Properties of chemical potential

Constant Pressure

The dependence of G/T on temperature at constant

pressure is also of interest, and it is derived by apply-

ing the ordinary rule of differentiation to (G/T)/ T, eq

26, and using the definition of enthalpy, H, to obtain

(e.g., Castellan 1983)

Equation 27 is known as the GibbsHelmholtz equa-

tion.

Vapor

For an equilibrium mixture (i.e., i is constant) at

constant temperature and pressure,

dG = ∑ i (dηi )

(28)

i

which can be integrated to obtain

∆G = ∑ i (∆ηi )

T' T

Tf'

T

(29)

f

b

i

For ηinitial = 0 = Ginitial, we obtain

Figure 1. Plot of as a function of temperature for a

pure solvent (solid lines). The dashed line represents

G = ∑ ηi i .

the chemical potential of the liquid solvent when sol-

(30)

ute i is present. (After Castellan 1983.)

i

Differentiating eq 23 and setting it equal to eq 4b results

 i 

in

 = V.



(34)

 P T

∑ ηi d i = -SdT + VdP.

(31)

i

Consider the equilibrium of a pure substance in two

Equation 31 is the GibbsDuhem equation. Note that

phases, a and b:

for constant temperature and pressure,

a(T, P) = b(T, P).

(35)

∑ ηid i = 0.

(32)

i

From eq 34 we know that a pressure increase, dP,

will result in a chemical potential increase, d. This

Chemical equilibrium between phases

will be accompanied by a change in equilibrium tem-

of a single component

For a system in chemical equilibrium containing

perature (e.g., Fig. 1). At (T + dT, P + dP), the new

more than one phase of a substance, the chemical poten-

equilibrium condition can be expressed as

tials of the substance in all phases must be equal. For a

a(T, P) + da = b(T, P) + db.

system containing a pure substance only, we know from

(36)

eq 19 that

Subtracting eq 36 from 35 results in da = db, or by

 i 

substituting each of these expressions into eq 19 and

 = -S.



(33)

 T P

setting these equal to each other,

(Sb - Sa )dT = (Vb - Va )dP

Thus, a plot of vs. T for any phase will have a slope

(37)

of -S and equilibrium between phases occurs when

or

the chemical potentials of both phases are equal (Fig.

1). Proceeding from solid to liquid to vapor, the nega-

 P   ∆S 

 =

(38)

.

tive slopes increase, reflecting the increase in entropy

 T   ∆V 

(eq 33). Figure 1 shows that if the chemical potential

Equation 38 is known as the Clapeyron equation, an

of the liquid phase is lowered (e.g., adding salt to water

important equation of equilibrium between two phases

lowers the chemical potential of the water--see eq 25),

of a substance. Phase diagrams, such as the one for pure

there will be an accompanying decrease in the freezing

water shown in Figure 2, consist of lines that represent

point and increase in the boiling point. From eq 19

6

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