Simple Nature - Light and Matter

h/Entropy can be understood

using the metaphor of a water

wheel. Letting the water levels

equalize is like letting the entropy

maximize. Taking water from the

high side and putting it into the

low side increases the entropy.

Water levels in this metaphor

correspond to temperatures in

the actual definition of entropy.

It turns out to be simpler and more useful to define changes

in entropy than absolute entropies. Suppose as an example that a

system contains some hot matter and some cold matter. It has a

relatively high grade of energy because a heat engine could be used

to extract mechanical work from it. But if we allow the hot and

cold parts to equilibrate at some lukewarm temperature, the grade

of energy has gotten worse. Thus putting heat into a hotter area

is more useful than putting it into a cold area. Motivated by these

considerations, we define a change in entropy as follows:

∆S =

Q

T

[change in entropy when adding

heatQto matter at temperatureT;

∆Sis negative if heat is taken out]

A system with a higher grade of energy has a lower entropy.

Entropy is additive. example 10

Since changes in entropy are defined by an additive quantity (heat)

divided by a non-additive one (temperature), entropy is additive.

Entropy isn’t changed by a Carnot engine. example 11

The efficiency of a heat engine is defined by

efficiency = 1−QL/QH,

and the efficiency of a Carnot engine is

efficiency = 1−TL/TH,

so for a Carnot engine we haveQL/QH =TL/TH, which can be

rewritten asQL/TL=QH/TH. The entropy lost by the hot reservoir

is therefore the same as the entropy gained by the cold one.

Entropy increases in heat conduction. example 12

When a hot object gives up energy to a cold one, conservation

of energy tells us that the amount of heat lost by the hot object

is the same as the amount of heat gained by the cold one. The

change in entropy is−Q/TH+Q/TL, which is positive because

TL<TH.

Entropy is increased by a non-Carnot engine. example 13

The efficiency of a non-Carnot engine is less than 1 - TL/TH,

soQL/QH>TL/THandQL/TL>QH/TH. This means that the

entropy increase in the cold reservoir is greater than the entropy

decrease in the hot reservoir.

A book sliding to a stop example 14

A book slides across a table and comes to a stop. Once it stops,

all its kinetic energy has been transformed into heat. As the book

and table heat up, their entropies both increase, so the total en-

tropy increases as well.

Section 5.3 Entropy as a macroscopic quantity 323