Thermodynamics and Chemistry

CHAPTER 9 MIXTURES

9.3 GASMIXTURES 238

We recognize each partial derivative as a partial molar quantity and rewrite the equation as

iDHiTSi (9.2.46)

This is analogous to the relationDG=nDHmTSmfor a pure substance.
From the total differential of the Gibbs energy, dGDSdTCVdpC

P

iidni(Eq.
9.2.34), we obtain the following reciprocity relations:

@i
@T



p;fnig

D



@S

@ni



T;p;nj§i



@i

@p



T;fnig

D



@V

@ni



T;p;nj§i

(9.2.47)

The symbolfnigstands for the set of amounts of all species, and subscriptfnigon a partial
derivative means the amount ofeachspecies is constant—that is, the derivative is taken at
constant composition of a closed system. Again we recognize partial derivatives as partial
molar quantities and rewrite these relations as follows:

@i
@T



p;fnig

DSi (9.2.48)



@i

@p



T;fnig

DVi (9.2.49)

These equations are the equivalent for a mixture of the relations.@=@T /pD Smand
.@=@p/TDVmfor a pure phase (Eqs.7.8.3and7.8.4).
Taking the partial derivatives of both sides ofUDHpVwith respect toniat constant
T,p, andnj§igives
UiDHipVi (9.2.50)
Finally, we can obtain a formula forCp;i, the partial molar heat capacity at constant
pressure of speciesi, by writing the total differential ofHin the form

dHD



@H

@T



p;fnig

dTC



@H

@p



T;fnig

dpC

X

i



@H

@ni



T;p;nj§i

dni

DCpdTC



@H

@p



T;fnig

dpC

X

i

Hidni (9.2.51)

from which we have the reciprocity relation.@Cp=@ni/T;p;nj§iD.@Hi=@T /p;fnig, or

Cp;iD



@Hi

@T



p;fnig

(9.2.52)

9.3 Gas Mixtures

The gas mixtures described in this chapter are assumed to be mixtures of nonreacting
gaseous substances.