measurable experimentally according to the

prescriptions imposed by the boundary conditions (a)-(d) and

the indicated definitions for each coefficient. Note that by

virtue of having set up phenomenologlcal equations in partially

inverted form the phenomenologlcal coefficients Lij in Eqs.

(6.10.i) assume a particularly simple form: L3s - PI, Ls4 - -

HzRI, LII - ~I/T, L13 -- ~I, LI4 - - HzNI, LIZ - - HzmM/T. On

inserting these relations into (6.10.I) one thus obtains a

complete description of irreversible processes for the system

under study. This permits a complete analysis to be made of

the 560 possible galvano-thermomagnetlc effects that can be

achieved in the rectangular parallelepiped geometry of Fig.

6.10.1.

EXERCISES

6.10.i (a) Develop phenomenologlcal equations for the condition jz _ VyT - J~, - 0 and prove that a temperature gradient along x is established as a consequence of current flow in that direction. (b) What is the resultant heat flux along y? (c) Express the resistivity and the Hall coefficient in terms of the appropriate Lij. Compare these with the results in the text.

6.10.2 Consider the phenomenological equations under the conditions JY- J~,- 0. (a) Express VxT in terms of jx. (b) Find VyT in terms of jx. (c) Express the resistivity and Hall coefficient in terms of appropriate Lij and compare these

CHEMICAL PROCESSES 5 7 9

results with those cited in the text. 6.10.3 Provide a detailed description of the mechanism

that shows precisely how transverse effects arise.

6. II CHEMICAL PROCESSES

(a) Irreversible phenomena pertaining to chemical processes may

be handled by the same techniques as previously employed. At

uniform temperatures and constant electrochemical potentials

Eq. (6.2.15) becomes 8 - T-1~(r)~=Ar >_ 0, which leads to a set of

reaction velocities (fluxes) ~r that result from the

corresponding driving forces Ar, the chemical affinities

introduced in Section 2.14. In what follows we closely adhere

to the treatment provided by Haase. 5

Note that at equilibrium ~r - Ar " O; however, situations

may arise where (1) ~r - O, A r ~ 0, corresponding to inhibited

reactions that may be remedied by introduction of a suitable

catalyst, (ii) ~r ~ 0, A r - 0, as in thought experiments in

which a reaction is carried out under near-equillbrlum

conditions.

If only one process is considered, (r- I) then ~A > O, so

that ~ and A must have the same sign. If two processes occur

simultaneously, ~IAI + ~2A2 _> 0; thus, for example, it is

possible to have ~IAI < 0 if ~2A2 > I~IAII.

(b) Consider two reactions of the type A = B and B = C.

If the third process A = C is not feasible the number of

elementary reactions is the same as the number of linearly

independent reaction equations; the reactions are said to be

uncoupled. If A = C is a feasible reaction the three processes

are coupled; generally, coupling occurs whenever there is a

redundancy in the number of reaction steps.

For situations not far removed from equilibrium (what this

implies will be fully documented later), one postulates the

usual linear relations between fluxes and forces. In the

present context (A, should not be confused with A)

5R. Haase, Thermodynamics of Irreversible Processes (Addison Wesley, Reading, Mass., 1968).


R

~r == X ar,A, (r - I, 2, . . . , R).

s-I

( 6 . 1 1 . 1 )

Coupled equations are characterized by nonvanlshing cross

coefficients: a=, ~ 0 for r ~ s. The dissipation function is

given by

R R

~r--~ s--~ ar'ArA" >- 0. ( 6 . 1 1 . 2 )

(c) It is instructive to specialize to the case of two

reactions (R - 2):

~I = a11A1 + a12A2

(~2- a21A1 + a22A2.

a21 - a12

( 6 . 1 1 . 3 )

Then

-- aliA21 + 2aI2AIA 2 + a22A ~ >_ 0, (6.11.4)

which requires a11 >_ 0, and a11a12- a22 >_ 0 (see Section 2.2).

Where there is no coupling, a12 - a21 - 0; in that event, ~IAI -

a11A12 >_ 0 and ~2A2 - a22A 2 >_ 0.

We next inquire as to the range of validity of the linear

approximation. For this purpose note that if a system is

characterized by n + I deformation coordinates xl, then in

general ~ - ~o(xl, .... ,xn+l) and A - Ao(xl,...,xn+l) ; one may

eliminate xn+ I between the two functions to obtain ~ -

~(xl,...,Xn,A) and A - A(xl,...,xn+l). But as A ~ 0, ~ ~ 0 as

well, so that the deviation of A from zero may be taken as a

measure of the deviation of the system from the equilibrium

conditions at which the x i assume their equilibrium values x I -

x~. It is therefore reasonable to expand ~ as a Taylor's series

in A while setting all x i - x~; on retaining only the term of

lowest order, one obtains

CHEMICAL PROCESSES S8 I

- ( a ~ / a A ) ~ r ~ o ~ A + . . . (i- 1,2,..., n). (6.11.5a)

On writing ~- aA, one finds that

a I (ao~/aA)=i.o i

(6. ll.5b)

identifies the coefficient a.

To check on the adequacy of the linear approximation we now

introduce the law of mass action in the form

~ == ~ C i ~',i

- - SO" C,j , (6. l l . 6 a )

corresponding to the schematic reaction

where, as usual, the v's are stolchlometry coefficients and the

A's are reacting species; K and m" are reaction rate constants

for the forward and reverse process as written in Eq.

(6.11.6b). Now rewrite Eq. (6.11.6a) as

= u ~ ( 1 - k~c,',), (6.11.6c)

in which ~ is the rate of the forward reaction, m~c~ i, and J L

k = ~' /~ . Referring back to Section 2.14 one notes that the affinity

may be reformulated as

A - - Z v lpl " RT ~n K- RT ~ c e , !

(6.11.7)

where K is the equilibrium constant appropriate to the reaction

(6.11.6b) when ~ is referred to the standard chemical poten-

5 8'2 6. IRREVERSIBLE THERNODYNANICS

tlal. It now follows that Ke -A/R7 - c e ; when this expression

is introduced in (6.11.6c) one flnds

- ~f(l - - kKe--AIRT). (6.11.S)

Now at equilibrium, ~ - A - O; according to (6.11.8) this means

that kK- i, so that one obtains the final expression in the

form

~0 - ~f(1 -- e --A/RT), (6.11.9)

which exhibits an exponential dependence of w on A.

It is now clear that the postulated linear dependence in

Eq. (6.11.5) obtains only if IA/RTI << i; in which case one may

approximate i - e --A/RT by A/RT. This specifies what one means

by "small departures from equilibrium" as a prerequisite for

the application of the linear phenomenological equation w- aA.

Comparing w - ~fA/RT with (6.11.5a) one notes that wf/RT -

(a~/aA)x~.~2. On the right-hand side the subscripts clearly

refer to equilibrium conditions; thus ~f may be computed from

the equilibrium values of the various concentrations, denoted

by c~ in the present context. Thus, one may set ~- ~~ (c~)Vt M G

-~. This finally leads to

- (w~/RT)A - aA (6.11.10)

as the phenomenological expression for the rate of a process;

the phenomenological coefficient a is also identified in this

procedure.

6.12 COUPLED REACTIONS : SPECIAL EXAMPLE

Consider the following case of three coupled reactions denoted

by A = B :(I), B = C :(II) and C = A :(III), and compare this

situation to the linearly independent reactions A = B:(1) and

B = C: (2). The rate of disappearance of the various reagents

COUPLED REACTIONS: SPECIAL EXAMPLE ~83

is related to the reaction rates, ~, by

- dnA/dt - wl - ~ -- &0Ill

- dnB/dt - ~z - ~I - wII - ~~

- dnc/dt - - ~z - c~ - - ~01I" (6.12.1)

Comparing the ~'s with Arabic and Roman subscripts, one obtains

O) 1 -- ~01 -- e)Ii I

~02 -- ~011 -- 0~ii I. (6.12.2)

The chemical affinity for reaction r is given by A r - ~(1)V~r~i,

where 2 refers to the various species that are involved. In

the present case

AI - A1 - ~B -- ~A

AII - - A2 - - ~ C - - ~ B

AIII-- (A1 + A2) - ~A- ~C, (6.12.3)

and the dissipation function is given by

= ~1AI + ~zA2 = ~IAI + a ~ i i A i i + ~IIIAIII . (6.12.4)

At equilibrium the ~ and A vanish for the linearly independent

reactions: ~I - ~z - 0 and A I - A z - 0. From (6.12.2) it now

follows that for the linearly dependent case

O~ I -- O~iI -- COil I (6.12.5a)

A I = AII - - AII I = 0 (equilibrium conditions). (6.12.5b)

Note that the ~'s in (6.12.5a) have been shown to be equal, but

84 6. IRREVERSIBLE T H E R M O D Y N A M I C S

at this stage they do not necessarily vanish. When Eq.

(6.11.7) is adapted to the present case one finds

A I = RT 2 n ( K I C A / C B ) , AII = RT 2 n ( K i I C B / C c ) ,

AII I - - RT 2 n (KIIICC/CA). ( 6 . 1 2 . 6 )

At this state we introduce the laws of mass action and use v to

denote the rate of the forward reaction. Also we set k- ~"/~,

where ~ and ~" are the rate constants for the forward and

reverse reactions as written. Then

~ I - ~ I C A - ~;'ICB - v I ( 1 - k i c k / c A )

~ I I = ~ i i c B - ~ ' i l C c = v i i ( 1 - k i i c c / e B )

w I I I = ' c I I I C c - ' ~ ' I I I C A - v i i i ( 1 -- k I I I C J C c ) - ( 6 . 1 2 . 7 )

When (6.22.6) is introduced in these relations one obtains the

expressions

~Z = v z [ i -- kzKze---z/RT ]

wzz = v i i [I - k I I K I I e - - A I I / R T ] (6.12.8)

~"zzz = v i i i [ i - kzzzKzzze--Azzz/R~] .

At equilibrium where the ~'s are equal (see Eq. 6.12.5a), and

when A vanishes,

v z ( l - k l A I ) ,,,, v l l ( I - kzzAzz) = V l l I ( I -- kzzzAzz z) . (6.12.9)

The principle o_f detailed balance or microscopic

reversibility is now invoked. It states that at equilibrium

each elementary process proceeds as readily in one direction as

in the other. According to this principle, one must demand

that ~I = ~II = ~III = 0; note that this enlarges on requirement

COUPLED REACTIONS: GENERAL CASE SSS

(6.12.5a). It is an immediate consequence of this requirement

that kiK i - i in Eq. (6.12.8), i - I, II, III. It then follows

that

(~I -- vI ( i -- e--AI/RT)

(~II -- VII (I - e-AII/RT) (6.12.10)

wIII -- Vli I(l -- e--AIII/RT).

These are the fundamental equations of interest. Close to

equilibrium, where IA/RTI <<I, one obtains the linearized forms

k I - (V~I/RT)AI, kIi- (v~II/RT)AII , kii I - (v~III/RT)AIII,

(6.12.11)

where the equilibrium values of the v's have been introduced in

accord with the discussion of Eq. (6.12.10). Now on account of

Eq. (6.12.2) and (6.12.3), Eq. (6.2.12) becomes

wl- (v~I + v~III)AI/RT + v~IIIAz/RT

w z -- V~IIIAI/RT + (v~ii + v~II)Az/RT. (6.12.12)

These expressions are in the form of linear phenomenological

equations ~i - aliA1 + alzA1 + azzAz, with alz - a21 - v~III/RT, all

- (v~I + ~II)/RT, azz- (v~i I + v~iIi)/RT.

This example illustrates the fundamental principle that if

one describes coupled reactions in terms of a set of linearly

independent steps, then sufficiently close to equilibrium the

reaction rates may be described in terms of phenomenological

equations involving the chemical affinities as driving forces.

6.13 COUPLED REACTIONS, GENERAL CASE

The preceding argumentation will be briefly extended to cover

the case of more complex reactions of the type


~m I V~mA~ = ~ ~j.Aj, (m- I, 2 .... ,M) ( 6 . 1 3 . 1 )

involving M elementary reactions indexed by m, of which R,

indexed by r, are linearly independent; M _> R. The notation

used below conforms to the scheme

~! m cilm C, " N ...~'jm. . . '-'j / F ~ �9 (6 13 2) e j i

The rates of the various reactions are specified by the law

of mass action

�9 u. Vtm) - - - o - v o ( 1 - ,

i j ! ( 6 . 1 3 . 3 )

wherein v m = ~mF~C[ Im is the forward rate of reaction m and k m �9 _ ~tm) m ~ m/~m. UsingiAm-- ~(,)U,m~m RT 2n(Km/F~c I one obtains

!

(Din- vm(l - kmKm e-~m/RT) ( m - 1 , 2 , . . . ,M). (6.13.4)

As already discussed, at equilibrium only the reaction

rates (Dr of the R linearly independent reactions can initially

be assumed to vanish. To handle the correspondence between the

linearly dependent and independent reactions we relate the A m

and A r affinities through linear equations of the form

R

A m -r~ibr~r , ~ ( 6.13.5 )

wherein the brm are appropriate combination coefficients.

Since the rate of entropy production must be independent

of the manner in which one writes out the reaction sequences,

one must have ~(m)(DmAm - ~(r)(DrAr; using (6.13.5), one obtains

R M R

This implies that

R

(Dr "~ Z m-1

~'~r" (6.13.6)

bm(D m ( r- 1,2,...,R). (6.13.7a)

COUPLED REACTIONS: GENERAL CASE ~87

For the set of ~r in (6.13.7), one may specify at equilibrium

that

R

~r -- X b m~ m - 0 (r - 1 2 ... R), m ~-~I ~ '

(6.13.7b)

or

Ar - 0 (equilibrium conditions). (6.13.8)

However, if one introduces the principle of detailed balance,

one can then further specify that ~m - 0 (m - 1,2,...,M); also,

Am- 0 at equilibrium. From Eq. (6.13.4) it now follows that

k~ - i, i.e.,

~m- vm(l - e-~/RT) (m- 1,2,...M), (6.13.9)

so that we have once more recovered an exponential dependence

of ~m on Am, even in the case of M coupled equations of any

degree of complexity. For ]Am/RT [ << i, ~m- ~~ ~m ~c[ Im (m

- 1,2,...,M) and Eq. (6.13.9) reduces to the followln~ linear

form

~m- (v~ (6.13.10)

where the superscripts o again refer to equilibrium conditions.

Because of the interrelations between coupled and uncoupled

reactions, one also obtains

M M Q

~)r -- (I/Rr)m~ lffi bmv~ - (I/RT) m-~l s-~l bmb'mv~

(r- 1,2,...R). (6.13.11)

This expression forms a set of linear phenomenologlcal

equations

R

arsA s, (6 13 12a) (Dr "S--i " "

8 8 6. IRREVERSIBLE THERMODYNAMICS

with

art - a,r -m~l bmb,=~=. ( 6 .13 .12b )

This set of equations obeying the Onsager reciprocity

conditions obtains only near equilibrium. It is generally the

case that the linear approximation for the reactions is valid

over a far more limited range than is the linear approximation

for the types of processes discussed in Sections 6.7-6.11.

chapter 6 irreversible thermodynamics 6.0...

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