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

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

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

(326)

Now by (263)

${\displaystyle a_{1}m_{1}+a_{2}m_{2}={\frac {pv}{t}}\cdot }$

(327)

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

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

(328)

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

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

(329)

we shall have

${\displaystyle a_{1}=2a_{2}.}$

(330)

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

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

whence

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

(331)

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

(332)

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

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

(333)

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

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

(334)

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

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

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

(335)