= Repeated ideal-stage enrichment
{title2=$x_{n+1}=2x_n/(1+x_n)$}
Stipulate a constant <relative volatility> $\alpha=2$. The <vapor-liquid equilibrium> model then becomes
$$
y=\frac{2x}{1+x},\qquad y-x=\frac{x(1-x)}{1+x}>0\quad(0<x<1).
$$
An initial <liquid> <mole fraction> $x_0=3/10$ gives an equilibrium <vapor> <mole fraction> $y_0=6/13\approx0.461538$. Now imagine completely condensing a small sample of that <vapor> and using this condensate as a fresh <liquid> for another equilibration. Repeating this formal operation gives
$$
x_0=\frac3{10}=0.300000
\;\longmapsto\;
x_1=\frac6{13}\approx0.461538
\;\longmapsto\;
x_2=\frac{12}{19}\approx0.631579
\;\longmapsto\;
x_3=\frac{24}{31}\approx0.774194.
$$
These are original model calculations of ideal-stage enrichment. They are not measured compositions, a <reflux> design, or successive compositions of one pot undergoing <simple distillation>. That pot loses the more volatile component and becomes depleted; each step here instead starts from the previous enriched condensate. The displayed sequence specifies compositions, not the recoverable amount at each step.
\Image[/distillation-ideal-enrichment-and-material-balance.png]
{title=Stipulated <vapor-liquid equilibrium> enrichment and a separate <distillation material balance>}
{id=image-distillation-ideal-enrichment-and-material-balance}
{description=Left: the model $y=2x/(1+x)$ and formal repeated equilibration from $x_0=0.30$, using complete condensation between steps. The staircase is not a <reflux> design or one-pot time history. Right: an independent product split with feed <amount of substance> $F=1\,\mathrm{mol}$ and specified <mole fractions>. Neither panel contains measured data or a <boiling>-<temperature> curve.}
The algebra makes the enrichment especially transparent. With composition odds $r_n=x_n/(1-x_n)$, every formal step doubles the odds, so
$$
r_n=2^n\frac37,\qquad x_n=\frac{3\,2^n}{7+3\,2^n}.
$$
For every finite $n$, $x_n<1$, although $x_n\to1$ as $n\to\infty$. This explains why repeated ideal equilibration can improve composition without making finite-step perfect separation a consequence of the model.
Back to article page
