Page:Scientific Papers of Josiah Willard Gibbs.djvu/119

which we are next to consider there are equations of condition due to a cause of a different nature.

Effect of a Diaphragm (Equilibrium of Osmotic Forces).

If the given mass, enclosed as before, is divided into two parts, each of which is homogeneous and fluid, by a diaphragm which is capable of supporting an excess of pressure on either side, and is permeable to some of the components and impermeable to others, we shall have the equations of condition

$\delta \eta '+\delta \eta ''=0,$

(72)

$\delta v'=0,\,\,\,\delta v''=0,$

(73)

and for the components which cannot pass the diaphragm

$\delta m'_{a}=0,\,\,\,\delta m''_{a}=0,\,\,\,\delta m'_{b}=0,\,\,\,\delta m''_{b}=0,\,\,etc.,$

(74)

and for those which can

$\delta m'_{k}+\delta m''_{k}=0,\,\,\,\delta m'_{l}+\delta m''_{l}=0,\,\,etc.$

(75)

With these equations of condition, the general condition of equilibrium (see (15)) will give the following particular conditions:—

$t'=t'',$

(76)

and for the components which can pass the diaphragm, if actual components of both masses,

$\mu _{k}'=\mu _{k}'',\,\,\,\mu _{l}'=\mu _{l}''\,\,\,etc.,$

(77)

but not

$p'=p''$

nor	$\mu _{a}'=\mu _{a}'',\,\,\,\mu _{b}'=\mu _{b}'',\,\,etc.$

Again, if the diaphragm is permeable to the components in certain proportions only, or in proportions not entirely determined yet subject to certain conditions, these conditions may be expressed by equations of condition, which will be linear equations between $\delta m'_{1},\delta m'_{2}$, etc., and if these be known the deduction of the particular conditions of equilibrium will present no difficulties. We will however observe that if the components $S_{1},S_{2}$, etc. (being actual components on each side) can pass the diaphragm simultaneously in the proportions $a_{1},a_{2}$, etc. (without other resistances than such as vanish with the velocity of the current), values proportional to $a_{1},a_{2}$, etc. are possible for $\delta m'_{1},\delta m'_{2}$, etc. in the general condition of equilibrium, $\delta m''_{1},\delta m''_{2}$, etc.,

having the same values taken negatively, so that we shall have for one particular condition of equilibrium

$a_{1}\mu _{1}'+a_{2}\mu _{2}'+etc.=a_{1}\mu _{1}''+a_{2}\mu _{2}''+etc.$

(78)

There will evidently be as many independent equations of this form as there are independent combinations of the elements which can pass the diaphragm.