Thermodynamics and Chemistry

CHAPTER 9 MIXTURES

9.2 PARTIALMOLARQUANTITIES 228

Physically, this means that when a small amount of crystalline MgSO 4 dissolves at

constant temperature in water, the liquid phase contracts. This unusual behavior is due

to strong attractive water–ion interactions.

9.2.2 The total differential of the volume in an open system

Consider an open single-phase system consisting of a mixture of nonreacting substances.
How many independent variables does this system have?
We can prepare the mixture with various amounts of each substance, and we are able
to adjust the temperature and pressure to whatever values we wish (within certain limits
that prevent the formation of a second phase). Each choice of temperature, pressure, and
amounts results in a definite value of every other property, such as volume, density, and
mole fraction composition. Thus, an open single-phase system ofCsubstances has 2 CC
independent variables.^3
For a binary mixture (C D 2 ), the number of independent variables is four. We may
choose these variables to beT,p,nA, andnB, and write the total differential ofV in the
general form

dV D



@V

@T



p;nA;nB

dTC



@V

@p



T;nA;nB

dp

C



@V

@nA



T;p;nB

dnAC



@V

@nB



T;p;nA

dnB (9.2.5)

(binary mixture)

We know the first two partial derivatives on the right side are given by^4

@V
@T



p;nA;nB

DV



@V

@p



T;nA;nB

DTV (9.2.6)

We identify the last two partial derivatives on the right side of Eq.9.2.5as the partial molar
volumesVAandVB. Thus, we may write the total differential ofVfor this open system in
the compact form

dV DVdTTVdpCVAdnACVBdnB (9.2.7)

(binary mixture)

If we compare this equation with the total differential ofV for a one-componentclosed
system, dV DVdTTVdp(Eq.7.1.6), we see that an additional term is required for
each constituent of the mixture to allow the system to be open and the composition to vary.
WhenTandpare held constant, Eq.9.2.7becomes

dV DVAdnACVBdnB (9.2.8)

(binary mixture,

constantTandp)

(^3) Cin this kind of system is actually the number ofcomponents. The number of components is usually the
same as the number of substances, but is less if certain constraints exist, such as reaction equilibrium or a fixed
mixture composition. The general meaning ofCwill be discussed in Sec.13.1.
(^4) See Eqs.7.1.1and7.1.2, which are for closed systems.