Geology Reference
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work (exergy) for a given amount of component A which must be expended in the
separation process.
dI
dx
A
= RT
0
[lnx
A
ln(1 x
A
)]
(9.7)
whereby the solution to the equation falls to zero when x
A
= 1x
A
, that is to say
when the level of mixing is 50%. It reaches 1 when x
A
! 0.
As such, Eq. (9.6) expresses that the minimum effort required to separate two
blended ideal gases at a 50/50 mix is zero but as one tries to isolate them further,
more work has to be done. As one arrives to absolute purity the quantity of work
required becomes infinite. In other words, to pass from 50-51% purity does in no
way entail the same amount of work required to go from 90-91%. In effect, the input
of resources follows an exponential law that means that the work involved to purify
from 90-91% is less than that needed to go from 99.0 to 99.1 and substantially lower
than that required to push towards a purity of 99.90 to 99.91 and so on. Worse still,
if the mixture had constituted more than two substances then the expression would
have been identical but the summation extended to the total number of substances
involved (Eq. (9.8)).
X
I = RT
0
x
i
lnx
i
(9.8)
i=1
With its graphical form plotted as I against x
i
, for different values of i, the form
is identical in appearance, albeit becoming steadily more elongated as n increases
(Fig. 9.2). This effectively means that the effort of separation escalates as one
moves closer towards absolute purity. Moreover, when mixing does not involve
ideal substances, as in any real process, the irreversibility values are considerably
larger.
Finally, this analysis provides an accurate depiction of the metaphorical sense
of the word “entropy” that is often applied by practitioners outside of the thermo-
dynamic realm. In effect, it facilitates the connection between generated entropy,
exergy and exergy cost (or embodied exergy). The difference between the latter
two is the sum of irreversibilities that occurred in the material's production. A low
entropic material means a product with a low entropic content and/or whose manu-
facture generated limited amounts of entropy. This is the case of natural materials
at their early stages of production. On the contrary, a highly entropic material
entails a large irreversibility in its manufacture, use and/or disposal and has a large
embodied exergy. The Gouy-Stodola theorem facilitates such an explanation since
it relates irreversibility or the loss of quality energy in a process with the entropy
generated alongside it. Any material is thus indissolubly linked with its production
process.
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