In equilibrium S is a maximum.

• Closed, isothermal and isochoric system: when volume and temper-

ature are kept constant (dV = 0, dT = 0), then dq = dU and T dS ≥ dU .

This implies that

dA ¤ 0. (16.45)

The system will spontaneously evolve in the direction of lowest Helmholtz

free energy. In equilibrium A is a minimum.

16.5 Phase behavior 433

• Closed, isothermal and isobaric system: When pressure and temper-

ature are kept constant (dp = 0, dT = 0), then dq = dH and T dS ≥ dH.

This implies that

dG ¤ 0. (16.46)

The system will spontaneously involve in the direction of lowest Gibbs free

energy. In equilibrium G is a minimum.

For open systems under constant p and T that are able to exchange material,

we can formulate the second law as follows: the system will spontaneously

evolve such that the thermodynamic potential of each component becomes

homogeneous. Since G = i ni μi , the total G would decrease if particles

would move from a region where their thermodynamic potential is high to a

region where it is lower. Therefore particles would spontaneously move until

their μ would be the same everywhere. One consequence of this is that the

thermodynamic potential of any component is the same in two (or more)

coexisting phases.

16.5 Phase behavior

A closed system with a one-component homogeneous phase (containing n

moles) has two independent variables or degrees of freedom, e.g., p and T .

All other state functions, including V , are now determined. Hence there is

relation between p, V, T :

¦(p, V, T ) = 0, (16.47)

which is called the equation of state (EOS). Examples are the ideal gas

EOS: p v = RT (where v is the molar volume V /n), or the van der Waals

gas (p + a/v 2 )(v ’ b) = RT . If two phases, as liquid and gas, coexist, there is

the additional restriction that the thermodynamic potential must be equal

in both phases, and only one degree of freedom (either p or T ) is left. Thus

there is a relation between p and T along the phase boundary; for the liquid“

vapor boundary boiling point and vapor pressure are related. When three

phases coexist, as solid, liquid and gas, there is yet another restriction which

leaves no degrees of freedom. Thus the triple point has a ¬xed temperature

and pressure.

Any additional component in a mixture adds another degree of freedom,

viz. the mole fraction of the additional component. The number of degrees

of freedom F is related to the number of components C and the number of

coexisting phases P by Gibbs™ phase rule:

F = C ’ P + 2, (16.48)

434 Review of thermodynamics

C

xC

xB

A B

xA

(a) (b)

Figure 16.1 (a) points representing mole fractions of 3 components A,B,C in a

cartesian coordinate system end up in the shaded triangle (b) Each vertex represents

a pure component and each mole fraction is on a linear 0-1 scale starting at the

opposite side. The dot represents the composition xA = 0.3, xB = 0.5, xC = 0.2.

as the reader can easily verify.

A phase diagram depicts the phases and phase boundaries. For a single

component with two degrees of freedom, a two-dimensional plot su¬ces, and

one may choose as independent variables any pair of p, T , and either V or

molar density ρ = n/V . Temperature“density phase diagrams contain a

coexistence region where a single phase does not exist and where two phases

(gas and liquid, or solid and liquid) are in equilibrium, one with low density

and one with high density. In simulations on a small system a density-

temperature combination in the coexistence region may still yield a stable

¬‚uid with negative pressure. A large amount of real ¬‚uid would separate

because the total free energy would then be lower, but the separation is a

slow process, and in a small system the free energy cost to create a phase

boundary (surface pressure) counteracts the separation.

For mixtures the composition comes in as extra degrees of freedom. Phase

diagrams of binary mixtures are often depicted as x, T diagrams. Ternary

mixtures have two independent mole fractions; if each of the three mole

fractions are plotted along three axes of a 3D cartesian coordinate system,

the condition i xi = 1 implies that all possible mixtures lie on the plane

through the points (1,0,0), (0,1,0) and (0,0,1) (Fig. 16.1a). Thus any com-

position can be depicted in the corresponding triangle (Fig. 16.1b).

Along a phase boundary between two phases 1 and 2 in the T, p plane we

know that the thermodynamic potential at every point is equal on both sides

of the boundary. Hence, stepping dT, dp along the boundary, dμ1 = dμ2 :

dμ1 = v1 dp ’ s1 dT = dμ2 = v2 dp ’ s2 dT. (16.49)

16.6 Activities and standard states 435

Therefore, along the boundary the following relation holds

dp ”s 1 ”h

= = , (16.50)

dT ”v T ”v

where ” indicates a di¬erence between the two phases. These relations

are exact. If we consider the boiling or sublimation line, and one phase

can be approximated by an ideal gas and the other condensed phase has a

negligible volume compared to the gas, we may set ”v = RT /p, and we

arrive at the Clausius“Clapeyron equation for the temperature dependence

of the saturation pressure

”hvap

d ln p

= . (16.51)

RT 2

dT

This equation implies that the saturation pressure increases with tempera-

ture as

’”hvap

p(T ) ∝ exp . (16.52)

RT

16.6 Activities and standard states

The thermodynamic potential for a gas at low pressure, which approaches

the ideal gas for which the molar volume v = RT /p, is given by

p

p

0 0

v dp = μ0 (p0 ) + RT ln

μ(p) = μ (p ) + . (16.53)

p0

p0

Here μ0 is the standard thermodynamic potential at some standard pressure

p0 . For real gases at non-zero pressure the thermodynamic potential does

not follow this dependence exactly. One writes

γp

μ(p) = μ0 + RT ln 0 , (16.54)

p

where γ is the activity coe¬cient and f = γp is called the fugacity of the

gas. It is the pressure the gas would have had it been ideal. For p ’ 0,

γ ’ 1.

For solutions the thermodynamic potential of a dilute solute behaves in

a similar way. When the concentration (or the mole fraction or molality)

of a component approaches zero, the thermodynamic potential of that com-

ponent becomes linear in the logarithm of the concentration (mole fraction,

molality):

μ(c) = μ0 + RT ln(γc c/c0 ), (16.55)

c

μ(m) = μ0 + RT ln(γm m/m0 ), (16.56)

m

μ(x) = μ0 + RT ln(γx x). (16.57)