Physical Chemistry Third Edition

130 3 The Second and Third Laws of Thermodynamics: Entropy

Removable partitions

V 1 V 2 V 3 Vc

Figure 3.10 A System for Carrying Out the Irreversible Mixing of Gases.

The Entropy Change of Mixing Ideal Gases

Theentropy change of mixingis the entropy change for producing a mixture from the

pure components. Consider a mixture of several ideal gases in whichn 1 is the amount

of substance 1,n 2 is the amount of substance 2, and so on. The number of substances

is denoted bys. We imagine an initial state with each substance confined in a separate

compartment of a container, as shown in Figure 3.10. We arrange the system so that

each gas is at the temperature and the pressure of the final mixture by letting

Vi

niRT

P

(i1, 2, 3,...,s) (3.3-13)

whereViis the volume of compartment numberi,niis the amount of substance number

iin compartment numberi, andTandPare the temperature and pressure of the final

mixture. The total volume of the container is denoted byV:

V

∑s

i 1

Vi (3.3-14)

The gases are mixed by withdrawing the partitions between compartments, so that

each gas mixes irreversibly with the others and fills the entire volume. According to

Dalton’s law of partial pressures each gas in a mixture of ideal gases acts as though it

were alone in the container. Gas numberiundergoes a process with the same initial

and final states as an isothermal reversible expansion from volumeVito volumeV.

The entropy changes of the individual gases are given by Eq. (3.3-3):

∆SiniRln

(

V

Vi

)

(i1, 2, 3,...,s) (3.3-15)

The entropy change of the system is the sum of these quantities:

∆S

∑s

i 1

niRln

(

V

Vi

)

(3.3-16)

We now express∆Sin terms of the mole fractions. Themole fractionof substance

numberiis defined by

xi

ni

n

(definition of the mole fractionxi) (3.3-17)