Page:Scientific Papers of Josiah Willard Gibbs.djvu/212

2pressure may by (263) be expressed by the formula ${\frac {P}{a_{s}t}}$, the relative density of a binary gas-mixture may be expressed by

$D=(m_{1}+m_{2}){\frac {a_{s}t}{pv}}\cdot$

(326)

Now by (263)⁠$a_{1}m_{1}+a_{2}m_{2}={\frac {pv}{t}}\cdot$

(327)

By giving to $m_{2}$ and $m_{1}$ successively the value zero in these equations, we obtain

$D_{1}={\frac {a_{s}}{a_{1}}},$⁠$D_{2}={\frac {a_{s}}{a_{2}}},$

(328)

where $D_{1}$ and $D_{2}$ denote the values of $D$ when the gas consists wholly of one or of the other component. If we assume that

$D_{2}=2D_{1},$

(329)

we shall have⁠$a_{1}=2a_{2}.$

(330)

From (326) we have⁠$m_{1}+m_{2}=D{\frac {pv}{a_{s}t}},$ and from (327), by (328) and (330),

$2m_{1}+m_{2}=D_{2}{\frac {pv}{a_{s}t}}=2D_{1}{\frac {pv}{a_{s}t}},$

whence⁠$m_{1}=(D_{2}-D){\frac {pv}{a_{s}t}},$

(331)

$m_{2}=2(D-D_{1}){\frac {pv}{a_{s}t}}\cdot$

(332)

By (327), (331), and (332) we obtain from (320)

$\log {\frac {(D_{2}-D)^{2}p}{2(D-D_{1})a_{s}}}={\frac {A}{a_{2}+{\frac {B}{a_{2}}}\log t-{\frac {C}{a_{s}t}}}}\cdot$

(333)

This formula will be more convenient for purposes of calculation if we introduce common logarithms (denoted by $\log _{10}$) instead of hyperbolic, the temperature of the ordinary centigrade scale $t_{c}$ instead of the absolute temperature $t$, and the pressure in atmospheres $p_{at}$ instead of $p$ the pressure in a rational system of units. If we also add the logarithm of $a_{s}$ to both sides of the equation, we obtain

$\log _{10}{\frac {(D_{2}-D)^{2}p_{at}}{2(D-D_{1})}}={\mathsf {A}}+{\frac {B}{a_{2}}}\log _{10}(t_{c}+273)-{\frac {\mathsf {C}}{t_{c}+273}},$

(334)

where ${\mathsf {A}}$ and ${\mathsf {C}}$ denote constants, the values of which are closely connected with those of $A$ and $C$.

From the molecular formulæ of peroxide of nitrogen NO₂ and N₂O₄, we may calculate the relative densities

$D_{1}={\frac {14+32}{2}}.0691=1.589,$⁠and⁠$D_{2}={\frac {28+64}{2}}.0691=3.178$

(335)