Topic790

Compressions: Isentropic: Solutions: Partial and Apparent Molar

Isentropic properties of aqueous solutions are defined in a manner

analogous to that used to define isothermal compressions and isothermal

compressibilities. The assertion is made that a system (e.g. an aqueous

solution) can be perturbed along a pathway where the affinity for

spontaneous change is zero by a small change in pressure δp, to a

neighbouring state having the same entropy. The (equilibrium) isentropic

compression is defined by equation (a).

0A);aq(SS ]p/)aq(V[)aq(K =∂∂−= (a)

The constraint on this partial differential refers to 'at constant S(aq)'.

The definition of KS (aq) uses non-Gibbsian independent variables. In

other words, isentropic parameters do not arise naturally from the

formalism which expresses the Gibbs energy in terms of independent

variables in the case of, for example, a simple solution, [T,p,n1,nj] [1].

The isothermal compression of a solution KT(aq) and partial molar

isothermal compressions of both solvent KT1(aq) and solute KTj (aq) are

defined using Gibbsian independent variables. Unfortunately the

corresponding equations cannot be simply carried over to the isentropic

property KS(aq). The volume of a solution is expressed in terms of the

amounts of solvent n1 and solute nj.

V aq n V aq n V aqj j( ) ( ) ( )= ⋅ + ⋅1 1 (b)

The latter equation is differentiated with respect to pressure at constant

entropy of the solution S(aq). The latter condition includes the condition

that the system remains at equilibrium where the affinity for spontaneous

change is zero. We emphasise a point. The entropy which remains

constant is that of the solution.

K aq n V aq p n V aq pS S aq A j j S aq A( ) [ ( ) / ] [ ( ) / ]( ); ( );= − ⋅ − ⋅= =1 1 0 0∂ ∂ ∂ ∂ (c)

KS (aq) is an extensive property of the aqueous solution. KS(aq) may also

be re-expressed using Euler’s theorem as a function of the composition

of the solution.

K aq n K aq n n K aq nS S T p n j j S j T p n( ) [ ( ) / ] [ ( ) / ], , ( ) , , ( )= ⋅ + ⋅1 1 1∂ ∂ ∂ ∂ (d)

Because KS(aq) is defined using non-Gibbsian independent variables,

two important inequalities follow.

− ≠[ ( ) / ] [ ( ) / ]( ) , , ( )∂ ∂ ∂ ∂V aq p K aq nS aq S T p n j1 1 (e)

)1(n,p,TjS)aq(Sj ]n/)aq(K[]p/)aq(V[ ∂∂≠∂∂− (f)

[ ( ) / ] , , ( )∂ ∂K aq nS T p n j1 and [ ( ) / ] , , (1)∂ ∂K aq nS j T p n are respectively the

partial molar properties of the solvent and solute. Because partial molar

properties should describe the effects of a change in composition on the

properties of a solution, we write equation (d) for an aqueous solution in

the following form.

)def;aq(Kn)def;aq(Kn)aq(K Sjj1S1S ⋅+⋅= (g)

Hence, )aq(SjSj ]p/)aq(V[)def;aq(K ∂∂−≠ (h)

In view of the latter inequality KSj(aq;def) is a non-Lewisian partial

molar property [2]. We could define a molar isentropic compression of

solute j as (minus) the isentropic differential dependence of partial molar

volume on pressure. This alternative definition is consistent with

equation (g) expressing a summation rule analogous to that used for

partial molar properties. However some other thermodynamic

relationships involving partial molar properties would not be valid in this

case. Therefore, )aq(Sj ]p/)aq(V[ ∂∂− is a semi-partial molar property. A

similar problem is encountered in defining an apparent molar

compression for solute j, φ(KSj ) in a solution where the solute has

apparent molar volume φ(Vj); cf. equation (h) [3,4]. We might assert that

φ(KSj) is related to the isentropic differential dependence of φ(Vj ) on

pressure, −[ ( ) / ] ( )φ ∂V pj S aq . Alternatively, using as a guide the apparent

molar properties φ(Epj ) and φ(KTj ), we could define φ(KSj;def) using

equation (i).

)def;K(n)l(Kn)aq(K Sjj*1S1S φ⋅+⋅= (i)

KSj (aq;def) as given by equation (d) and φ(KSj;def) are linked; equation

(j). )1(n,p,TjSjjSjSj ]n/)def;K([n)def;K()def;aq(K ∂∂φ⋅+φ= (j)

Equation (j) is of the general form encountered for other apparent and

partial molar properties. This form is also valid in the case of partial and

apparent molar isobaric expansions, isothermal compressions and

isobaric heat capacities. On the other hand, the semi-partial molar

isentropic compression defined by −[ ( ) / ] ( )∂ ∂V aq pj S aq and the semi-

apparent molar isentropic compression defined by −[ ( ) / ] ( )∂φ ∂V pj S aq are

related. The isentropic pressure dependence of Vj(aq) is given by

equation (k).

)aq(S)1(n,p,Tjjj)aq(Sj

)aq(Sj

}p/]n/)V([{n]p/)V([

]p/)aq(V[

∂∂∂φ∂⋅−∂∂φ−

=∂∂− (k)

However,

{ [ ( ) / ] / } { [ ( ) / ] / }, , ( ) ( ) ( ) , , ( )∂ ∂φ ∂ ∂ ∂ ∂φ ∂ ∂V n p V p nj j T p n S aq j S aq j T p n1 1≠ (l)

Hence, the analogue of equation (j) does not hold for these 'semi'

properties. The inequalities (e) and (f) highlight the essence of non-

Lewisian properties. Their origin is a combination of properties defined

in terms of Gibbsian and non-Gibbsian independent variables as in

equations (e) and (f). This combination is also the reason for the

inequality (l). We stress that the isentropic condition in equations (e) and

(f) refers to the entropy S(aq) of the solution defined as is the volume

V(aq) by the Gibbsian independent variables [ , , ]T, p n n j1 . But this is not

the entropy S1* ( )l of the pure solvent having volume V1

* ( )l . S(aq) at

fixed composition is not simply related to S1* ( )l as, for example, linear

functions of temperature and pressure.

The isentropic condition is involved in the definitions of isentropic

compression, KS1* ( )l and isentropic compressibility κ S1

* ( )l of the

solvent.

K V pS1 1* *( ) [ ( ) / ]l l= − ∂ ∂ at constant S1

* ( )l (m)

)(V/]p/)(V[

)(V/)(K)(*

1*

1

*1

*1S

*1S

ll

lll

∂∂−=

=κ at constant S1

* ( )l (n)

The different isentropic conditions in equation (a) and in equations (m)

and (n) signal a complexity in the isentropic differentiation of equation

(o) with respect to pressure [5,6].

)V(m)(VM)kg 1w;aq(V jj*

11

11 φ⋅+⋅== − l (o)

Footnotes

[1] J. C. R. Reis, M. J. Blandamer, M. I. Davis and G. Douheret, Chem.

Phys. Phys. Chem., 2001, 3,1465.

[2] J. C. R. Reis, J. Chem. Soc. Faraday Trans.,2,1982, 78,1565.

[3] M. J. Blandamer, M. I. Davis, G. Douheret and J. C. R. Reis,

Chem. Soc. Rev.,2001,30,8.

[4] J. C. R. Reis, J. Chem. Soc. Faraday Trans.,1998,94,2385.

[5] M. J. Blandamer, J. Chem. Soc. Faraday Trans.,1998,94,1057.

[6] M. J. Blandamer, Chem. Soc. Rev.,1998,27,73.