Concise Physical Chemistry

c05 JWBS043-Rogers September 13, 2010 11:25 Printer Name: Yet to Come

SPONTANEOUS PROCESSES 77

These derivations are given in more detail in Metiu (2006) along with results cal-

culated from accurate equations of state for real gases and comparisons to National

Institutes of Standards and Technology tabulations (webbook.nist.gov).

5.3 SPONTANEOUS PROCESSES

5.3.1 Mixing

Consider two chambers of the same volume, one containing 0.500 mol of ideal gas

A and the other containing 0.500 mol of ideal gas B. The gases are at the same

temperature and 1.0 bar pressure, and the chambers are connected by a valve. When

the valve is opened, the gases mix spontaneously just as the smell of perfume gradually

permeates all regions of a closed room. The diffusion process is like an expansion

for the gases considered individually because the volume in which the gas molecules

move is doubled relative to what it was before the valve was opened. For gas A we

can say

SA= 0. 500 Rln

V 2

V 1

= 0. 500 Rln

2

1

= 2 .88 J

The equivalent calculation for gas B gives the same result, so the total entropy change

is the sum

SA+
SA=2(2.88)= 5 .76 J

If the volumes of the chambers are changed arbitrarily so that they are not equal

and if quantities of gas are taken that are not 0.500 mol, nor are they equal, the entropy

increase for gases A and B is different from what we have just calculated, but it is

always anincreasebecause each gas sees a larger volume after the mixing process

than it did before. Mixing is always spontaneous and the entropy of mixing isalways

positivebecause, at constantp,V 2 is always larger thanV 1 from the point of view of

each gas.

5.3.2 Heat Transfer

Consider now two bricks in contact with one another. One is a hot brick and the other

is a cold brick. We know, from millennia of experience, that after sufficient time we

will have two warm bricks and that we never observe the reverse process, two warm

bricks spontaneously undergoing a transformation that produces a hot brick and a

cold brick. This is a crude statement of the second law. In fact an engine is merely a

device placed between a hot reservoir and a cold reservoir to siphon off some of the

heat flow and use it to do work.

Spontaneous flow of heat from hot to cold cannot be explained by the first law

because for each joule of heat lost by the hot brick there is exactly one joule gained