Energy and Human Ambitions on a Finite Planet, 2021a

6 Putting Thermal <strong>Energy</strong> to Work 92
never happen. Pieces of ceramic strewn about the floor will never
spontaneously assemble into a mug <strong>and</strong> leap from the floor! <strong>Energy</strong>
is not the barrier, because the total energy in all forms is the same 29
before <strong>and</strong> after. It’s entropy: the more ordered states are less likely
to spontaneously emerge. To appreciate how pervasive entropy is,
imagine how easy it is to spot a “fake” video run backwards.
These two laws of thermodynamics, plus a way to quantify entropy
changes that we will see shortly, are all we need to figure out the
maximum efficiency a heat engine can achieve in delivering work. If we
draw an amount of heat, ΔQ h from a hot bath 30 at temperature T h , <strong>and</strong>
allow part of this energy to be “exported” as useful work, ΔW, then
we must have the remainder flow as heat (ΔQ c ) into the cold bath at
temperature T c . Figure 6.4 offers a schematic of the process. The First
Law of Thermodynamics 31 requires that ΔQ h ΔQ c + ΔW, or that all
of the extracted heat from the hot bath is represented in the external
work <strong>and</strong> flow to the cold bath: nothing is lost.
29: . . . provided that the system boundary
is drawn large enough that no energy escapes
30: By “bath,” we mean a large reservoir at
a constant temperature that is large enough
not to appreciably change its temperature
upon extraction of some amount of thermal
energy, ΔQ.
31: . . . conservation of energy
large hot reservoir at T h
heat flows...
Q h
...from hot... 30
...to cold
Q c
20
10
W
large cold reservoir at T c
example quantities (J)
Q h
= Q c
+ W
useful work
extracted
Figure 6.4: Heat engine energy balance.
Heat flowing from the hot bath to the cold
bath can perform useful work, ΔW,inthe
process—subject to conservation of energy
(ΔQ h ΔQ c + ΔW), where ΔQ is a heat
flow. Entropy constraints limit how large
ΔW can be. Arrow widths are proportional
to energy, <strong>and</strong> red numbers are example
energy amounts, for use in the text.
So where does entropy come in? Extracting heat from the hot bath in
the amount ΔQ h results in an entropy change in the hot bath according
to Definition 6.4.5.
Definition 6.4.5 Entropy Change: when energy (heat, ΔQ, in J) is moved
into or out of a thermal bath at temperature T, the accompanying change in
the bath’s entropy, ΔS, obeys the relation:
ΔQ TΔS. (6.2)
When heat is removed, entropy is reduced. When heat is added, entropy
increases. The temperature, T, must be in Kelvin, <strong>and</strong> entropy is measured
in units of J/K.
So the extraction of energy from the hot bath results in a decrease of
entropy in the hot bath of ΔS h according to ΔQ h T h ΔS h . Meanwhile,
ΔS c of entropy is added to the cold bath according to ΔQ c T c ΔS c .
The Second Law of Thermodynamics enforces that the total change in
Table 6.3: Thermodynamic symbols.
Symbol Describes (units)
T temperature (K)
ΔT temp. change (K, ◦ C)
ΔQ thermal energy (J)
ΔW mechanical work (J)
ΔS entropy change (J/K)
ε efficiency
η entropy ratio
© 2021 T. W. Murphy, Jr.; Creative Commons Attribution-NonCommercial 4.0 International Lic.;
Freely available at: https://escholarship.org/uc/energy_ambitions.