168
EQUILIBRIUM OF HETEROGENEOUS SUBSTANCES.
different kinds of gas, and by
as before the total volume, the increase of entropy may be written in the form
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And if we denote by
, etc., the numbers of the molecules of the several different kinds of gas, we shall have
![{\displaystyle r_{1}=Cm_{1}a_{1},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4260f42215139571ce254c2e63e881f519513cfc) etc.,
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where
denotes a constant. Hence
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and the increase of entropy may be written
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(298)
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The Phases of Dissipated Energy of an Ideal Gas-mixture with Components which are Chemically Related.
We will now pass to the consideration of the phases of dissipated energy (see page 140) of an ideal gas-mixture, in which the number of the proximate components exceeds that of the ultimate.
Let us first suppose that an ideal gas-mixture has for proximate components the gases
, and
, the units of which are denoted by
and
that in ultimate analysis
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(299)
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and
denoting positive constants, such that
. The
phases which we are to consider are those for which the energy of the gas-mixture is a minimum for constant entropy and volume and constant quantities of
and
, as determined in ultimate analysis. For such phases, by (86),
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(300)
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for such values of the variations as do not affect the quantities of
and
as determined in ultimate analysis. Values of
proportional to
and only such, are evidently consistent with this restriction: therefore
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(301)
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If we substitute in this equation values of
taken from (276), we obtain, after arranging the terms and dividing by
,
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(302)
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where
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(303)
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(304)
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(305)
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