Part IV — Work
Lecture 12 — PV work
Recall the notion of a thermodynamic process: change of state via some external intervention.
Common processes include:
•
Adding or removing heat
•
Expanding or compressing the system
Often, the process involves requires or generates work. We say, work is done on the system, or done
by the system. Sign convention: work done on the system is positive, work done by the system is
negative. (We are keeping track of the energy of the system.)
Recall that being able to calculate the required amount of work for a process, and designing a
process so that the minimum amount of work is required, or the maximum amount of work is
generated, are central goals of thermodynamics.
An external force acting through a small displacement does a small amount of work
dW
=
F
ext
dx
=

P
ext
dV
We integrate to Fnd the total work done during the process
W
=

±
V
2
V
1
P
ext
dV
Sign convention: when we compress the system,
dx >
0 (displacement in direction of force applied)
and
dV <
0. In compressing, we do work on the system, increasing its energy (like compressing a
spring).
Work is
not
a state variable — it depends on the path (i.e., the details of the process). ±or example:
it is harder to compress a hot gas than a cold one. So consider a process in which we increase
T
by some amount and decrease
V
by some amount. If we warm Frst, the work will be greater than
if we warm after compressing. (sketch) We will see more speciFc examples of this next lecture.
To compute
W
, we need to know how
P
ext
varies as we do the volume change. If we do the change
slowly enough,
P
ext
=
P
(
V,T
) (the equilibrium system pressure) to a close approximation — nearly
balanced forces, very slow motion of the piston. This we call a reversible change — at every point
along the way, we are very close to equilibrium. Then, we can compute the work, if we have an
EOS to tell us
P
(
V,T
).
In general, if the process is done at a Fnite rate, it won’t be “reversible”. In that case,
W
=
W
rev
+
W
lost
30
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View Full DocumentWe always have
W
lost
>
0
(This is a
postulate
, that is consistent with our common experience — and detailed experiment
besides. We shall meet other equivalent forms of this assumption in later lectures.) If we are doing
work on the system (
W>
0), it takes more work than the minimum
W
rev
to e
f
ect the change. If
the system is doing work (
W<
0), we get less work than the maximum

W
rev
.
Example: we slowly heat an ideal gas conFned by a Fxed external pressure
P
ext
. “Slowly” means
P
≈
P
ext
during the process. Because the external pressure is Fxed, the work is simple to com
pute.
Pv
=
RT
implies
Δ
v
=
R
Δ
T/P
, hence
W
=

R
Δ
T
. The gas does positive work on the
surroundings (pushes in the direction of motion), so the surroundings do negative work on the gas
(
W<
0).
Example: we slowly expand an ideal gas at constant temperature. Again, “slowly” means we adjust
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 Spring '08
 PROF.MARANNAS
 Thermodynamics, Work

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