158 Chapter 5Integration
according as the external pressurep
extis greater than or less than p. Ifp
ext1 > 1 pthen
the fluid is compressed, and the work done by the external forceF
extin moving the
piston from ato bis
(5.64)
Now, the external force has magnitude|F
ext| 1 = 1 p
extA, and a length of cylinder |dx|
contains a volume|dV| 1 = 1 A|dx|. The work can therefore be written in ‘pressure–
volume’ form as
(5.65)
in which the limits of integration now refer to the volume, and the minus sign is
included to make the work positive for compression. It can be shown that this
expression for the mechanical work done on a thermodynamic system is independent
of the shape of the container.
To compress the fluid, it is necessary that the external pressure be greater than the
internal pressure of the fluid. Letp
ext1 = 1 (p 1 + 1 ∆p)where∆pis a positive excess pressure
that, for simplicity, can be assumed to be constant throughout the compression.
Then, with V
a1 > 1 V
b,
(5.66)
Conversely, to allow the fluid to expand fromV
btoV
ait is necessary that the external
pressure be smaller than the internal pressure. Ifp
ext1 = 1 (p 1 − 1 ∆p)then
(5.67)
>−Z
bapdV
WpdVpVV
babaab=−Z +∆()−
>−Z
abpdV
W p dV p dV p dV p V V
ababababab=−ZZZ−∆∆=− + ()−
WpdV
abab=−Z
extWFdx
abab=Z
ext. ......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................
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...Figure 5.23