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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
5 3 4 6. IRREVERSIBLE THERMODYNAMICS
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.
FIRST AND SECOND LAW IN LOCAL FORM 535
(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
FIRST AND SECOND LAW IN LOCAL FORM 541
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.
~4 6. IRREVERSIBLE THERMODYNAMICS
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).
5 6 8 6. IRREVERSIBLE THERMODYNAMICS
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
570 6. IRREVERSIBLE THERMODYNAMICS
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
~2 6. IRREVERSIBLE THERMODYNAMICS
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)
5 7 6 6. IRREVERSIBLE THERMODYNAMICS
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
578 6. IRREVERSIBLE THERMODYNAMICS
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).
5 8 0 6. IRREVERSIBLE THERMODYNAMICS
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
5 8 6 6. IRREVERSIBLE THERMODYNAMICS
~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.
top related