PHYSICAL CHEMISTRY IN BRIEF

CHAP. 6: THERMODYNAMICS OF HOMOGENEOUS MIXTURES [CONTENTS] 155

6.4 Thermodynamics of an open system and the chemical potential

6.4.1 Thermodynamic quantities in an open system

In an open system [see1.1.3], the first law of thermodynamics does not apply. The Gibbs
equations do not apply in the form defined in3.4.1but in the extended form

dU = TdS−pdV+

∑k

i=1

μidni, (6.61)

dH = TdS+Vdp+

∑k

i=1

μidni, (6.62)

dF = −SdT−pdV+

∑k

i=1

μidni, (6.63)

dG = −SdT+Vdp+

∑k

i=1

μidni, (6.64)

whereμiare thechemical potentialsof the system components.

6.4.2 Chemical potential

It follows from equations (6.64) and (6.41) that the chemical potential is identical with the
partial molar Gibbs energy of the component

μi=

(

∂G

∂ni

)

T,p,nj 6 =i

=Gi. (6.65)

However, it also follows from equations (6.61) to (6.64) that we must have

μi=

(

∂U

∂ni

)

S,V,nj 6 =i

=

(

∂H

∂ni

)

S,p,nj 6 =i

=

(

∂F

∂ni

)

T,V,nj 6 =i

. (6.66)