’ dn1 ’

dS1 + dS2 = + dn2 . (16.91)

T1 T2 T1 T2

i i

Now call the entropy production per unit volume and per unit time σ. If

we call the energy ¬‚ux per unit time and per unit of surface area Ju (in

J m’2 s’1 ) and the particle ¬‚ux Ji (in mol m’2 s’1 ), then

dU2 dU1

=’

Ju ”y ”z = , (16.92)

dt dt

dn2i dn1i

=’

Ji ”y ”z = , (16.93)

dt dt

so that

1 1 μ2i μ1i

’ ’ ’

σ ”x ”y ”z = Ju Ji ”y ”z. (16.94)

T2 T1 T2 T1

i

Equating the di¬erences between 1 and 2 to the gradient multiplied by ”x,

446 Review of thermodynamics

and then extending to three dimensions, we ¬nd

1 μi

σ = Ju · ∇ ’ Ji · ∇ . (16.95)

T T

i

This equation formulates the irreversible entropy production per unit volume

and per unit time as a sum of scalar products of ¬‚uxes J ± and conjugated

forces X ± :

J ± · X ± = JT X,

σ= (16.96)

±

where the gradient of the inverse temperature is conjugate to the ¬‚ux of

internal energy, and minus the gradient of the thermodynamic potential

divided by the temperature, is conjugate to the particle ¬‚ux. The last

form in (16.96) is a matrix notation, with J and X being column vector

representations of all ¬‚uxes and forces.

It is possible and meaningful to transform both ¬‚uxes and forces such

that (16.96) still holds. The formulation of (16.94) is very inconvenient:

for example, a temperature gradient does not only work through the ¬rst

term but also through the second term, including the temperature depen-

dence of the thermodynamic potential. Consider as independent variables:

p, T, ¦, xi , i = 1, . . . , n ’ 112 and use the following derivatives:

‚μi ‚μi ‚μi

= ’si ,

= vi , = zi F, (16.97)

‚p ‚T ‚¦

where zi is the charge (including sign) in units of the elementary charge of

species i. Equation (16.94) then transforms to

1 1 1 1

σ = Jq · ∇ ’ J v · ∇p + j · E ’ J i · (∇μi )p,T . (16.98)

T T T T

i

Here we have used the relation μi = hi ’ T si and the following de¬nitions:

• J q is the heat ¬‚ux (in J m’2 s’1 ), being the energy ¬‚ux from which the

contribution as a result of particle transport has been subtracted:

def

Jq = Ju ’ J i hi . (16.99)

i

Note that the energy transported by particles is the partial molar enthalpy,

not internal energy, because there is also work performed against the

pressure when the partial molar volume changes.

With n components there are n ’ 1 independent mole fractions. One may also choose n ’ 1

12

concentrations or molalities.

16.10 Thermodynamics of irreversible processes 447

• J v is the total volume ¬‚ux (in m/s), which is a complicated way to express

the bulk velocity of the material:

def

Jv = J i vi . (16.100)

i

• j is the electric current density (in A/m2 ):

def

j= J i zi F. (16.101)

i

Note that the irreversible entropy production due to an electric current is

the Joule heat divided by the temperature.

The last term in (16.98) is related to gradients of composition and needs

to be worked out. If we have n species in the mixture, there are only

n ’ 1 independent composition variables. It is convenient to number the

species with i = 0, 1, . . . , n ’ 1, with i = 0 representing the solvent, and

xi , i = 1, . . . , n ’ 1 the independent variables. Then

x0 = 1 ’ xi , (16.102)

i

where the prime in the summation means omission of i = 0. The Gibbs“

Duhem relation (16.16) relates the changes in chemical potential of the dif-

ferent species. It can be written in the form

x0 (∇μ0 )p,T + xi (∇μi )p,T = 0. (16.103)

i

Using this relation, the last term in (16.98) can be rewritten as

1 1 xi

’ J i · (∇μi )p,T = ’ Ji ’ · (∇μi )p,T . (16.104)

J0

T T x0

i i

Here a di¬erence ¬‚ux with respect to ¬‚ow of the solvent appears in the

equation for irreversible entropy production. If all particles move together

at the same bulk ¬‚ux J (total moles per m2 and per s), then Ji = xi J for

all i, and Jid = 0. So this de¬nition makes sense: a concentration gradient

produces a di¬erence ¬‚ux by irreversible di¬usion processes.

Note that relation (16.98), including (16.104), has the form of a product

of ¬‚uxes and forces as in (16.96). Hence also for these ¬‚uxes and forces

linear and symmetric Onsager relations (Section 16.10.3) are expected to be

valid.

448 Review of thermodynamics

16.10.2 Chemical reactions

Whenever chemical reactions proceed spontaneously, there is also an irre-

versible entropy production. According to (16.90), entropy is generated

when the composition changes according to ’(1/T ) i μi dni . In the pre-

vious subsection this term was evaluated when the number of molecules

changed due to ¬‚uxes. We have not yet considered what happens if the

numbers change due to a chemical reaction.

Assume the reaction13

0’ νi Ci (16.105)

i

proceeds for a small amount, ”n mole, as written. This means that νi ”n

moles of species Ci are formed (or removed when νi is negative). The irre-

versible entropy production ”S is

1

”S = ’ ( νi μi )”n. (16.106)

T

i

With the following de¬nitions:

def

• the a¬nity of the reaction A = ’ i νi μi ;

def