The Thermodynamic PotentialsFour Fundamental Thermodynamic Potentials

dU = TdS - pdV

dH = TdS + Vdp

dG = Vdp - SdT

dA = -pdV - SdT

The appropriate thermodynamic potentialto use is determined by the constraints imposed on the system. For example,since entropy is hard to control (adiabaticconditions are difficult to impose) G and Aare more useful. Also in the case of solidsp is a lot easier to control than V so G isthe most useful of all potentials for solids.

( , ); 0U U S V dU

( , ); 0H H S P dH

( , ); 0G G P T dG

( , ); 0A A V T dA

fixed V,S

fixed S,P

fixed P,T

fixed T,V

Equilibrium

Our discussion of these thermodynamic potentials has considered only“closed” (fixed size and composition) systems to this point. In this casetwo independent variables uniquely defines the state of the system.

For example for a system at constant P and T the condition, dG = 0 defines equilibrium, i.e., equilibrium is attained when the Gibbs potential or Gibbs FreeEnergy reaches a minimum value.

If the composition of the system is variable in that the number of moles of thevarious species present changes (e.g., as a consequence of a chemical reaction)then minimization of G at fixed P and T occurs when the system has a unique composition.

For example, for a system containing CO, CO2, H2 and H2O at fixed P and T,minimization of G occurs when the following reaction reaches equilibrium.

2 2 2CO H O CO H

Since G is an EXTENSIVE property, for multi-component or open systemit is necessary that the number of moles of each component be specified.i.e.,

1 2 3, , , , ,... iG G T P n n n n

Then

1 2 1 2 2 3 1 3

1 2, , ... , , ... 1 2, , , ... , , , ...

...P n n T n n P T n n P T n n

G G G GdG dT dP dn dn

T P n n

If the number of moles of each of the individual species remain fixed weknow that

dG SdT VdP 1 2 1 2, , ... , , ...

; P n n T n n

G GS V

T P

...1 , , j

k

ii i T P n

Gdn

n

...1 , , j

k

ii i T P n

GdG SdT VdP dn

n

Chemical potential: The quantity, , ...T P n j

i ii

GG

n

is called the chemical potential of component i. It correspond to the rate of change of G with ni when the component i is added to the system at fixed P,T

and number of moles of all other species.

1

k

i ii

dG SdT VdP dn

One can add the same open system term, for the other thermodynamicpotentials, i.e., U, H and A.

i idn

2 3 1 3

1 21 2, , , ... , , , ...

1 1 2 2

...

...

P T n n P T n n

V VdV dn dn

n n

dV V dn V dn

Physically this corresponds to how the volume in the system changes upon

addition of 1 mole of component ni at fixed P,T and mole numbers of other

components.

The chemical potential is the partial molal Gibbs Free Energy (or U,H, A) of

component i. Similar equations can be written for other extensive variables, e.g.,

Maxwell relations: These mathematical relations are used to connect experimentally measurable quantities to those that are not easilyaccessible

Consider the relation for the Gibbs Free Energy:

dG SdT VdP

at fixed T

T

GV

P

at fixed P

P

GS

T

Now take the derivative of these quantities at fixed P and T respectively,

T PP

G V

T P T

P TT

G S

P T P

T PP

G V

T P T

P TT

G S

P T P

If we compare the LHS of these equations, they must be equal since G is a state function and an exact differential and the order of differentiation is inconsequential,

T PP T

G G

T P P T

T P

S V

P T

So, the RHS of each of these equations must be equal,

Similarly we can develop a Maxwell relation from each of the other threepotentials:

T P

T V

S P

S V

S V

P T

S P

V T

T V

P S

T P

V S

A

B

C

D

Let’s see how these Maxwell relations ca be useful. Consider the followingfor the entropy.

,

V T

S S T V

S SdS dT dV

T V

Using the definition of the constant volume heat capacity and the definition ofentropy for a reversible process

VV V

q dUC

dT dT

revqdS

T

rev VTdS q dU nc dT

Dividing by dT, the entropy change with temperature at fixed P is

V

V

ncS

T T

V

T

nc SdS dT dV

T V

Then,

V T

S SdS dT dV

T V

For the entropy change with volume at fixed T we can use the Maxwell relation B

T V

S P

V T

V

V

nc PdS dT dV

T T

Now from the ideal gas law, PV = nRT

V

P nR

T V

Vnc nRdS dT dV

T V

Integrating between states 1 and 2,

2 22 1

1 1

ln lnV

T VS S nc nR

T V

This equation can be used to evaluate the entropy change at fixed T,problem 4.1.

Some important bits of information

For a mechanically isolated system kept at constant temperature and volumethe A = A(V, T) never increases. Equilibrium is determined by the state ofminimum A and defined by the condition, dA = 0.

For a mechanically isolated system kept at constant temperature and pressurethe G = G(p, T) never increases. Equilibrium is determined by the state ofminimum G and defined by the condition, dG = 0.

Consider a system maintained at constant p. Then

2

1

2

1

2

1

2

1

2

1

11212

12

T

T

T

Tp

T

Tp

T

Tp

T

T

TlndTCdTTSTTTGTGG

TlndTCTSTS

TlndTCdTTSS

SdTG

Temperature dependence of H, S, and G

Consider a phase undergoing a change in temp @ const P

ppp dT

dH

dT

qC

pH C ( t )dT

TdCT

dTC

T

qdS p

p ln

pS C (T )d lnT and SdTVdpdG

@ Const. P

SdTdG

2

1

T

T

G S(T )dT TdTCS p ln)(

2

1

ln)()()( 12

T

T

p TdTCTSTS

2

1

2

1

ln)()()()( 11212

T

T

p

T

T

TdTCdTTSTTTGTGG

2

1

1

T

T

G T S(T ) SdT

Temperature dependence of H, S, and G

Temperature Dependence of the Heat Capacity

Cp

1

R

CV

3

T (K)

Contributions to Specific Heat

1. Translational motion of free electrons ~ T1

2. Lattice vibrations ~ T3

3. Internal vib. within a molecule4. Rotation of molecules5. Excitation of upper energy levels6. Anomalous effects

Dulong and Petit value

Temp. dep. of H, S, and G

H

H0 298K

ref. state for H is arbitrarilyset@ H(298) = 0 and P = 1 atm for elemental substances

T

S0 ≡ 0 pure elemental solids, Third Law

S

T

T

G

G0

G H TS

T

G

H

TS

G

H

slope = Cp

slope = -S

Thermodynamic Description of Phase Transitions

1. Component Solidifica-tion

m

L TG

T

T* Tm

Gsolid

G

Gliquid

s s sG H TS l l lG H TS

STHTGTGTG ls )()()(

T

@ T= Tm

Gl =Gs

dGl = dGs ; G = 0

m m

H LΔS =

T T

Where L is the enthalpy change

of the transition or the heat of fu-

sion (latent heat).

For a small undercooling to say T*

ΔH and ΔS are constant (zeroth order approx.)

) (SH ΔG mT

HTHT

*

m m

T LΔG H(1 ) T

T T Where ΔT = T m – T*

* Note that L will be negative

The location of the transition temp Tm will change with pressure

dTSdpVdG lll

dTSdpVdG sss

sl dGdG

V

S

VV

SS

dT

dp

sl

sl

mT

LS

m

dp L

dT T V Clapeyron equation

For a small change in melting point ΔT, we can assume that ΔS & ΔH are constant so

PH

VTT

The Clapeyron eq. governs the vapor pressure in any first order transition.

Melting or vaporization transitions are called first-order transitions

(Ehrenfest scheme) because there is a discontinuity in entropy, volume etc

which are the 1st partial derivatives of G with respect to Xi i.e.,

ST

G

p

;T

GV

p

0S 0V

There are phase transition for which ΔS = 0 and ΔV = 0 i.e., the first

derivatives of G are continuous. Such a transition is not of first-order.

According to Ehrenfest an nth order transition if at the transition point.

n

n

n

n

T

G

T

G

21

Whereas all lower derivatives are equal.

There are only two transitions known to fit this schemegas – liquid2nd order trans. in superconductivity

Notable exceptions are;Curie pt. trans in ferromag.Order-disorder trans. in binary alloysλ – transition in liquid helium