Physical Chemistry , 1st ed.

There is another way of generalizing equation 3.21. If two or more gas sam-

ples have the same pressure and temperature, then their volumes are directly

proportional to the number of moles of gas present. The mole fraction of gas i,

xi,is defined as the ratio of the number of moles of gas i,ni, and the total num-

ber of moles of gas,ntot:

xi

n

n

to

i

t

 (3.22)

It can be shown that



V

V

to

i

t



n

n

to

i

t

xi

so that the expression for the overall entropy can be expressed as

S(n 1 Rln x 1 ) (n 2 Rln x 2 )

The negative signs are introduced because in order to substitute the mole frac-

tion into the expression, we have to take the reciprocal of the volume fraction.

For any number of gases being mixed:

mixSR 

no. of gases

i 1

niln xi (3.23)

where mixSis referred to as the entropy of mixing.Because xiis always less

than 1 (for two or more components), its logarithm is always negative. The

negative sign as part of equation 3.23 means that the entropy of mixing is

always a sum of positive terms and the overall mixSis always positive.

Example 3.4

Calculate the entropy of mixing 10.0 L of N 2 with 3.50 L of N 2 O at 300.0 K

and 0.550 atm. Assume that the volumes are additive; that is,Vtot13.5 L.

Solution

We need to determine the number of moles of each component in the re-

sulting mixture. Given all of the conditions, we can use the ideal gas law to

calculate them:

78 CHAPTER 3 The Second and Third Laws of Thermodynamics

Remove barrier

Gas 1

n

VVV 11

n

Remove barrier

Gas 2

VV

n

VVV 22

n 2

Figure 3.5 The mixing of two gases can be separated into two individual processes, where

gas 1 expands into the right side and gas 2 expands into the left side.