BioPHYSICAL chemistry

CHAPTER 4 PHASE DIAGRAMS AND MIXTURES 75

μA=μ^0 A+RTlnXA and μB=μ^0 B+RTlnXB (4.6)

Since there are only two components, the sum of the mole fractions must
be equal to one (XA+XB=1). Hence the chemical potential of A can be
expressed in terms of the mole fraction of B:

Since XA+XB=1, μA=μ^0 A+RTlnXA=μ^0 A+RTln(1 −XB)

≈ (4.7)

ln(1 −X) ≈X+X^2 /2

Expressing the mole fraction XBin terms of the solute concentration cBfor
a dilute solution gives:

(4.8)

and the standard free energy becomes:

(4.9)

where B is thesecond 9 iral coefficient. For an ideal solution, the second viral
coefficient can be written as:

(4.10)

When B >Bidealthe chemical potential decreases more rapidly than the
ideal case and the system behaves as a good solvent. When B <<0, the
chemical potential decreases less rapidly and the system behaves as a poor
solvent and phase separation occurs. For lipids and detergents, this phase
separation can be in the form of the micelles, bilayers, or other aggregates.

Lipid and detergent formation into micelles and bilayers

The actual packing of the lipids will be determined by the geometric aspects
and interactions between head groups. For lipids, the formation of the
bilayers (Figure 4.4) is a rapid and spontaneous process once the con-
centration reaches a critical point. Other types of lipid arrangement are

B

V

ideal M

= A

(^2) B
2
μμAA AB
B

=− +B

⎛

⎝

⎜⎜

⎞

⎠

(^02) RTV ⎟⎟
c
M
Bc

X

c

B M

B

B

Moles B

Moles A

gml

gmol

==

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

−

−

1

1 VV

cV

A M

BA

B

ml mol−^1 =

μAB^0 B

2

2

++

⎛

⎝

⎜⎜

⎞

⎠

RT X ⎟⎟

X