BioPHYSICAL chemistry

84 PARTI THERMODYNAMICS AND KINETICS

This last expression can be written in terms of mole fractions, yielding an

expression for the Gibbs energy of mixing in terms of the mole fractions:

(4.20)

n=nA+nB

ΔG=nRT(XAlnXA+XBlnXB)

This final relationship expresses a simple idea that, by simply allowing differ-

ent substances to mix together, the Gibbs energy will decrease. Although

this was derived for ideal gases, the same idea will hold for real gases pro-

vided that the interactions between the gas molecules are negligible.

Thus, for ideal gases, the change in the Gibbs energy is sum of the con-

tributions from each molecule. Since the sum of the mole fractions is equal

to one, the change in Gibbs energy is uniquely determined by the mole

fraction for any one gas (Figure 4.13). The difference ranges from zero,

when only one type of gas is present, to a minimum value when the two

gases are present in equal amounts. Because mole fractions are never

greater than one, the logarithms are always negative and the change in

the Gibbs energy is negative for all ideal gases. The decrease in energy

results in a spontaneous mixing of the two gases. For non-ideal

gases, the gas molecules will interact and there are additional

contributions to the energy difference.

If the two gases in the example are the same gas then the change

in the Gibbs energy is zero because the final mole fraction is one.

The identity of the gases influences the outcome because the

driving force for mixing is actually the entropy change of the

system upon mixing. To probe the role of entropy, consider

mixing two buckets of balls together. If the balls are identical,

then the amount of disorder, or entropy, does not change upon

mixing. However, if the balls are distinguishable, the entropy

does change. For example, consider two different-sized balls that

are initially placed in separate chambers. The content of the

two chambers is mixed by a central paddle that allows the small

balls to pass but not the large balls. In the course of mixing, work

is performed and the entropy changes (Figure 4.14).

X

n

n

P

B P

==B B

X

n

n

P

A P

==AA

lnlnln

a

b

c

b

a

b

b

c

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟−

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟=⋅

⎛

⎝

⎜⎜

⎞⎞

⎠

⎟⎟=

⎛

⎝

⎜⎜

⎞

⎠

ln ⎟⎟

a

c

ΔG

mixing

(J mol

^1

)

Mole fraction

Figure 4.13The
Gibbs energy of
mixing two ideal
gases.