BioPHYSICAL chemistry

86 PARTI THERMODYNAMICS AND KINETICS

So the chemical potential of molecule A in a mixture can be determined

by the ratio of the partial pressure and the pressure of the pure state.

The relationship between the vapor pressures and chemical composition

of liquids was established by the French scientist Francois Raoult, who

established the following relationship between the partial pressure PAdue

to molecule A in a mixture and the mole fraction XA:

PA=PpureXA (4.24)

This relationship states that the partial pressure of each of the components

is directly proportional to the vapor pressure of the corresponding pure

substance with the proportionality constant being the mole fraction. The

relationship also provides the basis for deriving the chemical potential for

the mixture. Substituting this relationship for partial pressure (eqn 4.22)

into the relationship for chemical potential (eqn 4.21) yields an expression

for the chemical potential of a mixture in terms of the chemical potential

of the pure state and the mole fraction of the mixture:

(4.25)

Now the various thermodynamic parameters are eliminated and the

only parameters left are those that can be defined for the liquid phase.

The determination of this simple relationship hinges on

Raoult’s law. The molecular basis for Raoult’s law can

be understood by considering the surface of a liquid

with molecules leaving and returning (Figure 4.15).

The rate at which the molecules leave the surface, or

equivalently the rate of vaporization, is proportional

to the number of molecules on the surface, which is

proportional to the number in solution:

Rate of leaving =clea 9 ingXA (4.26)

where clea 9 ingis the proportionality constant. Assuming

that the surface is large and molecules leaving do not

interact with molecules returning, the rate at which the

molecules return, or the rate of condensation, can be

written as:

μμA() () lnliquid =+pure liquid pure A

pu

RT

PX

Prre pure

RT X

⎛

⎝

⎜

⎜

⎞

⎠

⎟

⎟=+μ () lnliquid A

ln()ln()lnAB

A

B

−=

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

Blocked

Figure 4.15A
molecular view of
Raoult’s law.