THERMODYNAMICS OF SOLUTIONS

In order to calculate the equilibrium composition of a system consisting

of

one

or

more phases in equilibrium with an aqueous solution of electrolytes, a review of the basic thermodynamic functions and the conditions of equilibrium is impor- tant. This is particularly true inasmuch as the study of aqueous solutions requires consideration of chemical and/or ionic reactions in the aqueous phase as well as a thermodynamic framework which is, for the most part, quite different from those definitions associated with nonelectrolytes. Therefore, in this section we will review the definition of the basic thermodynamic functions. the partial molar quantities, chemical potentials, conditions of equilibrium, activities, activity coefficients, standard states, and composition scales encountered in describing aqueous solutions.

Basic Thermodynamic Functions

The thermodynamic properties of a system at equilibrium consists of two types of properties, intensive and extensive properties. The most common intensive properties encountered are the temperature, T. and pressure,

P,

which are independent of the size of a measurement sample and are constant throughout the system. In fact, our definition of true equilibrium, to be described later requires T and P to be uniform throughout the system and the constituent phases.

The most common extensive properties are volume, V and mass. As one would suspect, these extensive properties are proportional to the size of a measurement sample.

The thermodynamic properties most often encountered in describing phase equilibria of a system are functions of the state of the system. This is important since the calculation of these thermodynamic properties depends only on the existing state of the system and not the route by which this state has been reached. The following energy and energy related properties are extensive properties if they refer to the system as a whole:

The intrinsic energy E

The enthalpy H = E + P V

AQUEOUS ELECTROLYTE THERMODYNAMICS

The entropy S

The Gibbs free energy

The heat capacity at constant pressure G =

H - TS

Cp =

(g)p

The above thermodynamic properties are intensive properties when their values are expressed on a per mole basis. Other useful relationships which we will encounter are obtained through differentiation of these basic thermodynamic functions and include :

dE = TdS

-

PdV (2.1)

riH = TdS

+

VdP (2.2)

dG = -SdT + VdP (2.3)

These relationships express the first and second laws of thermodynamics for a closed system. Furthermore, from the last expression above we can also obtain

Rewriting the

T h i s equation which will be constants and

(%)*

⁼

(

% )

P =

-s

definition of the Gibbs free energy,

can be arranged to give the Gibbs-Helmholtz equation, an expression very useful in calculating the effect of temperature on equilibrium is given by:

-H

-

Solutions

-

Basic Definitions and Concepts

The pure substances from which a solution can be made are called the components, or constituents of a solution. The extensive properties of a solution are deter- mined by the pressure, temperature, and the amount of each constituent. The intensive properties of a solution are determined by the pressure, temperature and the relative amounts of each constituent. or in other words b y the pressure, temperature and composition of the solution. For aqueous solutions. the most commonly used measurement of composition of the solution is the molality, m.

Molality is defined as the number of moles of a solute in one kilogram of the solvent, and for aqueous solutions the solvent is water. One of the advantages of using the molality scale for concentration is that it is independent of temperature

11. Thermodynamics of Solutions

and thus, the density of the solution does not need to be known in order to determine the composition on a mole basis as would be required with the unit of concentration, molarity. The molality of a solute i in water is given by 1000gi/(M go) or 1000

ni/go

where gi and go are the number

of

grams of solute and solvent,

M

is the solute mo!ecule weight and ni is the number of gm-moles of solute.

The thermodynamic analysis of solutions is facilitated by the introduction of quantities that measure how the extensive thermodynamic quantities

(V,

E,

HI)

G ,

...I

of the system depend on the state variab!es T , P, and ni. This leads to the definition of partial molar quantities where, i f , w e let Y be any extensive thermodynamic property, we can define the partial molar value of Y for the ith component as:

‘i = ( e ) T , P , n j + j ^(2.8)

where nj stands for all the mole quantities except ni. It is important to note that the partial molar

(or

partial mold, which has the same meaning) quantities pertain to the individual components of the system and are also properties of the system ^as a whole. Furthermore

dY = (%)T.P,nj+l + ( e ) T , P , n jS2 dn2 +

...

-

- -

dY = Y1 dnl

+

Y2 dn2 + Y3 dn3 +

...

or dY =

c - Yi

dni (2.9)

i

Partial molar quantities are intensive properties of the solution since they depend only on the composition of the solution, not upon the total amount of each component. If w e add the several components simultaneously, keeping their ratios constant, the partial molal quantities remain the same. W e can thus integrate the above expression keeping “1, “2,

...

ⁱⁿ constant proportions and find, while holding temperature and pressure constant, that

-

-

Y = Yl”l + Y2%

+ ...

or Y

= c

Yini i

Furthermore, w e can differentiate this expression

to

obtain d Y = Y1 dnl

+

r q d Y +

Y2

dn2 + npY2 +

...

(2.10)

15

AQUEOUS ELECTROLYTE THERMODYNAMICS

Since

-

dY =

c

Yi dni i

a s has previously been shown, our total differential, equation

O = C n i i Since Y represents any G , for

Y we

obtain the O = C n i

i

C ni dYi

-

i ^(2.11)

( 2 - 9 )

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