Thermodynamics and Chemistry

CHAPTER 9 MIXTURES

9.3 GASMIXTURES 241

chemical potential. Conceptually, a standard state should be a well-defined state of the

system, which in the case of a gas is thepureideal gas atpDp. Thus, although a

constituent of an ideal gas mixture with a partial pressure of 1 bar is not in its standard

state, it has the same chemical potential as in its standard state.

Equation9.3.5will be taken as the thermodynamicdefinitionof an ideal gas mixture.
Any gas mixture in which each constituentiobeys this relation betweeniandpiat all
compositions is by definition an ideal gas mixture. The nonrigorous nature of the assump-
tion used to obtain Eq.9.3.5presents no difficulty if we consider the equation to be the basic
definition.
By substituting the expression foriinto.@i=@T /p;fnig D Si (Eq.9.2.48), we
obtain an expression for the partial molar entropy of substanceiin an ideal gas mixture:

SiD



@i(g)

@T



p;fnig

Rln

pi

p

DSiRln

pi

p

(9.3.6)

(ideal gas mixture)

The quantitySiDå@i(g)=@T çp;fnigis thestandard molar entropyof constituenti. It
is the molar entropy ofiin its standard state of pure ideal gas at pressurep.
Substitution of the expression forifrom Eq.9.3.5and the expression forSifrom Eq.
9.3.6intoHiDiCTSi(from Eq.9.2.46) yieldsHiDi(g)CTSi, which is equivalent
to

HiDHi (9.3.7)

(ideal gas mixture)

This tells us that the partial molar enthalpy of a constituent of an ideal gas mixture at a given
temperature isindependentof the partial pressure or mixture composition; it is a function
only ofT.
From.@i=@p/T;fnigDVi(Eq.9.2.49), the partial molar volume ofiin an ideal gas
mixture is given by

ViD



@i(g)

@p



T;fnig

CRT



@ln.pi=p/

@p



T;fnig

(9.3.8)

The first partial derivative on the right is zero becausei(g) is a function only ofT. For
the second partial derivative, we writepi=pDyip=p. The mole fractionyiis constant
when the amount of each substance is constant, so we haveå@ln.yip=p/=@pçT;fnigD1=p.
The partial molar volume is therefore given by

ViD

RT

p

(9.3.9)

(ideal gas mixture)

which is what we would expect simply from the ideal gas equation. The partial molar
volume is not necessarily equal to the standard molar volume, which isViDRT=pfor
an ideal gas.