1

Chapter 9The chemical potential and open systems

2

Closed system: System does not exchange matter with surroundings.

Open system: Quantity of matter not fixed.

Chemical Potential.We give a brief introduction to this concept as we will

need to refer to it later in the course. As the name suggests, it is very important for studying chemical reactions and your will encounter this concept in a first course in physical chemistry. Just as the temperature governs the flow of energy between two systems, the chemical potential governs the flow of particles. When the chemical potential of the two systems are equal, they are in diffusive equilibrium.

We are now going to consider the introduction of matter into a system. If we introduce, say, dn kilomoles of matter into a system, there will be a change in energy of the system and dU will be proportional to dn

3

For a pure substance, with a constant number of particles, we had, for a quasi-static process: )1.......(PdVQdU

and for a reversible process, dU=TdS –PdV ………(2)

We now consider a more general case where a system may consist of several different constituents. Furthermore, the system is no longer necessarily closed so that the number of moles of the various constituents may vary.

in = number of moles of constituent i

We now generalize equation (2) to allow for the possibility of adding of removing particles from our system. Before we had U(S,V) and now

sonVSU i ),,( )3.....(dnn

UdV

V

UdS

S

UdU i

n,V,Siin,Sn,V j

In the first two derivatives, all the n are constant and in the third derivative all the n are constant except in

đ

4

Comparing equations (2) and (3)

nSnVV

UP

S

UT

,,

and now we define the chemical potentials by

jn,V,Si

i n

U

Equation (3) can now be written as )4.....(ii

idnPdVTdSdU

Solving for dS: ii

i dnT

dVT

PdU

TdS

1

and sojnVUi

i

nUnVn

S

TV

S

T

P

U

S

T,,,,

1

The Helmholtz function is F=U-TS

5

SdTTdSdnPdVTdSSdTTdSdUdF ii

i )(

soanddnSdTPdVdF ii

i

jnTVii

nVnTn

F

T

FS

V

FP

,,,,

EXAMPLE: Mixture of two ideal monatomic gases We will take as given the Helmholtz Thermodynamic Potential.

),,,( 21 nnTVF From the viewpoint of thermodynamics this comes from

experiment. In statistical mechanics this is derived from a model in which two types of molecules are present.

22

222222

21

111111

2ln

23

)ln(ln23

ln

2ln

23

)ln(ln23

ln

h

kmnnNnnTnVn

h

kmnnNnnTnVn

RTF

A

A

(reciprocity relations)

6

m=molecular mass =Avogadro’s number h=Planck’s constant AN

nRTPVRTnnPVV

n

V

nRTP

V

FP

nT

)( 2121

,

If we have 02 n

If we have01 n

V

nRTP 1

1

V

nRTP 2

2

so 21 PPP

Dalton’s Law The pressure of a mixture of ideal gases is equal to the sum of the partial pressures. The partial pressure of a gas is the pressure that it would exert if it alone occupied the volume V at temperature T.

)(23

21

,

nnRT

FS

T

FS

nV

7

2,,11

nTVn

F

21

11

2ln

23

11)ln(ln23

lnh

kmNnTVRT A

21

11

2ln

23

)ln(ln23

lnh

kmNnTVRT A

)(

23

21 nnRT

FTFTSFU

2121 )(23

UUUnnRTU

From the fact sheet we obtain G=F+PV so RTnnFG )( 21

8

22

22222

21

11111

2ln

23

)ln(ln23

ln

2ln

23

)ln(ln23

ln

h

kmnNnnTnVn

h

kmnNnnTnVn

RTG

A

A

21 GGG

Comparison with the expression for and a similar expression for gives

1 22211 nnG In general i

iinG

For a system consisting of just one constituent (one phase) we have

gn

GornG

We see that for such a simple system, the chemical potential is just the Gibb’s function.

{We mentioned earlier that the Gibbs Potential was particularly important in physical chemistry.}