S.I. units.

16.2 De¬nitions 427

Table 16.1 Thermodynamic state functions and their de¬nitions and units.

All variables with upper case symbols are extensive and all variables with

lower case symbols are intensive, with the following exceptions: n and m

are extensive; T and Mi are intensive.

De¬nition Name S.I. unit

m3

V volume

p see text pressure Pa

T see text temperature K

n total amount of moles -

m total mass of system kg

densitya kg/m3

ρ m/V

U see text internal energy J

S see text entropy J/K

H U + pV enthalpy J

U ’ TS

A Helmholtz free energy J

G A + pV Gibbs free energy J

= H ’ TS or Gibbs function

ni moles of ith component mol

Mi molar mass of ith component kg/mol

xi ni /n mole fraction of ith component -

concentrationb of ith componenet mol/m3

ci ni /V

mi ni /ms molality of ith componenet mol/kg

(mol solute per kg solvent) = molal

CV (‚U/‚T )V,ni isochoric heat capacity J/K

Cp (‚H/‚T )p,ni isobaric heat capacity J/K

J mol’1 K’1

cV CV /n molar isochoric heat capacity

J mol’1 K’1

cp Cp /n molar isobaric heat capacity

J kg’1 K’1

cV CV /m isochoric speci¬c heat

J kg’1 K’1

cp Cp /m isobaric speci¬c heat

K’1

± (1/V )(‚V /‚T )p,ni volume expansion coe¬.

Pa’1

’(1/V )(‚V /‚p)T,ni

κT isothermal compressibility

Pa’1

’(1/V )(‚V /‚p)S,ni

κS adiabatic compressibility

μJT (‚T /‚p)H,ni Joule“Thomson coe¬cient K/Pa

a The symbol ρ is sometimes also used for molar density or concentration, or for number density:

particles per m3 .

b The unit “molar”, symbol M, for mol/dm3 = 1000 mol/m3 is usual in chemistry.

16.2.1 Partial molar quantities

In Table 16.1 derivatives with respect to composition have not been included.

The partial derivative yi of an extensive state function Y (p, T, ni ), with

respect to the number of moles of each component, is called the partial

428 Review of thermodynamics

molar Y :

‚Y

yi = . (16.7)

‚ni p,T,nj=i

For example, if Y = G, we obtain the partial molar Gibbs free energy, which

is usually called the thermodynamic potential or the chemical potential :

‚G

μi = , (16.8)

‚ni p,T,nj=i

and with the volume V we obtain the partial molar volume vi . Without fur-

ther speci¬cation partial molar quantities are de¬ned at constant pressure

and temperature, but any other variables may be speci¬ed. For simplicity of

notation we shall from now on implicitly assume the condition nj=i = con-

stant in derivatives with respect to ni .

If we enlarge the whole system (under constant p, T ) by dn, keeping the

mole fractions of all components the same (i.e., dni = xi dn), then the system

enlarges by a fraction dn/n and all extensive quantities Y will enlarge by a

fraction dn/n as well:

Y

dY = dn. (16.9)

n

But also:

‚Y

dY = dni = yi xi dn. (16.10)

‚ni p,T,nj=i

i i

Hence

Y =n xi yi = n i yi . (16.11)

i i

Note that this equality is only valid if the other independent variables are

intensive state functions (as p and T ) and not for, e.g., V and T . The most

important application is Y = G:

G= ni μi . (16.12)

i

This has a remarkable consequence: since

dG = μi dni + ni dμi , (16.13)

i i

but also, as a result of (16.8),

(dG)p,T = μi dni , (16.14)

i

16.3 Thermodynamic equilibrium relations 429

it follows that

ni (dμi )p,T = 0. (16.15)

i

This equation is the Gibbs“Duhem relation, which is most conveniently ex-

pressed in the form

xi (dμi )p,T = 0. (16.16)

i

The Gibbs“Duhem relation implies that not all chemical potentials in a

mixture are independent. For example, consider a solution of component s

(solute) in solvent w (water). If the mole fraction of the solute is x, then

xw = (1 ’ x) and xs = x, (16.17)

and

x dμs + (1 ’ x) dμw = 0. (16.18)

This relation allows derivation of the concentration dependence of μs from

the concentration dependence of μw . The latter may be determined from

the osmotic pressure as a function of concentration.

There are numerous other partial molar quantities. The most important

ones are

‚V

vi = , (16.19)

‚ni p,T

‚U

ui = , (16.20)

‚ni p,T

‚H

hi = , (16.21)

‚ni p,T

‚S

si = , (16.22)

‚ni p,T

which are related by

μi = hi ’ T si and hi = ui + p vi . (16.23)