Page:Scientific Papers of Josiah Willard Gibbs.djvu/450

In the upper regions,

${\text{F}}'(p)={\frac {1}{\gamma }}={\frac {at}{p}};$ $\therefore {\text{F}}'(p)=at\log p,$

where $t$ denotes temperature, and $a$ the constant of the law of Boyle and Charles. Hence,

$at\log p_{\text{A}}-at\log p_{\text{B}}=g({\text{C}}_{\text{A}}-{\text{C}}_{\text{B}}).$

Moreover, if $1\,:\,n$ represents the constant ratio in which the S- and D-inolecules are mixed in the A-column, we shall have in the upper regions, where the S-molecules have the same density in the two columns,

$\gamma _{\text{A}}=(1+n)\gamma _{\text{B}},$$p_{\text{A}}=(1+n)p_{\text{B}},$ $g({\text{C}}_{\text{A}}-{\text{C}}_{\text{B}})=at\log(1+n).$

Therefore, at any height,

${\text{F}}(p_{\text{A}})-{\text{F}}(p_{\text{B}})=at\log(1+n).$

This equation gives the required relation between the pressures in ${\text{A}}$ and ${\text{B}}$ and the composition of the fluid in ${\text{A}}$. It agrees with van't Hoff's law, for when $n$ is small the equation may be written

${\text{F}}'(p_{\text{A}})(p_{\text{A}}-p_{\text{B}})=atn$

or

$p_{\text{A}}-p_{\text{B}}=atn\gamma _{\text{A}}.$

But we must not suppose, in any literal sense, that this difference of pressure represents the part of the pressure in ${\text{A}}$ which is exerted by the D-molecules, for that would make the total pressure calculable by the law of Boyle and Charles.

To show that the case is substantially the same, at least for any one temperature, when the fluid is not volatile, we may suppose that we have many kinds of molecules, ${\text{A, B, C}}$, etc., which are identical in all properties except in regard to passing diaphragms. Let us imagine a row of vertical cylinders or tubes closed at both ends. Let the first contain A-molecules sufficient to give the pressure $p'$ at a certain level. Then let it be connected with the second cylinder through a diaphragm impermeable to B-molecules, freely permeable to all others. Let the second cylinder contain such quantities of A- and B-molecules as to be in equilibrium with the first cylinder, and to have a certain pressure $p''$ at the level of $p'$ in the first cylinder. At a higher level this second cylinder will have the pressure which we have called $p'$. There let it be connected with the third cylinder through a diaphragm impermeable to C-molecules, and to them alone. Let this third cylinder contain such quantities of A-, B-, and C-molecules as to be in equilibrium with the second cylinder, and have the pressure $p''$ at the diaphragm; and so on, the connections being so made, and the