Chemistry Reference
In-Depth Information
particular case may give rise to further connections between the intensive variables,
reducing the variance further. Thus, starting with
n
substances distributed over
f
phases, where there are additional conditions such as
r
independent chemical
reactions coming to equilibrium and establishing connections between the chemical
potentials, together with
s
stoichiometric constraints connecting the proportions
of products of chemical reactions in a particular phase or several phases
(these conditions may be intraphase relationships or interphase relationships), the
variance
¼n
-
f
+2 -
r
-
s
. The number of independent substances,
c,
can be
identified with
n
-
r
-
s
, but although this number is unique, which they are is
usually not, and there are frequently several ways of selecting
c
independent sub-
stances from all those apparently present. However chosen, they may not all be
present in all the phases, as assumed in textbook proofs such as that given here
(footnote 3). Duhem (
1898
) pointed out that the well-known equilibrium
established when calcium carbonate is heated in a closed container was such a
case and proceeded to give a rigorous proof of the phase rule which doesn
t rely on
this assumption. Another case, studied by Zernike (
1951
,
1954
), involves heating
ammonium bicarbonate, NH
4
HCO
3
, in an initially otherwise empty container,
when a dissociated vapour phase and a liquid phase appear in addition to solid
ammonium bicarbonate. There is an
interphase
stoichiometric condition at play in
this case, affecting the composition of several phases. The vapour phase contains
ammonia, carbon dioxide and water, each of which have different solubilities in
liquid ammonium bicarbonate. Consequently, neither the vapour phase (from which
the substances dissolved in the liquid derive) nor the liquid has a composition
corresponding to that represented by the formula NH
4
HCO
3
. The additional con-
dition giving the proportions of the products of dissociation in the liquid and gas
phases, which can be readily calculated, is counted as one of the conditions
s
that
has to be taken into account in applying the phase rule. A further consideration is
that enantiomorphs, usually counted different substances (see later), have the same
values for their intensive properties except for rotatory power (equal in magnitude
but opposite in sign). This may render some of the Gibbs-Duhem equations for
different phases in a system no longer independent and therefore affect the variance
(Scott
1977
; Wheeler
1980
).
The calcium carbonate equilibrium was one in which nineteenth-century
scientists had to revise their apparent observation that only two phases—a white
solid and a colourless gas—where involved because this wasn
'
t consistent with a
variance of one in the light of the phase rule. Usually the number of phases is a
straightforward matter. But this still leaves several variables, often raising questions
of interpretation which must be addressed before the number of substances
present in a mixture can be determined from an application of the phase rule.
Such interpretation usually calls on ideas about distinctions of substance drawn
from other quarters. The microscopic realm is an obvious source, but as we will
see, the idea of molecular structure is equally one which doesn
'
t stand on its own
feet. The notion of sameness of substance is an eclectic affair involving diverse
theoretical inputs. A simple, unproblematic, universal criterion of sameness of
substance is something of a pipedream.
'
